A handy formula for the Fredholm index of Toeplitz plus Hankel operators
aa r X i v : . [ m a t h . F A ] D ec A handy formula for the Fredholm index ofToeplitz plus Hankel operators
Steffen Roch and Bernd SilbermannDedicated to the memory of Israel Gohberg
Abstract
We consider Toeplitz and Hankel operators with piecewise continuousgenerating functions on l p -spaces and the Banach algebra generated bythem. The goal of this paper is to provide a transparent symbol calculusfor the Fredholm property and a handy formula for the Fredholm index foroperators in this algebra. Keywords:
Toeplitz plus Hankel operators, Fredholm index
Primary: 47B35, secondary: 47B48
Throughout this paper, let 1 < p < ∞ . For a non-empty subset I of the set Z of the integers, let l p ( I ) denote the complex Banach space of all sequences x = ( x n ) n ∈ I of complex numbers with norm k x k p = ( P n ∈ I | x n | p ) /p < ∞ . Weconsider l p ( I ) as a closed subspace of l p ( Z ) in the natural way and write P I forthe canonical projection from l p ( Z ) onto l p ( I ). For I = Z + , the set of the non-negative integers, we write l p and P instead of l p ( I ) and P I , respectively. By J wedenote the operator on l p ( Z ) acting by ( J x ) n := x − n − , and we set Q := I − P .For every Banach space X , let L ( X ) stand for the Banach algebra of allbounded linear operators on X , and write K ( X ) for the closed ideal of L ( X )of all compact operators. The quotient algebra L ( X ) /K ( X ) is known as theCalkin algebra of X . Its importance in this paper stems from the fact that theinvertibility of a coset A + K ( X ) of an operator A ∈ L ( X ) in this algebra isequivalent to the Fredholm property of A , i.e., to the finite dimensionality of thekernel ker A = { x ∈ X : Ax = 0 } and the cokernel coker A = X/ im A of A , withim A = { Ax : x ∈ X } referring to the range of A . If A is a Fredholm operatorthen the difference ind A := dim ker A − dim coker A is known as the Fredholmindex of A . 1ur goal is a criterion for the Fredholm property and a formula for the Fred-holm index for operators in the smallest closed subalgebra of L ( l p ) which containsall Toeplitz and Hankel operators with piecewise continuous generating function.The precise definition is as follows. Let T be the complex unit circle. For eachfunction a ∈ L ∞ ( T ), let ( a k ) k ∈ Z denote the sequence of its Fourier coefficients, a k := 12 π Z π a ( e iθ ) e − ikθ dθ. The
Laurent operator L ( a ) associated with a ∈ L ∞ ( T ) acts on the space l ( Z ) ofall finitely supported sequences on Z by ( L ( a ) x ) k := P m ∈ Z a k − m x m . (For every k ∈ Z , there are only finitely many non-vanishing summands in this sum.) Wesay that a is a multiplier on l p ( Z ) if L ( a ) x ∈ l p ( Z ) for every x ∈ l ( Z ) and if k L ( a ) k := sup {k L ( a ) x k p : x ∈ l ( Z ) , k x k p = 1 } is finite. In this case, L ( a ) extends to a bounded linear operator on l p ( Z ) whichwe denote by L ( a ) again. The set M p of all multipliers on l p ( Z ) is a Banachalgebra under the norm k a k M p := k L ( a ) k . We let M h p i stand for M if p = 2and for the set of all a ∈ L ∞ ( T ) which belong to M r for all r in a certain openneighborhood of p if p = 2.It is well known that M = L ∞ ( T ). Moreover, every function a with boundedtotal variation Var( a ) is in M p for every p , and the Stechkin inequality k a k M p ≤ c p ( k a k ∞ + Var( a ))holds with a constant c p independent of a . In particular, every trigonometricpolynomial and every piecewise constant function on T are multipliers for every p .We denote the closure in M p of the algebra P of all trigonometric polynomials andof the algebra P C of all piecewise constant functions by C p and P C p , respectively.Thus, C p and P C p are closed subalgebras of M p for every p . Note that C is justthe algebra C ( T ) of all continuous functions on T , and P C is the algebra P C ( T )of all piecewise continuous functions on T . It is well known that C p ⊆ C ( T ) and C p ⊆ P C p ⊆ P C ( T ) for every p . In particular, every multiplier a ∈ P C p possessesone-sided limits at every point t ∈ T (see [2] for these and further properties ofmultipliers). For definiteness, we agree that T is oriented counter-clockwise, andwe denote the one-sided limit of a at t when approaching t from below (fromabove) by a ( t − ) (by a ( t + )).Let a ∈ M p . The operators T ( a ) := P L ( a ) P and H ( a ) := P L ( a ) QJ , thoughtof as acting on im P = l p are called the Toeplitz and Hankel operator withgenerating function a , respectively. It is well known that k T ( a ) k = k a k M p and k H ( a ) k ≤ k a k M p for every multiplier a ∈ M p .For a subalgebra A of M p , we let T ( A ) and TH ( A ) stand for the smallestclosed subalgebra of L ( l p ) which contains all operators T ( a ) with a ∈ A and all2perators T ( a ) + H ( b ) with a, b ∈ A , respectively. We will be mainly concernedwith the algebras C p , P C p , and with their intersections with M h p i , in place of A .Now we can state the goal of the paper more precisely: we will state a criterionfor the Fredholm property of operators in TH ( P C p ) and derive a formula for theFredholm index of operators T ( a ) + H ( b ) with a, b ∈ P C p .The study of the Fredholm property of operators in TH ( P C p ) has a long andinvolved history. We are going to mention only some of its main stages.The Fredholm properties of operators in the algebra T ( P C p ) are well under-stood thanks to the work of I. Gohberg/N. Krupnik and R. Duduchava; see [2]and the literature cited there. We will need these results later on; therefore werecall them in Section 2. Different approaches to these algebras were developedin [2] and [11]; our presentation will be mainly based on the latter.The structure of the algebras TH ( P C p ) is much more involved than that of T ( P C p ). For instance, the Calkin image T π ( P C ) := T ( P C ) /K ( l ) of T ( P C ) isa commutative algebra, whereas that one of TH ( P C ) is not. The Calkin im-age of TH ( P C ) was first described by Power [16]. An alternative approach wasdeveloped by one of the authors in [21], where it was shown that the algebra TH π ( P C ) := TH ( P C ) /K ( l ) possesses a matrix-valued Fredholm symbol. In thepresent paper, we take up the approach from [21] in order to study the Fredholmproperties of operators in TH ( P C p ) for p = 2.It should be mentioned that the algebras TH ( P C p ) have close relatives whichlive on other spaces than l p , such as the Hardy spaces H p ( R ) and the Lebesguespaces L p ( R + ). The corresponding algebras were examined (with different meth-ods) in the report [20], see also the recent monograph [19]. Despite these fairlycomplete results for the Fredholm property, a general, transparent and satisfyingformula for the Fredholm index of operators in TH ( P C p ) (or on related algebras)was not available until now. Among the particular results which hold underspecial assumptions we would like to emphasize the following. In [12], there isderived an index formula for operators of the form λI + H where λ ∈ C and H is a Hankel operator on H p ( R ). Already earlier, some classes of Wiener-Hopfplus Hankel operators were studied in connection with diffraction problems; see[13, 14]. Note also that the (very hard) invertibility problem for Toeplitz plusHankel operators is treated in [1, 3].Finally we would like to mention that algebras like TH ( P C p ) can also beviewed of as subalgebras of algebras generated by convolution-type operatorsand Carleman shifts changing the orientation. First results in that direction werepresented in [8, 9] where, in particular, a matrix-valued Fredholm symbol wasconstructed.The goal of the present paper is to provide a transparent symbol calculus forthe Fredholm property as well as a handy formula for the Fredholm index foroperators in the algebra TH ( P C p ). The techniques developed and used in thispaper also allow to handle the corresponding questions for the related algebrason the spaces H p ( R ) and L p ( R + ). 3 The Fredholm property
In what follows, we fix p ∈ (1 , ∞ ) and consider all operators as acting on l p unless stated otherwise.As already mentioned, we start with recalling the basic results of the Fredholmtheory of operators in the algebra T ( P C p ), which are due Gohberg/Krupnikand Duduchava. The functions f ± ( t ) := t ± are multipliers for every p . Itis easy to check that the algebra generated by the Toeplitz operators T ( f ± )contains a dense subalgebra of K ( l p ). Thus, the ideal K ( l p ) is contained in T ( C p ), hence also in T ( P C p ), and it makes sense to consider the quotient algebra T ( P C p ) /K ( l p ). Clearly, if A ∈ T ( P C p ) and if the coset A + L ( l p ) is invertible in T ( P C p ) /K ( l p ), then it is also invertible in the Calkin algebra L ( l p ) /K ( l p ), hence A is a Fredholm operator. The more interesting question is if the converse holds,i.e., if the invertibility of A + L ( l p ) in the Calkin algebra implies the invertibilityof A + K ( l p ) in T ( P C p ) /K ( l p ). If this implication holds for every A ∈ T ( P C p ),one says that T ( P C p ) /K ( l p ) is inverse closed in L ( l p ) /K ( l p ).Let R denote the two-point compactification of the real line by the points ±∞ (thus R is homeomorphic to a closed interval) and let the function µ p : R → C be defined by µ p ( λ ) := (1 + coth( π ( λ + i/p ))) / λ ∈ R and by µ p ( −∞ ) = 0 and µ p (+ ∞ ) = 1. Note that when λ runs from −∞ to ∞ then µ p ( λ ) runs along a circular arc in C which joins 0 to 1 and passesthrough the point (1 − i cot( π/p )) /
2. An easy calculation gives µ p ( − λ ) = 1 − µ q ( λ ), where 1 /p +1 /q = 1. Thus, for fixed t ∈ T , the values Γ( T ( a )+ K ( l p ))( t, λ )defined in the following theorem run from a ( t −
0) to a ( t + 0) along a circular arcwhen λ runs from −∞ to ∞ . Theorem 1 ( a ) T ( P C p ) /K ( l p ) is a commutative unital Banach algebra. ( b ) The maximal ideal space of T ( P C p ) /K ( l p ) is homeomorphic with the cylinder T × R , provided with an exotic (non-Euclidean) topology. ( c ) The Gelfand transform
Γ : T ( P C p ) /K ( l p ) → C ( T × R ) of the coset T ( a ) + K ( l p ) with a ∈ P C p is Γ( T ( a ) + K ( l p ))( t, λ ) = a ( t − − µ q ( λ )) + a ( t + 0) µ q ( λ ) . ( d ) T ( P C p ) /K ( l p ) is inverse closed in L ( l p ) /K ( l p ) . The topology mentioned in assertion ( b ) will be explicitly described in Section3. Note that this topology is independent of p . Since the cosets T ( a ) + K ( l p )with a ∈ P C p generate the algebra T ( P C p ) /K ( l p ), the Gelfand transform on T ( P C p ) /K ( l p ) is completely described by assertion ( c ). Thus, if A ∈ T ( P C p ),then the coset A + K ( l p ) is invertible in T ( P C p ) /K ( l p ) if and only if the functionΓ( A + K ( l p )) does not vanish on T × R . Together with assertion ( d ) this shows that4 ∈ T ( P C p ) is a Fredholm operator if and only if Γ( A + K ( l p )) does not vanishon T × R . It is therefore justified to call the function smb p A := Γ( A + K ( l p ))the Fredholm symbol of A .The index of a Fredholm operator in T ( P C p ) can be determined my means ofits Fredholm symbol. First suppose that a ∈ P C p is a piecewise smooth functionwith only finitely many jumps. Then the range of the functionΓ( T ( a ) + K ( l p ))( t, λ ) = a ( t − )(1 − µ q ( λ )) + a ( t + ) µ q )( λ )is a closed curve with a natural orientation, which is obtained from the (essential)range of a by filling in the circular arcs C q ( a ( t − ) , a ( t + )) := { a ( t − )(1 − µ q ( λ )) + a ( t + ) µ q )( λ ) : λ ∈ R } at every point t ∈ T where a has a jump. (If the function a is continuous at t ,then C q ( a ( t − ) , a ( t + )) reduces to the singleton { a ( t ) } .) If this curve does not passthrough the origin, then we let wind Γ( T ( a ) + K ( l p )) denote its winding numberwith respect to the origin, i.e., the integer 1 / (2 π ) times the growth of the argu-ment of Γ( T ( a ) + K ( l p )) when t moves along T in positive (= counter-clockwise)direction. If this condition is satisfied then T ( a ) is a Fredholm operator, andind T ( a ) = − wind Γ( T ( a ) + K ( l p ))(see [2], Section 2.73 and Proposition 6.32 for details). Moreover, as in Section5.49 of [2], one can extend both the definition of the winding number and theindex identity to the case of an arbitrary Fredholm operator in T ( P C p ). Moreprecisely, one has the following. Proposition 2
Let A ∈ T ( P C p ) be a Fredholm operator. Then ind A = − wind Γ( A + K ( l p )) . We would like to emphasize an important point. The algebra T ( P C ) /K ( l )is a commutative C ∗ -algebra, hence the Gelfand transform is an isometric ∗ -isomorphism from T ( P C ) /K ( l ) onto C ( T × R ). In particular, the radical of T ( P C ) /K ( l ) is trivial, and the equality smb A = 0 for some operator A ∈ T ( P C ) implies that A is compact. For general p it is not known if the radical of T ( P C p ) /K ( l p ) is still trivial; it is therefore not known if smb p A = 0 implies thecompactness of A .In order to state our results on the Fredholm property of operators in theToeplitz+Hankel algebra TH ( P C p ) /K ( l p ) we need some notation. Let T + be theset of all points in T with non-negative imaginary part and set T := T + \{− , } .Further let the function ν p : R → C be defined by ν p ( λ ) := (2 i sinh( π ( λ + i/p ))) − if λ ∈ R and by ν p ( ±∞ ) = 0. Recall that 1 /p + 1 /q = 1.5 heorem 3 ( a ) Let a, b ∈ P C p . Then the operator T ( a ) + H ( b ) is Fredholm ifand only if the matrix smb p ( T ( a ) + H ( b ))( t, λ ) := (1) (cid:18) a ( t + ) µ q ( λ ) + a ( t − )(1 − µ q ( λ )) ( b ( t + ) − b ( t − )) ν q ( λ )( b (¯ t − ) − b (¯ t + )) ν q ( λ ) a (¯ t − )(1 − µ q ( λ )) + a (¯ t + ) µ q ( λ ) (cid:19) is invertible for every ( t, λ ) ∈ T × R and if the number smb p ( T ( a ) + H ( b ))( t, λ ) := (2) a ( t + ) µ q ( λ ) + a ( t − )(1 − µ q ( λ )) + it ( b ( t + ) − b ( t − )) ν q ( λ ) is not zero for every ( t, λ ) ∈ {± } × R . ( b ) The mapping smb p defined in assertion ( a ) extends to a continuous algebrahomomorphism from TH ( P C p ) to the algebra F of all bounded functions on T + × R with values in C × on T × R and with values in C on {± } × R . Moreover,there is a constant M such that k smb p A k := sup ( t,λ ) ∈ T + × R k smb p A ( t, λ ) k ∞ ≤ M inf K ∈ K ( l p ) k A + K k (3) for every operator A ∈ TH ( P C p ) . Here, k B k ∞ refers to the spectral norm of thematrix B . ( c ) An operator A ∈ TH ( P C p ) has the Fredholm property if and only if the func-tion smb p A is invertible in F . ( d ) The algebra TH ( P C p ) /K ( l p ) is inverse closed in L ( l p ) /K ( l p ) . Before going into the details of the proof, we remark two consequences of Theorem3 which will be needed in the next section.
Corollary 4
Let a, b ∈ P C p and T ( a ) + H ( b ) a Fredholm operator on l p . Then ( a ) the function a is invertible in P C p , and ( b ) if b is continuous at ± , then T ( a ) − H ( b ) is a Fredholm operator on l p . Proof. If T ( a ) + H ( b ) is a Fredholm operator, then the diagonal matricessmb p ( T ( a ) + H ( b ))( t, ±∞ ) = diag ( a ( t ± ) , a ( t ± ))are invertible for every t ∈ T and the numbers smb p ( T ( a ) + H ( b ))(1 , ±∞ ) = a (1 ± ) and smb p ( T ( a ) + H ( b ))( − , ±∞ ) = a (( − ± ) are not zero by assertion( a ) of Theorem 3. Hence, a is invertible as an element of P C . Since the algebra
P C p is inverse closed in P C by Proposition 6.28 in [2], assertion ( a ) follows. Theproof of assertion ( b ) is also immediate from the form of the symbol described inTheorem 3 ( a ). 6he remainder of this section is devoted to the proof of Theorem 3. We will needtwo auxiliary ingredients which we are going to recall first. Let A be a unitalBanach algebra. The center of A is the set of all elements a ∈ A such that ab = ba for all b ∈ A . A central subalgebra of A is a closed subalgebra C of the center of A which contains the identity element. Thus, C is a commutative Banach algebrawith compact maximal ideal space M ( C ). For each maximal ideal x of C , considerthe smallest closed two-sided ideal I x of A which contains x , and let Φ x refer tothe canonical homomorphism from A onto the quotient algebra A / I x .In contrast to the commutative setting, where C /x ∼ = C for all x ∈ M ( C ), thequotient algebras A / I x will depend on x ∈ M ( C ) in general. In particular, it canhappen that I x = A for certain maximal ideals x . In this case we define thatΦ x ( a ) is invertible in A / I x for every a ∈ A . Theorem 5 (Allan’s local principle)
Let C be a central subalgebra of the uni-tal Banach algebra A . Then an element a ∈ A is invertible if and only if the cosets Φ x ( a ) are invertible in A / I x for each x ∈ M ( C ) . Here is the second ingredient. Recall that an idempotent is an element p of analgebra such that p = p . Theorem 6 (Two idempotents theorem)
Let A be a Banach algebra withidentity element e , let p and q be idempotents in A , and let B denote the smallestclosed subalgebra of A which contains p, q and e . Suppose that and belongto the spectrum σ B ( pqp ) of pqp in B and that and are cluster points of thatspectrum. Then ( a ) for each point x ∈ σ B ( pqp ) , there is a continuous algebra homomorphism Φ x : B → C × which acts at the generators of B by Φ x ( e ) = (cid:18) (cid:19) , Φ x ( p ) = (cid:18) (cid:19) , Φ x ( q ) = (cid:18) x p x (1 − x ) p x (1 − x ) 1 − x (cid:19) where p x (1 − x ) denotes any complex number with ( p x (1 − x )) = x (1 − x ) . ( b ) an element a ∈ B is invertible in B if and only if the matrices Φ x ( a ) areinvertible for every x ∈ σ B ( pqp ) . ( c ) if σ B ( pqp ) = σ A ( pqp ) , then B is inverse closed in A . We proceed with the proof of Theorem 3, which we split into several steps.
Step 1: Localization.
For every operator A ∈ L ( l p ), we denote its coset A + K ( l p ) in the Calkin algebra by A π , and for every multiplier a ∈ M p , we put˜ a ( t ) := a (1 /t ). The identities T ( ab ) = T ( a ) T ( b ) + H ( a ) H (˜ b ) and H ( ab ) = T ( a ) H ( b ) + H ( a ) T (˜ b ) , (4)which hold for arbitrary a, b ∈ M p , together with the compactness of the Hankeloperators H ( c ) for c ∈ C p show that the set C p of all cosets T ( c ) π with c ∈ C p and7 = ˜ c forms a central subalgebra of the algebra TH ( M p ) /K ( l p ) and, in particular,of the algebra TH ( P C p ) /K ( l p ). One can, thus, reify Allan’s local principle with TH ( P C p ) /K ( l p ) and C p in place of A and C , respectively. It is not hard tosee that the maximal ideal space of C p is homeomorphic to the arc T + , with t ∈ T + corresponding to the maximal ideal { c ∈ C p : c ( t ) = 0 } of C p . We let J t denote the smallest closed ideal of TH ( P C p ) /K ( l p ) which contains the maximalideal t and write A πt for the coset A π + J t of A ∈ TH ( P C p ). Instead of T ( a ) πt and H ( b ) πt we often write T πt ( a ) and H πt ( b ), respectively, and the local quotientalgebra ( TH ( P C p ) /K ( l p )) / J t is denoted by TH πt ( P C p ) therefore. By Allan’s localprinciple, we then have σ TH ( P C p ) /K ( l p ) ( A π ) = ∪ t ∈ T + σ TH πt ( P C p ) ( A πt ) (5)for every A ∈ TH ( P C p ). Step 2: Local equivalence of multipliers.
Let a, b ∈ P C p and t ∈ T + . Weshow that if a ( t ± ) = b ( t ± ) and a ( t ± ) = b ( t ± ), then T πt ( a ) = T πt ( b ) and H πt ( a ) = H πt ( b ). This fact will be used in what follows in order to replace multipliers bylocally equivalent ones. It is clearly sufficient to prove that if a ∈ P C p satisfies a ( t ± ) = a ( t ± ) = 0, then T π ( a ) , H π ( a ) ∈ J t . We will give this proof for t ∈ T ;the proof for for t = ± ε >
0, let f ∈ P C such that k a − f k M p < ε . Then there is an open arc U := ( e − iδ t, e iδ t ) ⊂ T + such that | a ( s ) | < ε almost everywhere on U ∪ U and suchthat f has at most one discontinuity in each of U and U . Then | f ( s ) | < ε for s ∈ U ∪ U . Now choose a real-valued function ϕ ∈ C ∞ ( T ) such that ϕ ( t ) = 1,the support of ϕ is contained in U , and ϕ is monotonously increasing on thearc ( e − iδ t, t ) and monotonously decreasing on ( t, e iδ t ). Set ϕ := ϕ + f ϕ . Then ϕ = e ϕ , and T π ( f ) − T π ( f ϕ ) = T π ( f (1 − ϕ )) = T π ( f ) T π (1 − ϕ ) ∈ J t ,H π ( f ) − H π ( f ϕ ) = H π ( f (1 − ϕ )) = H π ( f ) T π (1 − ϕ ) ∈ J t . Since k f ϕ k ∞ < ε and Var( f ϕ ) < ε , we conclude that k f ϕ k M p < c p ε fromStechkin’s inequality. Thus, k T π ( f ϕ ) k < c p ε and k H π ( f ϕ ) k < c p ε , with aconstant c p depending on p only. Thus, T π ( a ) differs from the element T π ( f ) − T π ( f ϕ ) ∈ J t by the element T π ( a − f ) + T π ( f ϕ ), which has a norm less than(1 + 10 c p ) ε . Since ε > J t is closed, this implies T π ( a ) ∈ J t .Analogously, H π ( a ) ∈ J t . Step 3: The local algebras at t ∈ T . We start with describing the localalgebras TH πt ( P C p ) at points t ∈ T . Let χ t denote the characteristic functionof the arc in T which connects t with ¯ t and runs through the point -1. Clearly, χ t ∈ P C p . The crucial observation, which is a simple consequence of the identities(4), is that the operator T ( χ t ) + H ( χ t ) is an idempotent. Further, let ϕ t ∈ C p be8ny multiplier such that 0 ≤ ϕ t ≤ ϕ t ( t ) = 1, ϕ (¯ t ) = 0 and ϕ t + e ϕ t = 1. Againby (4), the coset T πt ( ϕ t ) is an idempotent.We claim that the idempotents p t := T πt ( ϕ t ) and q t := T πt ( χ t ) + H πt ( χ t )together with the identity element e := I πt generate the local algebra TH πt ( P C p ).Let a, b ∈ P C p . Then, using step 2, T πt ( a ) = a ( t + ) T πt ( χ t ϕ t ) + a ( t − ) T πt ((1 − χ t ) ϕ t ) + a (¯ t − ) T πt ( χ t (1 − ϕ t ))+ a (¯ t + ) T πt ((1 − χ t )(1 − ϕ t )) . (6)It is not hard to check that T πt ( χ t ϕ t ) = p t q t p t ,T πt ((1 − χ t ) ϕ t ) = p t ( e − q t ) p t ,T πt ( χ t (1 − ϕ t )) = ( e − p t ) q t ( e − p t ) ,T πt ((1 − χ t )(1 − ϕ t )) = ( e − p t )( e − q t )( e − p t ) . (7)Let us verify the first of these identities, for example. By definition, p t q t p t = T πt ( ϕ t ) T πt ( χ t ) T πt ( ϕ t ) + T πt ( ϕ t ) H πt ( χ t ) T πt ( ϕ t ) . Since T ( ϕ t ) commutes with T ( χ t ) modulo compact operators and H ( e ϕ t ) is com-pact, we can use the identities (4) to conclude T πt ( ϕ t ) T πt ( χ t ) T πt ( ϕ t ) = T πt ( χ t ) T πt ( ϕ t ) = T πt ( χ t ϕ t ) . Further, due to the compactness of H ( ϕ t ) and H ( e ϕ t ), T πt ( ϕ t ) H πt ( χ t ) T πt ( ϕ t ) = H πt ( ϕ t χ t ) T πt ( ϕ t ) = H πt ( ϕ t χ t e ϕ t ) . Since ϕ t χ t e ϕ t is a continuous function, H πt ( ϕ t χ t e ϕ t ) = 0. This gives the first of theidentities (7). The others follow in a similar way. Thus, (6) and (7) imply that T πt ( a ) belongs to the algebra generated by e , p t and q t . Similarly, we write H πt ( b ) = b ( t + ) H πt ( χ t ϕ t ) + b ( t − ) H πt ((1 − χ t ) ϕ t ) + b (¯ t − ) H πt ( χ t (1 − ϕ t ))+ b (¯ t + ) H πt ((1 − χ t )(1 − ϕ t )) (8)and use the identities H πt ( χ t ϕ t ) = p t q t ( e − p t ) ,H πt ((1 − χ t ) ϕ t ) = − p t q t ( e − p t ) ,H πt ( χ t (1 − ϕ t )) = ( e − p t ) q t p t ,H πt ((1 − χ t )(1 − ϕ t )) = − ( e − p t ) q t p t (9)to conclude that H πt ( b ) also belongs to the algebra generated by e , p t and q t .Thus, the algebra TH πt ( P C p ) is subject to the two idempotents theorem.9n order to apply this theorem we have to determine the spectrum of the coset p t q t p t = T πt ( χ t ϕ t ) in that algebra. We claim that σ TH πt ( P C p ) ( T πt ( χ t ϕ t )) = { µ q ( λ ) : λ ∈ R } (10)with 1 /p + 1 /q = 1. Let a t ∈ P C p be a multiplier with the following properties:( a ) a t is continuous on T \ { t } and has a jump at t ∈ T .( b ) a t ( t + ) = χ t ( t + ) = 1 and a t ( t − ) = χ t ( t − ) = 0.( c ) a t takes values in { µ q ( λ ) : λ ∈ R } only.( d ) a t is zero on the arc joining − t to t which contains the point 1.Then, by Theorem 1, the essential spectrum of the Toeplitz operator T ( a t ) in eachof the algebras L ( l p ) /K ( l p ) and T ( P C p ) /K ( l p ) is equal to the arc { µ q ( λ ) : λ ∈ R } .Hence, the essential spectrum of T ( a t ), now considered as an element of thealgebra TH ( P C p ) /K ( l p ), is also equal to this arc. Hence, σ TH πt ( P C p ) ( T πt ( a t )) ⊆ { µ q ( λ ) : λ ∈ R } by Allan’s local principle. Since T πt ( a t ) = T πt ( χ t ϕ t ), this settles the inclusion ⊆ in (10). For the reverse inclusion, let b t ∈ P C p be a multiplier with the followingproperties:( a ) b t is continuous on T \ { t } and has a jump at t ∈ T .( b ) b t ( t ± ) = χ t ( t ± ).( c ) b t takes values not in { µ q ( λ ) : λ ∈ R } on the arc joining − t to t whichcontains the point − d ) b t is zero on the arc joining − t to t which contains the point 1.Then, again by Theorem 1, the essential spectrum of the Toeplitz operator T ( b t )in each of the algebras L ( l p ) /K ( l p ) and T ( P C p ) /K ( l p ) is equal to the union ofthe arc { µ q ( λ ) : λ ∈ R } and the range of b t . Hence, the essential spectrum of T ( b t ), now considered as an element of the algebra TH ( P C p ) /K ( l p ), is also equalto this union. Since b t is continuous on T \ { t } by property ( a ), we have σ TH πs ( P C p ) ( T πs ( b t )) = { b t ( s ) , b t (¯ s ) } for s ∈ T \ { t } . Since the points b t ( s ) and b t (¯ s ) do not belong to { µ q ( λ ) : λ ∈ R } by property ( c ), we conclude that the open arc { µ q ( λ ) : λ ∈ R } is contained inthe local spectrum of T ( b t ) at t . Since spectra are closed, this implies { µ q ( λ ) : λ ∈ R } ⊆ σ TH πt ( P C p ) ( T πt ( b t )) . Since T πt ( b t ) = T πt ( χ t ϕ t ) by property ( b ), this settles the inclusion ⊇ in (10).Since ν q ( λ ) = µ q ( λ )(1 − µ q ( λ )), we can choose p µ q ( λ )(1 − µ q ( λ )) = ν q ( λ ).With this choice and identities (6) – (9) it becomes evident that the two idem-potents theorem associates with the coset T πt ( a ) + H πt ( b ) the matrix function λ (cid:18) a ( t + ) µ q ( λ ) + a ( t − )(1 − µ q ( λ )) ( b ( t + ) − b ( t − )) ν q ( λ )( b (¯ t − ) − b (¯ t + )) ν q ( λ ) a (¯ t − )(1 − µ q ( λ )) + a (¯ t + ) µ q ( λ ) (cid:19) R . Step 4: The local algebra at ∈ T + . Next we are going to consider the localalgebra TH π ( P C p ) at the fixed point 1 of the mapping t ¯ t . Let f : T → C denote the function e is − s/π where s ∈ [0 , π ). This function belongs to P C p , and it has its only jump at the point 1 ∈ T where f (1 ± ) = ±
1. Using ideasfrom [17], it was shown in [18] by one of the authors that the Hankel operator H ( f ) belongs to the Toeplitz algebra T ( P C p ) and that its essential spectrum isgiven by σ ess ( H ( f )) = { i ν q ( λ ) : λ ∈ R } . (11)(in fact, this identity was derived in [18] with p in place of q , which makes nodifference since ν p ( − λ ) = ν q ( λ ) for every λ .) Let χ + denote the characteristicfunction of the upper half-circle T + . Since every coset T π ( a ) with a ∈ P C p isa linear combination of the cosets I π and T π ( χ + ) and every coset H π ( b ) is amultiple of the coset H π ( f ), the local algebra TH π ( P C p ) is singly generated (as aunital algebra) by the coset T π ( χ + ). In particular, TH π ( P C p ) is a commutativeBanach algebra, and its maximal ideal space is homeomorphic to the spectrumof its generating element. Similar to the proof of (10) one can show that σ TH π ( P C p ) ( T πt ( χ + )) = { µ q ( λ ) : λ ∈ R } (12)It is convenient for our purposes to identify the maximal ideal space of the algebra TH π ( P C p ) with R . The Gelfand transform of T πt ( χ + ) is then given by λ µ q ( λ )due to (12). Let h denote the Gelfand transform of H π ( f ). From (4) we obtain H π ( f ) = T π ( f ˜ f ) − T π ( f ) T π ( ˜ f ) . The function f ˜ f is continuous at 1 ∈ T and has the value − f + ˜ f is continuous at 1 ∈ T and has the value 0 there. Thus, H π ( f ) = − I π + T π ( f ) . Since T π ( f ) = T π (2 χ + −
1) = 2 T π ( χ + ) − I π we conclude that h ( λ ) = (2 µ q ( λ ) − − π ( λ + i/q ))) − if λ ∈ R and by h ( ±∞ ) = 0. By (11), this equality necessarily implies that h ( λ ) = (sinh( π ( λ + i/q ))) − = 2 iν q ( λ )if λ ∈ R and h ( ±∞ ) = 0. Combining these results we find that the Gelfandtransform of T π ( a ) + H π ( b ) is the function λ a (1 + ) µ q ( λ ) + a (1 − )(1 − µ q ( λ )) + i ( b (1 + ) − b (1 − )) ν q ( λ ) . Step 5: The local algebra at − ∈ T + . It remains to examine the local algebra TH π − ( P C p ) at the point −
1. Let Λ : l → l denote the mapping ( x n ) n ≥ − n x n ) n ≥ . Clearly, Λ − = Λ, and one easily checks (perhaps most easily onthe level of the matrix entries, which are Fourier coefficients) thatΛ − T ( a )Λ = T (ˆ a ) and Λ − H ( a )Λ = − H (ˆ a )for a ∈ P C p , where ˆ a ( t ) := a ( − t ). Thus, the mapping A Λ − A Λ is an automor-phism of the algebra TH ( P C p ), which maps compact operators to compact oper-ators and induces, thus, an automorphism of the algebra TH ( P C p ) /K ( l p ). Thelatter maps the local ideal at 1 to the local ideal at − TH π ( P C p ) and TH π − ( P C p ),which sends T π ( χ + ) to T π − (1 − χ + ) and H π ( χ + ) to − H π − (1 − χ + ) = H π − ( χ + ),respectively. Step 6: From local to global invertibility.
We have identified the right-handsides of (1) and (2) as the functions which are locally associated with the oper-ator T ( a ) + H ( b ) via the two idempotents theorem and via Gelfand theory forcommutative Banach algebras, respectively. It follows from the two idempotentstheorem and from Gelfand theory that the so-defined mappings smb p ( t, λ ) ex-tend to a continuous homomorphism from TH ( P C p ) to C × or C , respectively,which combine to a continuous homomorphism from TH ( P C p ) to the algebra F . Allan’s local principle then implies that the coset A + K ( l p ) of an operator A ∈ TH ( P C p ) is invertible in TH ( P C p ) /K ( l p ) if and only if its symbol does notvanish. The proof of estimate (3) will base on Mellin homogenization arguments.We therefore postpone it until Section 5; see estimate (26). Step 7: Inverse closedness.
It remains to show that TH ( P C p ) /K ( l p ) is aninverse closed subalgebra of the Calkin algebra L ( l p ) /K ( l p ). We shall prove thisfact by using a thin spectra argument as follows: If A is a unital closed subalgebraof a unital Banach algebra B , and if the spectrum in A of every element in a densesubset of A is thin, i.e. if its interior with respect to the topology of C is empty,then A is inverse closed in B . See, e.g., [19], Corollary 1.2.32, for a simple proofof this argument.Let A be the set of all operators of the form A := l X i =1 k Y j =1 ( T ( a ij ) + H ( b ij )) with A ij , b ij ∈ P C , (13)and write σ T Hess ( A ) for the spectrum of A in TH ( P C p ) /K ( l p ). Then A /K ( l p ) isdense in TH ( P C p ) /K ( l p ), and the assertion will follow once we have shown that TH ( P C p ) /K ( l p ) is thin for every A ∈ A .Given A of the form (13), let Ω denote the set of all discontinuities of thefunctions a ij and b ij , and put e Ω := (Ω ∪ Ω) ∩ T + . Clearly, e Ω is a finite set. Bywhat we have shown above, σ T Hess ( A ) = ∪ ( t,λ ) ∈ T + × R σ (smb p ( A )( t, λ ))12here σ ( B ) stands for the spectrum (= set of the eigenvalues) of the matrix B .We write σ T Hess ( A ) as Σ ∪ Σ ∪ Σ whereΣ := ∪ ( t,λ ) ∈{− , }× R σ (smb p ( A )( t, λ )) , Σ := ∪ ( t,λ ) ∈ ( T \ e Ω) × R σ (smb p ( A )( t, λ )) , Σ := ∪ ( t,λ ) ∈ ( e Ω \{− , } ) × R σ (smb p ( A )( t, λ )) . It is clear that Σ is a set of measure zero. It is also clear that each setΣ ,t := ∪ λ ∈ R σ (smb p ( A )( t, λ )) with t ∈ T \ e Ωhas measure zero. Since the functions a ij and b ij are piecewise constant, themapping t Σ ,t is constant on each connected component of T \ e Ω, and thenumber of components is finite. Thus, Σ is actually a finite union of sets ofmeasure zero. Since e Ω is finite, it remains to show that each of the setsΣ ,t := ∪ λ ∈ R σ (smb p ( A )( t, λ )) with t ∈ e Ω \ {− , } has measure zero. For this goal it is clearly sufficient to show that each setΣ ,t := ∪ λ ∈ R σ (smb p ( A )( t, λ )) with t ∈ e Ω \ {− , } has measure zero. Let t ∈ e Ω \ {− , } , and write smb p ( A )( t, λ ) as ( c ij ( λ )) i,j =1 .The eigenvalues of this matrix are s ± ( λ ) = ( c ( λ ) + c ( λ )) / ± p r ( λ ) where r ( λ ) = ( a ( λ ) + a ( λ )) / − ( a ( λ ) a ( λ ) − a ( λ ) a ( λ ))and where p r ( λ ) is any complex number the square of which is r ( λ ). Since r is composed by the meromorphic functions coth and 1 / sinh, the set of zerosof r is discrete. Hence, R \ { λ ∈ R : r ( λ ) = 0 } is an open set, which as theunion of an at most countable family of open intervals. Let I be one of theseintervals. Then I can be represented as the union of countably many compactsubintervals I n such that the intersection I n ∩ I m consists of at most one pointwhenever n = m and each set r ( I n ) is contained in a domain where a continuousbranch, say f n , of the function z
7→ √ z exists. Then ± f n ◦ r : I n → C isa continuously differentiable function, which implies that ( ± f n ◦ r )( I n ) is a setof measure zero. Consequently, the associated sets s ± ( I n ) of eigenvalues havemeasure zero, too. Since the countable union of sets of measure zero has measurezero, we conclude that each set Σ ,t has measure zero, which finally implies that σ T Hess ( A ) = Σ ∪ Σ ∪ Σ has measure zero and is, thus, thin. This settles the proofof the inverse closedness and concludes the proof of Theorem 3.We would like to mention that there is another proof of the inverse closednessassertion in the previous theorem which is based on ideas from [5] and whichworks also in other situations. 13 An extended Toeplitz algebra
In the proof of the announced index formula for Toeplitz plus Hankel operators,we shall need an extension of the results of the previous section to certain matrixoperators. For k ∈ N and X a linear space, we let X k and X k × k stand for thelinear spaces of all vectors of length k and of all k × k -matrices with entries in X , respectively. If X is an algebra, then X k × k becomes an algebra under thestandard matrix operations. If X is a Banach space, then X k and X k × k becomeBanach spaces with respect to the norms k ( x j ) kj =1 k = k X j =1 k x j k and k ( a ij ) ki,j =1 k = k sup ≤ i,j ≤ k k a ij k . (14)If, moreover, X is a Banach algebra, then X k × k is a Banach algebra with respect tothe introduced norm. Actually, any other norm on X k and any other compatiblematrix norm on X k × k will do the same job. Note also that if X is a C ∗ -algebrathere is a unique norm (different from the above mentioned) which makes X k × k to a C ∗ -algebra. Since we will not employ C ∗ -arguments, the choice (14) will besufficient for our purposes.Let T ( P C p ) denote the smallest closed subalgebra of L ( l p ( Z )) which containsthe projection P and all Laurent operators L ( a ) with a ∈ P C p . The algebra T ( P C p ) contains T ( P C p ) in the sense that the operator P L ( a ) P : im P → im P can be identified with the Toeplitz operator T ( a ). For k ∈ N , the matrix algebra T ( P C p ) k × k will be also denoted by T k × k ( P C p ). One can characterize T k × k ( P C p )also as the smallest closed subalgebra of L ( l p ( Z ) k ) which contains all operators ofthe form L ( a )diag P + L ( b )diag Q with a, b ∈ ( P C p ) k × k , where Q := I − P , diag A stands for the operator on L ( l p ( Z ) k ) which has A ∈ L ( l p ( Z )) at each entry of itsmain diagonal and zeros at all other entries, and where L ( a ) = ( L ( a ij )) ki,j =1 refersto the matrix Laurent operator with generating function a = ( a ij ) ki,j =1 . Note that K ( l p ( Z ) k ) is contained in T k × k ( P C p ).The Fredholm theory for operators in T k × k ( P C p ) is well known. We willpresent it in a form which is convenient for our purposes. Our main tools are againAllan’s local principle (Theorem 5) and a matrix version of the two idempotentstheorem (Theorem 6) due to [5]. Here is the result. Theorem 7
Let a, b ∈ ( P C p ) k × k . ( a ) The operator A := L ( a )diag P + L ( b )diag Q is Fredholm on l p ( Z ) k if and onlyif the matrix (smb p A )( t, λ ) = (cid:18) a ( t − ) + ( a ( t + ) − a ( t − ))diag µ q ( λ ) ( b ( t + ) − b ( t − ))diag ν q ( λ )( a ( t + ) − a ( t − ))diag ν q ( λ ) b ( t + ) − ( b ( t + ) − b ( t − ))diag µ q ( λ ) (cid:19) is invertible for every pair ( t, λ ) ∈ T × R . b ) The mapping smb p defined in assertion ( a ) extends to a continuous algebrahomomorphism from T k × k ( P C p ) to the algebra F of all bounded functions on T × R with values in C k × k . Moreover, there is a constant M such that k smb p A k := sup ( t,λ ) ∈ T + × R k smb p A ( t, λ ) k ∞ ≤ M inf K ∈ K ( l p ( Z ) k ) k A + K k (15) for every operator A ∈ T k × k ( P C p ) . ( c ) An operator A ∈ T k × k ( P C p ) has the Fredholm property on l p ( Z ) k if and onlyif the function smb p A is invertible in F . ( d ) The algebra T k × k ( P C p ) /K ( l p ( Z ) k ) is inverse closed in the Calkin algebra L ( l p ( Z ) k ) /K ( l p ( Z ) k ) . ( e ) If A ∈ T k × k ( P C p ) is a Fredholm operator, then ind A = − wind (det smb p A ( t, λ ) / (det a ( t, ∞ ) det a ( t, −∞ ))) where smb p A = ( a ij ) i,j =1 with k × k -matrix-valued functions a ij . It is a non-trivial fact that the function W : T × R , ( t, λ ) det smb p A ( t, λ ) / (det a ( t, ∞ ) det a ( t, −∞ ))forms a closed curve in the complex plane. Thus, the winding number of W iswell defined if A is a Fredholm operator.The remainder of this section is devoted to the proof of Theorem 7. We shallmainly make use of results from Sections 2.3 - 2.5 in [11] and Chapter 6 in [2].We will be quite sketchy when the arguments are close to those from the proofof Theorem 3. Step 1: Spline spaces.
We start with recalling some facts about spline spacesand operators thereon from [11]. Let χ [0 , denote the characteristic function ofthe interval [0 , ⊂ R and, for n ∈ N , let S n denote the smallest closed subspaceof L p ( R ) which contains all functions ϕ k,n ( t ) := χ [0 , ( nt − k ) , t ∈ R , where k ∈ Z . The space l p ( Z ) can be identified with each of the spaces S n inthe sense that a sequence ( x k ) is in l p ( Z ) if and only if the series P k ∈ Z x k ϕ k,n converges in L p ( R ) and that (cid:13)(cid:13)(cid:13)X x k ϕ k,n (cid:13)(cid:13)(cid:13) L p ( R ) = n − /p k ( x k ) k l p ( Z ) in this case. Thus, the linear operator E n : l p ( Z ) → S n ⊂ L p ( R ) , ( x k ) n /p X x k ϕ k,n , E − n : L p ( R ) ⊃ S n → l p ( Z ) are isometries for every n . Further wedefine operators L n : L p ( R ) → S n , u n X k ∈ Z h u, ϕ k,n i ϕ k,n with respect to the sesqui-linear form h u, v i := R R uvdx , where u ∈ L p ( R ) and v ∈ L q ( R ) with 1 /p + 1 /q = 1. It is easy to see that every L n is a projectionoperator with norm 1 and that the L n converge strongly to the identity operatoron L p ( R ) as n → ∞ . Finally we set Y t : l p ( Z ) → l p ( Z ) , ( x k ) ( t − k x k ) for t ∈ T . Clearly, Y t is an isometry, and Y − t = Y t − . One easily checks that Y − t L ( a ) Y t = L ( a t ) with a t ( s ) = a ( ts ) for every multiplier a , which implies in particular that Y − t T ( P C p ) Y t = T ( P C p ). Step 2: Some homomorphisms.
In Sections 2.3.3 and 2.5.2 of [11] it is shownthat, for every A ∈ T ( P C p ) and every t ∈ T , the strong limitsmb t A := s-lim n →∞ E n Y − t AY t E − n L n exists and that the mapping smb t is a bounded unital algebra homomorphism.This homomorphism can be extended in a natural way to the matrix algebra T k × k ( P C p ). We denote this extension by smb t A again.In order to characterize the range of the homomorphism smb t , we have tointroduce some operators on L p ( R ). Let χ + stand for the characteristic functionof the interval R + = [0 , ∞ ) and χ + I for the operator of multiplication by χ + .Further, S R refers to the singular integral operator( S R f )( t ) := 1 πi Z ∞−∞ f ( s ) s − t ds, with the integral understood as a Cauchy principal value. Both χ + I and S R are bounded on L p ( R ), and S R = I . Thus, the operators P R := ( I + S R ) / Q R := I − P R are bounded projections on L p ( R ). We let Σ pk ( R ) standfor the smallest closed subalgebra of L ( L p ( R ) k ) which contains the operatorsdiag χ + I , diag S R , and all operators of multiplication by constant k × k -matrix-valued functions. Theorem 8
Let t ∈ T . Then ( a ) smb t diag P = diag χ + I . ( b ) smb t L ( a ) = a ( t + )diag Q R + a ( t − )diag P R for a ∈ ( P C p ) k × k . ( c ) smb t K = 0 for every compact operator K . ( d ) smb t maps the algebra T k × k ( P C p ) onto Σ pk ( R ) . ( e ) The algebra Σ pk ( R ) is inverse closed in L ( L p ( R ) k ) . c ) of the previous theorem implies that every mapping smb t induces anatural quotient homomorphism from T ( P C p ) /K ( l p ( Z )) to Σ p ( R ). We denotethis quotient homomorphism by smb t again. It now easily seen that the estimate(15) holds for every A ∈ T k × k ( P C p ) (with the constant M = 1 for k = 1). Step 3: The Fredholm property.
Since the commutator L ( a ) P − P L ( a ) iscompact for every a ∈ C p , the algebra C p := { diag L ( a ) : a ∈ C p } /K ( l p ( Z ) k ) liesin the center of the algebra A := T k × k ( P C p ) /K ( l p ( Z ) k ). It is not hard to see that C p is isomorphic to C p ; hence the maximal ideal space of C p is homeomorphic tothe unit circle T . In accordance with Allan’s local principle, we introduce thelocal ideals J t and the local algebras A t := A / J t at t ∈ T .By Theorem 8 ( b ), the local ideal J t lies in the kernel of smb t . We de-note the related quotient homomorphism by smb t again. Thus, smb t is an al-gebra homomorphism from A t onto Σ pk ( R ), which sends the local cosets con-taining the operators diag P and L ( a ) with a ∈ ( P C p ) k × k to diag χ + I and a ( t + ) diag Q R + a ( t − ) diag P R , respectively. By Theorem 2.3 in [11], this homo-morphism is injective, i.e., it is an isomorphism between A t and Σ pk ( R ).Since P R and diag χ + I are projections, the algebra Σ pk ( R ) is subject to thetwo projections theorem with coefficients, as derived in [5]. Alternatively, thisalgebra can be described by means of the Mellin symbol calculus, see Section 2.1in [11]. In each case, the result is that an operator of the form( a + diag χ + I + a − diag χ − I ) diag P R + ( b + diag χ + I + b − diag χ − I ) diag Q R (16)where χ − := 1 − χ + and a ± , b ± ∈ C k × k is invertible if and only if the (2 k ) × (2 k )-matrix-valued function λ (cid:18) a + diag (1 − µ p ( λ )) + a − diag µ p ( λ ) ( b + − b − ) diag ν p ( λ )( a + − a − ) diag ν p ( λ ) b + diag µ p ( λ ) + b − diag (1 − µ p ( λ )) (cid:19) is invertible at each point λ ∈ R . Note that the function λ a + diag (1 − µ p ( λ )) + a − diag µ p ( λ )is continuous on R and that this function connects a + with a − if λ runs from −∞ to + ∞ . For the sake of index computation, one would prefer to work witha function which connects a − with a + if λ increases. Since µ p ( − λ ) = 1 − µ q ( λ )and ν p ( − λ ) = ν q ( λ ) with q satisfying 1 /p + 1 /q = 1, we obtain that the operator A in (16) is invertible if and only if the matrix function λ (cid:18) a + diag µ q ( λ ) + a − diag (1 − µ q ( λ )) ( b + − b − ) diag ν q ( λ )( a + − a − ) diag ν q ( λ ) b + diag (1 − µ q ( λ )) + b − diag µ q ( λ ) (cid:19) is invertible on R . This observation, together with the local principle, impliesthat the coset L ( a )diag P + L ( b )diag Q + K ( l p ( Z ) k ) is invertible in the quotient17lgebra T k × k ( P C p ) /K ( l p ( Z ) k ) if and only if the matrix function in assertion ( a )of Theorem 7 is invertible. In particular, this gives the “if”-part of assertion ( a ).The “only if”-part of this assertion follows from the inverse closedness assertion( d ), which can be proved using ideas from [5], where inverse closedness issues oftwo projections algebras with coefficients are studied. The proof of assertions ( b )and ( c ) of Theorem Theorem 7 is then standard. Step 4: The index formula.
It remains to prove the index formula ( e ). Firstwe have to equip the cylinder T × R with a suitable topology, which will be dif-ferent from the usual product topology. We provide T with the counter-clockwiseorientation and R with the natural orientation given by the order < . Then thedesired topology is determined by the system of neighborhoods U ( t , λ ) of thepoint ( t , λ ) ∈ T × R , defined by U ( t , −∞ ) = { ( t, λ ) ∈ T × R : | t − t | < δ, t ≺ t } ∪ { ( t , λ ) ∈ T × R : λ < ε } ,U ( t , + ∞ ) = { ( t, λ ) ∈ T × R : | t − t | < δ, t ≺ t } ∪ { ( t , λ ) ∈ T × R : ε < λ } if λ = ±∞ and by U ( t , λ ) = { ( t , λ ) ∈ T × R : λ − δ < λ < λ + δ } if λ ∈ R , where ε ∈ R and δ, δ , δ are sufficiently small positive numbers, andwhere t ≺ s means that t precedes s with respect to the chosen orientation of T . Note that the cylinder T × R , provided with the described topology, is justa homeomorphic image of the cylinder T × [0 , T ( P C p ) /K ( l p ); see [6] and [2], Proposition 6.28. If one identifies T × [0 ,
1] with T × R , then the Gelfand transform of a coset A + K ( l p ) of A ∈ T ( P C p ) is justthe function Γ( A ) defined in Theorem 1.It is an important point to mention that while the function smb p A for A ∈∈ T k × k ( P C p ) is not continuous on T × R (just consider the south-east entry ofsmb p ( L ( a ) P + L ( b ) Q )), the function( t, λ ) det smb p A ( t, λ ) / (det a ( t, ∞ ) det a ( t, −∞ ) is continuous on T × R . This non-trivial fact was observed by Gohberg andKrupnik in a similar situation when studying the Fredholm theory for singularintegral operators with piecewise continuous coefficients (see [7]; an introductionto this topic is also in Chapter V of [15]).We will establish the index formula by employing a method which also goesback to Gohberg and Krupnik and is known as linear extension. This method hasfound its first applications in the Fredholm theory of one-dimensional singular18ntegral equations; see [10, 15]. We will use this method in the slightly differ-ent context of Toeplitz plus Hankel operators. Therefore, and for the readers’convenience, we recall it here.Let B be a unital ring with identity element e . With every h × r -matrix β := ( b jl ) h,rj,l =1 with entries in B , we associate the elementel( β ) = h X j =1 b j . . . b jr ∈ B (17)generated by β and call the b jl the generators of el( β ). For each element of thisform, there is a canonical matrix ext( β ) ∈ B s × s with s = h ( r + 1) + 1 with entriesin the set { , e, b jk : 1 ≤ j ≤ h, ≤ k ≤ r } and with the property that el( β ) isinvertible in B if and only if ext( β ) is invertible in B s × s . Actually, a matrix withthis property can be constructed as follows. Letext( β ) := (cid:18) Z XY (cid:19) = (cid:18) e h ( r +1) W e (cid:19) (cid:18) e h ( r +1)
00 el( β ) (cid:19) (cid:18) Z X e (cid:19) (18)where e l denotes the unit element of B l × l , Z := e h ( r +1) + B · · · B · · · · · · B r · · · with B j := diag ( b j , b j , . . . , b hj ), X is the column − (0 , . . . , , e, . . . , e ) T with hr zeros followed by h identity elements, Y is the row ( e, . . . , e, , . . . ,
0) with h identity elements followed by hr zeros, and W := ( M , M , . . . , M r ) with M := ( e, . . . , e ) consisting of h identity elements and M j := ( b b . . . b j , b b . . . b j , . . . , b h b h . . . b hj )for j = 1 , . . . , r . The matrix ext( β ) in (18) is called the linear extension of el( β ).Since the outer factors on the right-hand side of (18) are invertible, it followsindeed that el( β ) is invertible in B if and only if its linear extension ext( β ) isinvertible in B s × s . As a special case we obtain that if the b jl are bounded linearoperators on some Banach space B , then el( β ) is a Fredholm operator on B if and only if ext( β ) is a Fredholm operator on L ( B ) s × s = L ( B s ). Moreover,ind el( β ) = ind ext( β ) is this case.We shall apply this observation for B = l p ( Z ) k and for the generating opera-tors b jl := L ( c jl ) diag P + L ( d jl ) diag Q with c jl , d jl ∈ ( P C p ) k × k . (19)19ut β := ( b jl ) h,rj,l =1 , γ := ( L ( c jl )) h,rj,l =1 and δ := ( L ( d jl )) h,rj,l =1 . The linear extensionsof γ and δ are Laurent operators again; thus ext( γ ) = L ( c ) and ext( δ ) = L ( d )with piecewise continuous multipliers c and d . Moreover,ext( β ) = L ( c ) diag P + L ( d ) diag Q. (20)If el( β ) is a Fredholm operator then, by Theorem 7 ( a ), the matrices c ( t ± ) and d ( t ± ) are invertible for every t ∈ T . Hence, c and d are invertible in ( P C p ) ks × ks .This fact together with the above observation implies that the operator el( β )is Fredholm on l p ( Z ) k if and only if its linear extension ext( β ) is Fredholm on l p ( Z ) ks , which on its hand holds if and only if the Toeplitz operator T ( d − c ) isFredholm on l pks , and that the Fredholm indices of the operators el( β ), ext( β )and T ( d − c ) coincide in this case. The symbol of the Toeplitz operator T ( d − c )is the functionsmb p ( T ( d − c ))( t, λ ) = ( d − c )( t + )diag µ q ( λ ) + ( d − c )( t − )diag (1 − µ q ( λ ))(which stems from the matrix-version of Theorem 1), and smb p (ext( β )) =:( a ij ) i,j =1 is related with smb p ( T ( d − c )) viadet smb p ( T ( d − c ))( t, λ ) = det(smb p ext( β ))( t, λ ) / (det a ( t, ∞ ) det a ( t, −∞ ))as can be checked directly; see [10, 15] for details. This fact can finally be usedto derive the index formula for Fredholm operators of the form el( β ) with theentries of β given by (19). For details we refer to [10, 15] again, where a similarsetting is considered.Since the operators el( β ) lie dense in T k × k ( P C p ), the index formula for aFredholm operator in this algebra follows by a standard approximation argument.To carry out this argument one has to use the estimate k smb p el( β ) k ≤ M inf K ∈ K ( l p ( Z ) k ) k el( β ) + K k with M independent of β , which is an immediate consequence of (15). T + H -operators Our next goal is to provide an index formula for Fredholm operators of theform T ( a ) + H ( b ) on l p where a, b are multipliers in P C p with a finite set ofdiscontinuities. We start with a couple of lemmata. Lemma 9 If a ∈ C ( T ) ∩ M h p i , then H ( a ) is compact on l p . Proof.
It is shown in Proposition 2.45 in [2] that C ( T ) ∩ M h p i ⊆ C p (in fact itis shown there that the closure of C ( T ) ∩ M h p i in the multiplier norm equals C p )20nd in Theorem 2.47 that H ( a ) is compact on l p if a ∈ C p .For a subset Ω of T , let P C (Ω) stand for the set of all piecewise continuousfunctions which are continuous on T \ Ω, and put
P C h p i (Ω) := P C (Ω) ∩ M h p i .Thus, C h p i := P C h p i ( ∅ ) = C ( T ) ∩ M h p i . From 6.27 in [2] one concludes that P C h p i (Ω) ⊆ P C p if Ω is finite.In what follows, we specify Ω := { τ , . . . , τ m } to be a finite subset of T \{± } and put Ω := Ω ∪ {± } . Let ϕ ∈ C h p i be a multiplier which satisfies ϕ = ˜ ϕ ,takes its values in [0 , {− , } and identically 0 on a certain neighborhood of Ω ∪ Ω . Moreover, we supposethat ϕ + ϕ = 1 where ϕ := 1 − ϕ . Lemma 10
Let c ∈ P C h p i ( {− , } ) and d ∈ P C h p i (Ω ) . Then the operators H ( c ) T ( d ) − H ( cdϕ ) and T ( c ) H ( d ) − H ( cdϕ ) are compact on l p . Proof.
We write H ( c ) T ( d ) = H ( c ) T ( d ) T ( ϕ ) + H ( c ) T ( d ) T ( ϕ ) with H ( c ) T ( d ) T ( ϕ ) = H ( c ) ( T ( dϕ ) − H ( d ) H ( f ϕ ))= H ( cdϕ ) − T ( c ) H ( g dϕ ) − H ( c ) H ( d ) H ( f ϕ ) ,H ( c ) T ( d ) T ( ϕ ) = H ( c ) T ( ϕ ) T ( d ) + H ( c ) ( T ( d ) T ( ϕ ) − T ( ϕ ) T ( d ))= ( H ( cϕ ) − T ( c ) H ( f ϕ )) T ( d )+ H ( c ) (cid:16) H ( d ) H ( f ϕ ) − H ( ϕ ) H ( ˜ d ) (cid:17) . The operators H ( g dϕ ), H ( f ϕ ), H ( cϕ ), H ( ϕ ) and H ( f ϕ ) are compact by Lemma9, which gives the first assertion. The proof of the second assertion proceedssimilarly. Lemma 11
Let a , b ∈ P C h p i ( {− , } ) and a , b ∈ P C h p i (Ω ) . Then the op-erator ( T ( a ) + H ( b ))( T ( a ) + H ( b )) − ( T ( a a ) + H ( a b ϕ ) + H ( a b ϕ )) is compact on l p . Proof.
We write ( T ( a ) + H ( b ))( T ( a ) + H ( b )) as T ( a ) T ( a ) + T ( a ) H ( b ) + H ( b ) T ( a ) + H ( b ) H ( b )= T ( a a ) + K + H ( a b ϕ ) + K + H ( b a ϕ ) + K + K where K := T ( a ) T ( a ) − T ( a a ) and K := H ( b ) H ( b ) = T ( b ) T ( e b ) − T ( b e b )are compact on l p by Proposition 6.29 in [2], and K := T ( a ) H ( b ) − H ( a b ϕ )and K := H ( b ) T ( a ) − H ( b a ϕ ) are compact by Lemma 10.The following proposition provides us with a key observation; it will allow us toseparate the discontinuities in Ω and {− , } .21 roposition 12 Let a, b ∈ P C h p i (Ω) . If the operator T ( a ) + H ( b ) is Fredholmon l p , then there are functions a , b ∈ P C h p i ( {− , } ) and a , b ∈ P C h p i (Ω ) such that T ( a ) + H ( b ) and T ( a ) + H ( b ) are Fredholm operators on l p and thedifference ( T ( a ) + H ( b ))( T ( a ) + H ( b )) − ( T ( a ) + H ( b )) is compact. Proof. If T ( a ) + H ( b ) is Fredholm on l p , then a is invertible in P C p by Corollary4 ( a ). Since the maximal ideal space of P C p is independent on p and a ∈ P C h p i ,one even has a − ∈ P C h p i .Let U and V be open neighborhoods of {− , } and Ω ∪ Ω , respectively,such that clos U ∩ clos V = ∅ . We will assume moreover that U = U − ∪ U is theunion of two open arcs such that ± ∈ U ± , and that V = V + ∪ V − is the union oftwo open arcs such that V + ⊆ T and V − ⊆ T \ T . Note that these conditionsimply that clos U − ∩ clos U = ∅ .Now we choose a continuous piecewise (with respect to a finite partition of T )linear function c on T which is identically 1 on clos V , coincides with a on ∂U ,and does not vanish on T \ U . This function is of bounded total variation; thus c ∈ C ( T ) ∩ M h p i , whence c ∈ C p as mentioned in the proof of Lemma 9. Put a := aχ U + cχ T \ U . Then a ∈ P C h p i and a − ∈ P C h p i . Further, set a := a − a .The function a is identically 1 on U and coincides with a on V . Since P C h p i isan algebra, a belongs to P C h p i . Finally, set b := bϕ and b := bϕ , with ϕ and ϕ as in front of Lemma 10.The above construction guarantees that a , b ∈ P C h p i ( {− , } ) and a , b ∈ P C h p i (Ω ), and the operator( T ( a ) + H ( b ))( T ( a ) + H ( b )) − ( T ( a a ) + H ( a b ϕ ) + H ( a b ϕ ))is compact on l p by Lemma 11. The functions ( a − b ϕ and ( a − b ϕ vanish identically on a certain neighborhood of Ω by their construction. Hence,the Hankel operators H (( a − b ϕ ) and H (( a − b ϕ ) are compact by Lemma9, which implies that the operator( T ( a ) + H ( b ))( T ( a ) + H ( b )) − ( T ( a a ) + H ( b ϕ ) + H ( b ϕ ))is compact. Since a a = a and b ϕ + b ϕ = b ( ϕ + ϕ ) = b , and since T ( a ) + H ( b ) and T ( a ) + H ( b ) are Fredholm operators on l p by Theorem 3,the assertion follows.By the previous proposition,ind ( T ( a ) + H ( b )) = ind ( T ( a ) + H ( b )) + ind ( T ( a ) + H ( b )) . Since H ( b ) ∈ T ( P C p ) as already mentioned, and since an index formula forFredholm operators in T ( P C p ) is known (see, e.g., 6.40 in [2]), the determinationof ind ( T ( a ) + H ( b )) is no serious problem. The following theorem provides uswith a basic step on the way to compute the index of T ( a ) + H ( b ).22 heorem 13 Let a, b ∈ P C h p i (Ω ) . If one of the operators T ( a ) ± H ( b ) isFredholm on l p , then the other one is Fredholm on l p , too, and the Fredholmindices of these operators coincide. Proof.
By Corollary 4 ( b ), the operators T ( a ) + H ( b ) and T ( a ) − H ( b ) areFredholm operators on l p only simultaneously. It remains to prove that theirindices coincide. Recall from the introduction that T ( a ) = P L ( a ) P and H ( a ) = P L ( a ) QJ . Thus, the index equality will follow once we have constructed a Fred-holm operator D such that the difference D ( P L ( a ) P + P L ( b ) QJ + Q ) − ( P L ( a ) P − P L ( b ) QJ + Q ) D (21)is compact. The following construction of D is a modification of an idea in [12].(Note that the compactness of the operator (21) also provides an alternate proofof the simultaneous Fredholm property of the operators T ( a ) ± H ( b ).)A function c ∈ M p is called even (resp. odd) if c = ˜ c (resp. c = − ˜ c )or, equivalently, if J L ( c ) J = L ( c ) (resp. J L ( c ) J = − L ( c )). Every function c ∈ C p can be written as a sum of an even and an odd function in a uniqueway: c = ( c + ˜ c ) / c − ˜ c ) /
2. Let θ o and θ e be an odd and an even function in C ( T ) ∩ M h p i , respectively, and assume that θ e vanishes at all points of Ω (and,hence, at all points of Ω ). Put D := P L ( θ o + θ e ) P + QL ( θ o − θ e ) Q. (22)We will later specify the functions θ o and θ e such that D becomes a Fredholmoperator. First note that J P L ( θ o + θ e ) P J = − QL ( θ o − θ e ) Q, J QL ( θ o − θ e ) QJ = − P L ( θ o + θ e ) P, whence J DJ = − D and J D + DJ = 0. Next we show that D commutes with theoperator P L ( a ) P + P L ( b ) Q + Q up to a compact operator. Since the Toeplitzoperators P L ( θ o + θ e ) P and P L ( a ) P commute modulo a compact operator, itremains to show that D commutes with P L ( b ) Q up to a compact operator. Thelatter fact follows easily from the identity DP L ( b ) Q − P L ( b ) QD = P L ( θ o + θ e ) P L ( b ) Q − P L ( b ) QL ( θ o − θ e ) Q = P L ( θ o + θ e ) L ( b ) Q − P L ( θ o + θ e ) QL ( b ) Q − P L ( b ) L ( θ o − θ e ) Q + P L ( b ) P L ( θ o − θ e ) Q = 2 P L ( θ e b ) Q − P L ( θ o + θ e ) QL ( b ) Q + P L ( b ) P L ( θ o − θ e ) Q and from the compactness of the operators P L ( θ e b ) Q and P L ( θ o ± θ e ) Q by Lemma9 (note that θ e b ∈ C ( T ) ∩ M h p i ). The compactness of the operator (21) is then a23onsequence of the identity D ( P L ( a ) P + P L ( b ) QJ + Q ) − ( P L ( a ) P − P L ( b ) QJ + Q ) D = DP L ( a ) P − P L ( a ) P D + DP L ( b ) QJ + P L ( b ) QJ D = DP L ( a ) P − P L ( a ) P D + (
DP L ( b ) Q − P L ( b ) QD ) J and of the compactness of the commutators [ D, P L ( a ) P ] and [ D, P L ( b ) Q ].Finally we show that the functions θ e and θ o can be specified such that theoperator D in (22) is a Fredholm operator on l p . Set ˆ θ o ( t ) := | t − | for t ∈ T .Then ˆ θ o is an even function in C ∞ ( T ) and θ o := χ T + ˆ θ o − χ T − ˆ θ o is an odd functionin C ( T ) ∩ M h p i . Further, θ e ( t ) := i m Y j =1 | t − τ j | | t − τ j | , t ∈ T defines an even function θ e ∈ C ( T ) ∩ M h p i which vanishes at the points of Ω .Since θ o and iθ e are real-valued functions, we conclude that θ o ± θ e are invertiblein C ( T ) ∩ M h p i , which implies that D is a Fredholm operator as desired.Now we are in a position to derive an index formula for a Fredholm operator ofthe form T ( a ) + H ( b ) with a, b ∈ P C h p i (Ω ). We make use of the well-knownidentity (cid:18) P L ( a ) P + P L ( b ) QJ + Q P L ( a ) P − P L ( b ) QJ + Q (cid:19) = 12 (cid:18) I JI − J (cid:19) (cid:18) P L ( a ) P + Q P L ( b ) QJ P L ( b ) QJ J ( P L ( a ) P + Q ) J (cid:19) (cid:18) I IJ − J (cid:19) , (23)where the outer factors in (23) are the inverses of each other. Thus, if one ofthe operators T ( a ) ± H ( b ) = P L ( a ) P ± P L ( b ) QJ is a Fredholm operator, thenso is the other, and the Fredholm indices of these operators coincide. Hence themiddle factor (cid:18) P L ( a ) P + Q P L ( b ) QJ P L ( b ) QJ J ( P L ( a ) P + Q ) J (cid:19) = (cid:18) P L ( a ) P + Q P L ( b ) QQL (˜ b ) P QL (˜ a ) Q + P (cid:19) in (23) is a Fredholm operator, andind ( T ( a ) + H ( b )) = 12 ind (cid:18) P L ( a ) P + Q P L ( b ) QQL (˜ b ) P QL (˜ a ) Q + P (cid:19) = 12 ind (cid:18) P L ( a ) P P L ( b ) QQL (˜ b ) P QL (˜ a ) Q (cid:19) . For the latter identity note that the operator A := (cid:18) P L ( a ) P + Q P L ( b ) QQL (˜ b ) P QL (˜ a ) Q + P (cid:19) ∈ L ( l p ( Z ) )24as the complementary subspaces L := { ( Qx , P x ) : ( x , x ) ∈ l p ( Z ) } and L := { ( P x , Qx ) : ( x , x ) ∈ l p ( Z ) } of l p ( Z ) as invariant subspaces and that A acts on L as the identity operator and on L as the operator A := (cid:18) P L ( a ) P P L ( b ) QQL (˜ b ) P QL (˜ a ) Q (cid:19) . Let the function W : T × R → C be defined by W ( t, λ ) = det smb p A ( t, λ ) / (˜ a ( t, ∞ )˜ a ( t, −∞ )) . Since T ( a ) + H ( b ) is Fredholm, W does not pass through the origin, and Theorem7 entails that ind A = − wind W . Thus,ind ( T ( a ) + H ( b )) = −
12 wind W. We are going to show that actuallyind ( T ( a ) + H ( b )) = − wind T + W, (24)where the right-hand side is defined as follows. The compression of W onto T + × R is a continuous function the values of which form a closed oriented curve in C which starts and ends at 1 ∈ C and does not contain the origin. The windingnumber of this curve is denoted by wind T + W . Analogously, we define wind T − W .For the proof of (24) we suppose for simplicity that a and b have jumps onlyat the points t and t where t ∈ T . If t moves along T + from 1 to t (resp.on T − from 1 to t ), then the values of W ( t, λ ) = a ( t ) / ˜ a ( t ) = a ( t ) /a ( t ) movecontinuously from 1 to a ( t − ) /a ( t ) (resp. from 1 to a ( t ) /a ( t − )). Using that W ( t, λ ) = W ( t, λ ) − for t ∈ T \ {− , } , one easily concludes that[arg W ] → t ⊂ T + = [arg W ] t → ⊂ T − where the numbers on the left- and right-hand side stand for the increase of theargument of W if t moves in positive direction along the arc from 1 to t in T + and along the arc from t to 1 in T − , respectively. Analogously,[arg W ] − → t ⊂ T − = [arg W ] t →− ⊂ T + . Consider W ( t , λ ) / ( a ( t ) a ( t − ))= [ a ( t +1 ) µ q ( λ ) + a ( t − )(1 − µ q ( λ ))] [ a ( t ) µ q ( λ ) + a ( t − )(1 − µ q ( λ ))] − ( b ( t +1 ) − b ( t − ))( b ( t ) − b ( t − )) µ q ( λ )(1 − µ q ( λ ))and the related expression for W ( t , λ ) / ( a ( t +1 ) a ( t − )). Then[arg W ] C q ( a ( t − ) , a ( t +1 )) = [arg W ] C q ( a ( t − ) , a ( t )) W ( t , λ ) / ( a ( t ) a ( t − )) = W ( t , λ ) / ( a ( t +1 ) a ( t − )). So we arrive at theequality wind T + W = wind T − W , whence (24) follows.Now suppose that a, b ∈ P C h p i are continuous on T \ {− , } . Then we definea function W : T + × R by W ( t, λ ) = (cid:0) a ( t + ) µ q ( λ ) + a ( t − )(1 − µ q ( λ )) + it ( b ( t + ) − b ( t − )) ν q ( λ ) (cid:1) a − ( ± ∓ )if t = ± W ( t, λ ) = a ( t ) /a ( t ) if t ∈ T . The function W is continuousand determines a closed curve which starts and ends at 1 ∈ C . If T ( a ) + H ( b )is a Fredholm operator, then this curve does not pass through the origin andpossesses, thus, a well defined winding number.Since T ( a )+ H ( b ) is in T ( P C p ) and the symbol V : T × R → C of this operatorrelative to the algebra T ( P C p ) is known (it is just given by V ( t, λ ) = a ( t + ) µ q ( λ ) + a ( t − )(1 − µ q ( λ )) + it ( b ( t + ) − b ( t − )) ν q ( λ )if t = ± V ( t, λ ) = a ( t ) if t ∈ T \{− , } ) and since ind T ( a ) = − wind T V ,one can again prove that wind T V = wind T + W by comparing the increments ofthe arguments as above.Now we look at the factorization given by Proposition 12 and denote by W and W the above defined function W : T + × R for the operators T ( a ) + H ( b )and T ( a ) + H ( b ), respectively. It is easy to see that W W coincides with thefunction W for the operator T ( a ) + H ( b ). Summarizing, we get Theorem 14
Let a, b ∈ P C h p i and T ( a ) + H ( b ) a Fredholm operator on l p . Then ind ( T ( a ) + H ( b )) = − wind T + W − wind T + W = − wind T + W with W , W and W defined as above. In this section we want to sketch an approach to derive an index formula for anarbitrary Fredholm operator A ∈ TH ( P C p ). With A , we associate the function W ( A ) : T + × R → C defined by W ( A )( t, λ ) = (cid:26) smb p A ( t, λ ) / smb p A ( t, ∓∞ ) if t = ± p A ( t, λ ) / ( a ( t, ∞ ) a ( t, −∞ )) if t = ± p A ( t, λ ) = ( a ij ( t, λ )) i,j =1 for t ∈ T . For A = T ( a ) + H ( b ),this definition coincides with that one from the previous section. Theorem 15 If A ∈ TH ( P C p ) is a Fredholm operator, then ind A = − wind T + W ( A ) . (25)26he remainder of this section is devoted to the proof of this theorem. It willbecome evident from this proof that W ( A ) traces out a closed oriented curvewhich does not pass through the origin; so the winding number of W ( A ) is welldefined.We start with the observation that Theorem 3 remains true for matrix-valuedmultipliers a, b ∈ ( P C p ) k × k : just replace µ q , 1 − µ q and ν q by the corresponding k × k -diagonal matrices diag µ q , diag (1 − µ q ) and diag ν q , respectively. AlsoProposition 2 holds in the matrix setting: If T ( a ) + H ( b ) := (diag P ) L ( a )(diag P ) + (diag P ) L ( b )(diag QJ )is a Fredholm operator, then the identityind ( T ( a ) + H ( b )) = − wind W ( T ( a ) + H ( b ))still holds if one replaces in the above definition of W all scalars by the determi-nants of the corresponding matrices. These facts follow in a similar way as theirscalar counterparts.Now let a jl , b jl ∈ P C p , consider the h × r -matrix β := ( T ( a jl ) + H ( b jl )) h,rj,l =1 ,and associate with β the operator A := el( β ) = h X j =1 ( T ( a j ) + H ( b j )) . . . ( T ( a jr ) + H ( b jr )) ∈ TH ( P C p )as in (17). Further set γ := ( L ( a jl )) h,rj,l =1 and δ := ( L ( b jl )) h,rj,l =1 . The linearextensions of γ and δ are Laurent operators again; thus ext( γ ) = L ( a ) andext( δ ) = L ( b ) with certain multipliers a, b ∈ ( P C p ) s × s with s = h ( r + 1) + 1.Moreover, these extensions are related with the extension of β byext( β ) = T (ext( γ )) + H (ext( δ )) = T ( a ) + H ( b ) ∈ L ( l ps )(note that H (1) = 0). In Section 3 we noticed that if el( β ) is Fredholm, then(and only then) ext( β ) is Fredholm and ind el( β ) = ind ext( β ). Further, if el( β )is a Fredholm operator, then the matrices a ( t ± ) are invertible for every t ∈ T .Hence, a is invertible in ( P C p ) s × s . Now considersmb p el( β ) = h X j =1 smb p ( T ( a j ) + H ( b j )) . . . smb p ( T ( a jr ) + H ( b jr )) . Let t = ±
1. Then smb p ( T ( a ) + H ( b ))( t, λ ) is a matrix of size 2 s × s . We putthe rows and columns of this matrix in a new matrix according to the followingrules: If j ≤ h ( r + 1) + 1, then the j th row of the old matrix becomes the 2 j − j > h ( r + 1) + 1, the j th row of the oldmatrix becomes the 2( j − h ( r + 1) −
1) th row of the new matrix. The columns of27mb p ( T ( a ) + H ( b ))( t, λ ) are re-arranged in the same way. The matrix obtainedin this way is just smb p el( β )( t, λ ). By these manipulations,smb p el( β )( t, λ ) = P smb p ( T ( a ) + H ( b ))( t, λ ) P T with a certain permutation matrix P and its transpose P T . Hence,det smb p ( T ( a ) + H ( b ))( t, λ ) = det smb p (el( β ))( t, λ )for t = ±
1. For t = ± p ( T ( a ) + H ( b ))( t, λ ).For t = ±
1, we write smb p ( T ( a ) + H ( b )( t, λ ) = ( a mn ( t, λ )) m,n =1 andsmb p ( T ( a jl ) + H ( b jl ))( t, λ )) = ( a jlmn ( t, λ )) m,n =1 . Then smb p el( β )( t, ±∞ ) = h X j =1 r Y l =1 (cid:18) a jl ( t, ±∞ ) 00 a jl ( t, ±∞ ) (cid:19) , and it follows that det a ( t, ±∞ ) = det ext( ρ ( t, ±∞ ))where ρ ( t, ±∞ ) := ( a jl ( t, ±∞ )) hrj,l =1 . It is now easy to see that W (el( β ))( t, λ ) = W ( T ( a ) + H ( b ))( t, λ ) = W (ext( β ))( t, λ )for all ( t, λ ) ∈ T + × R , which implies that ind el( β ) = − wind T + W (el( β )) and,thus, settles the proof of the index formula (25) for a dense subset of Fredholmoperators in TH ( P C p ).Finally, we are going to prove estimate (3), i.e., we will show that there is aconstant M such that k smb p A k ∞ ≤ M inf {k A + K k : K compact } (26)for every operator A ∈ TH ( P C p ). Once this estimate is shown, the validity of theindex formula (25) for an arbitrary Fredholm operator in TH ( P C p ) will follow bystandard approximation arguments as at the end of Section 3.To prove (26), we consider TH ( P C p ) as a subalgebra of the smallest closedsubalgebra T J ( P C p ) of L ( l p ( Z )) which contains all Laurent operators L ( a ) with a ∈ P C p , the projection P , and the flip J . The homomorphism smb t defined inSection 3 cannot be extended to the algebra T J ( P C p ) unless t = ±
1. Instead, weare going to use ideas from [4] and introduce a related family of homomorphismssmb t,t with t ∈ T from T J ( P C p ) onto (Σ p ( R )) × . A crucial observation ([4]) isthat the strong limitsmb t,t A := s-lim n →∞ (cid:18) A t,n, , A t,n, , A t,n, , A t,n, , (cid:19) (27)with A t,n,i,j := E n Y − t L ( χ T + ) J i AJ j L ( χ T + ) Y t E − n L n exists for every operator A ∈ T J ( P C p ) and every t ∈ T . 28 heorem 16 Let t ∈ T . Then the mapping smb t,t is a bounded homomorphismfrom T J ( P C p ) onto (Σ p ( R )) × . In particular, ( a ) smb t,t P = diag ( χ + I, χ − I ) with χ − = 1 − χ + , ( b ) smb t,t L ( a ) = diag ( a ( t + ) Q R + a ( t − ) P R , a ( t − ) Q R + a ( t + ) P R ) for a ∈ P C p , ( c ) smb t,t K = 0 for every compact operator K , ( d ) smb t,t J = (cid:18) II (cid:19) . Sketch of the proof.
The existence of the strong limits of the operators in( a ) - ( d ) and their actual values follow by straightforward computation. Let uscheck assertion ( a ), for instance. For A = P , the strong limits of the diagonalelements of the matrix (27) exist and are equal to χ + I and χ − I by Theorem8 ( a ) and since J P J = Q . Now consider the 01-entry of that matrix. It is L ( χ T + ) P J = J L ( χ T − ) Q and thus E n Y − t L ( χ T + ) P J L ( χ T + ) Y t E − n L n = (cid:0) E n Y − t J Y t E − n (cid:1) (cid:0) E n Y − t L ( χ T − ) QL ( χ T + ) Y t E − n L n (cid:1) . (28)The first factor on the right-hand side is uniformly bounded with respect to n ,whereas the second one tends strongly to 0 by Theorem 8 (note that χ T − ( t ) = 0for t ∈ T ). Thus, the sequence of the operators (28) tends strongly to zero. Thestrong convergence of the 10-entry to zero follows analogously.Another straightforward calculation shows that the mappings smb t,t are alge-bra homomorphisms and that these mappings are uniformly bounded with respectto t ∈ T . Thus, the mappings smb t,t are well-defined on a dense subalgebra of T J ( P C p ), and they extend to (uniformly bounded with respect to t ) homomor-phisms on all of T J ( P C p ) by continuity.By assertion ( c ) of the previous theorem, every mapping smb t,t induces a quo-tient homomorphism on T J ( P C p ) /K ( l p ( Z )) in a natural way. We denote thishomomorphism by smb t,t again.Now we are ready for the last step. Let t ∈ T and a, b ∈ P C p . FromTheorem 16 we conclude that then the operator smb t,t ( T ( a ) + H ( b )) is given bythe matrix (cid:18) χ + ( a ( t + ) Q R + a ( t − ) P R ) χ + I χ + ( b ( t + ) Q R + b ( t − ) P R ) χ − Iχ − ( b ( t − ) Q R + b ( t + ) P R ) χ + I χ − ( a ( t − ) Q R + a ( t + ) P R ) χ − I (cid:19) acting on L p ( R ) . This matrix operator has the complementary subspaces L := { ( χ − f , χ + f ) : f , f ∈ L p ( R ) } , L := { ( χ + f , χ − f ) : f , f ∈ L p ( R ) } of L p ( R ) as invariant subspaces, and it acts as the zero operator on L . So wecan identify smb t,t ( T ( a ) + H ( b )) with its restriction to L , which we denote by A for brevity. 29he space L can be identified with L p ( R ) in a natural way. Under thisidentification, the operator A can be identified with the operator A := χ + ( a ( t + ) Q R + a ( t − ) P R ) χ + I + χ + ( b ( t + ) Q R + b ( t − ) P R ) χ − I + χ − ( b ( t − ) Q R + b ( t + ) P R ) χ + I + χ − ( a ( t − ) Q R + a ( t + ) P R ) χ − I which belongs to Σ p ( R ). It is well known (see Section 4.2 in [19]) and not hard tocheck that the algebra Σ p ( R ) is isomorphic to Σ p × ( R + ), where the isomorphism η acts on the generating operators of Σ p ( R ) by η ( S R ) = (cid:18) S R + H π − H π − S R + (cid:19) and η ( χ + I ) = (cid:18) (cid:19) , with H π referring to the Hankel operator( H π ϕ )( s ) := 1 πi Z R + ϕ ( t ) t + s dt on L p ( R + ). The entries of the matrix η ( A ) are Mellin operators, and the valueof the Mellin symbol of η ( A ) at ( t, λ ) ∈ T × R is the matrix (cid:18) a ( t + ) µ q ( λ ) + a ( t − )(1 − µ q ( λ )) ( b ( t + ) − b ( t − )) ν q ( λ )( b ( t − ) − b ( t + )) ν q ( λ ) a ( t − )(1 − µ q ( λ )) + a ( t + ) µ q ( λ ) (cid:19) , which evidently coincides with smb p ( T ( a )+ H ( b ))( t, λ ) given in (1). Summarizingthe above arguments we conclude that the homomorphisms A + K ( l p ) (smb p A )( t, λ )are uniformly bounded with respect to ( t, λ ) ∈ T × R , which finally implies theestimate (26). References [1]
E. Basor, T. Ehrhardt , Factorization of a class of Toeplitz + Hankeloperators and the A p -condition. – J. Oper. Th. (2006), 2, 269 – 283.[2] A. B¨ottcher, B. Silbermann , Analysis of Toeplitz Operators. – Sprin-ger-Verlag, Berlin, Heidelberg, New York 1990. Second edition: 2006.[3]
T. Ehrhardt , Invertibility theory for Toeplitz plus Hankel operators andsingular integral operators with flip. – J. Funct. Anal. (2004), 1, 64 –106.[4]
T. Ehrhardt, B. Silbermann , Approximate identities and stability ofdiscrete convolutions with flip. – Oper. Theory: Adv. Appl. , Birkh¨auser,Basel 1999, 103 – 132. 305]
T. Finck, S. Roch, B. Silbermann , Two projections theorems andsymbol calculus for operators with massive local spectra. – Math. Nachr. (1993), 167 – 185.[6]
I. Gohberg, N. Krupnik , On the algebra generated by Toeplitz matrices.– Funkts. Anal. Prilozh. (1969), 2, 46 – 56 (Russian, Engl. transl.: Funct.Anal. Appl. (1969), 119 – 127).[7] I. Gohberg, N. Krupnik , Singular integral operators with piecewise con-tinuous coefficients and their symbols. – Izv. Akad. Nauk SSSR, Ser. Mat. (1971), 4, 955 – 979 (Russian, Engl. transl.: Math. USSR Izvestia (1971),4, 955 – 979).[8] I. Gohberg, N. Krupnik , One-dimensional singular integral operatorswith shift. – Izv. Akad. Nauk Armen. SSR, Ser. Mat. (1973), 1, 3 – 12(Russian, Engl. transl.: Operator Theory: Adv. Appl. , Birkh¨auser Ver-lag, Basel 2010, 201 – 211).[9] I. Gohberg, N. Krupnik , Algebras of singular integral operators withshift. – Matem. Issled. (1973), 2(28), 170 – 175 (Russian, Engl. transl. in:Operator Theory: Adv. Appl. , Birkh¨auser Verlag, Basel 2010, 213 –217).[10] I. Gohberg, N. Krupnik
One-dimensional Linear Singular Integral Equa-tions. Volume I: Introduction, Volume II: General Theory and Applications.– Birkh¨auser, Basel 1992.[11]
R. Hagen, S. Roch, B. Silbermann , Spectral Theory of ApproximationMethods for Convolution Equations. – Birkh¨auser Verlag, Basel, Boston,Berlin 1995.[12]
N. K. Karapetiants, S. G. Samko , On Fredholm properties of a classof Hankel operators. – Math. Nachr. (2000), 75 – 103.[13]
A. Lebre, E. Meister, F. Teixeira , Some results on the invertibility ofWiener-Hopf-Hankel operators. – Z. f. Anal. Anwend. (1992), 57 – 76.[14] E. Meister, F.-O. Speck, F. Teixeira , Wiener-Hopf-Hankel operatorsfor some wedge diffraction problems with mixed boundary conditions. – J.Int. Eq. Appl. (1992), 2, 229 – 255.[15] S. G. Mikhlin, S. Pr¨ossdorf , Singular Integral Operators. – Springer,Berlin, Heidelberg 1985.[16]
S. C. Power , C ∗ -algebras generated by Hankel and Toeplitz operators. –J. Funct. Anal. (1979), 52 – 68.3117] S. Pr¨ossdorf, A. Rathsfeld , Mellin techniques in the numerical analysisfor one-dimensional singular integral equations. – Report R-MATH 06/88,Karl-Weierstraß-Institut, Berlin 1988.[18]
S. Roch , Local algebras of Toeplitz operators. – Math. Nachr. (1991),69 – 81.[19]
S. Roch, P. A. Santos, B. Silbermann , Non-commutative GelfandTheories. – Springer, London 2011.[20]
S. Roch, B. Silbermann , Algebras of convolution operators and theirimage in the Calkin algebra. – Report R-Math-05-90, Karl-Weierstrass-Institut, Berlin 1990.[21]