Dual-band general Toeplitz operators
aa r X i v : . [ m a t h . F A ] D ec Truncated Toeplitz operators on multiband spaces
M. Cristina Cˆamara, ∗ Ryan O’ Loughlin † and Jonathan R. Partington ‡ January 1, 2021
Abstract
We relate multiband truncated Toeplitz operators to block trun-cated Toeplitz operators and, via equivalence after extension, withToeplitz operators with 4 × Keywords:
Truncated Toeplitz operator, Toeplitz operator, multiband sig-nal, Riemann–Hilbert method, Wiener–Hopf factorization
MSC (2010):
Multiband spaces, which can be regarded informally as the orthogonal di-rect sum of two shifts of a model space (precise definitions are given below),occur naturally in applications. First, multiband signals are seen in speechprocessing (see [2, 4, 6, 23] for example), as an alternative to the Paley–Wiener space
P W ( b ) of inverse Fourier transforms of functions in L ( − b, b ), ∗ Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Su-perior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. [email protected] † School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K. [email protected] ‡ School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K. [email protected] < a < b andconsider the inverse Fourier transform of the space L (( − b, − a ) ∪ ( a, b )),which is a space M ⊂ L ( R ). Indeed M = P W ( b ) ⊖ P W ( a ).If we define the inverse Fourier transform formally byˆ f ( s ) = 12 π Z ∞−∞ f ( t ) e ist dt, then this extends to an isomorphism between L (0 , ∞ ) and the Hardy space H ( C + ) on the upper half-plane, and L (0 , b − a ) corresponds to the modelspace K θ := H ( C + ) ⊖ θH ( C + ) where θ is the inner function θ ( s ) = e i ( b − a ) s .It is now clear that the space M above has the orthogonal decomposition M = φK θ ⊕ ψK θ , (1.1)where φ ( s ) = e − ibs and ψ ( s ) = e ias . (1.2)More generally, let θ be an inner function in H ∞ ( C + ), and φ , ψ uni-modular functions in L ∞ ( R ) such that φK θ ⊥ ψK θ . Then we shall considertruncated Toeplitz operators (TTO) A Mg , g ∈ L ∞ ( R ), defined on the space M := φK θ ⊕ ψK θ by A Mg u = P M ( gu ) , where P M is the orthogonal projection onto M ; in the case of the two-intervalexample given earlier, these are unitarily equivalent to convolution operatorsrestricted to the union of two intervals. Clearly, the same definition can alsobe made in the more usual situation of H ( D ), except that now φ, ψ areunimodular in L ∞ ( T ) and g ∈ L ∞ ( T ).Two degenerate cases may also be considered: the decomposition M = θK θ ⊕ K θ gives the Paley–Wiener space as a special case, and the decom-position K θ = K θ ⊕ θK θ is also a special case.Traditional single-band truncated Toeplitz operators of the form A θg u = P θ ( gu ), where P θ denotes the orthogonal projection onto K θ , have beenmuch studied, since being formally defined by Sarason [31], although theyoccur much earlier, for example in [1, 30]. Some recent surveys on thesubject are in [17, 19]. 2n this paper we relate multiband TTO to block TTO and, by usingthe concept of equivalence after extension, to Toeplitz operators with 4 × Our results are presented in the context of L and L ∞ spaces on the circle T , or H and H ∞ spaces on the disc D , but they apply also to the case of L and L ∞ spaces on the real line R and H and H ∞ spaces on the upperhalf-plane C + . Proposition 2.1.
Let θ ∈ H ∞ be inner, and let K θ = H ⊖ θH be thecorresponding model space. Then for φ, ψ ∈ L ∞ unimodular, the spaces φK θ and ψK θ are orthogonal if and only if the truncated Toeplitz operator A θφψ is the zero operator.Proof. If f = ψk and g = φk with k , k ∈ K θ , then h f, g i = h ψk , φk i = h φψk , k i = h P θ φψk , k i and this is zero for all f, g of that form if and only if A θφψ = 0.We have A θφψ = 0 if and only if φψ ∈ θH + θH [31, Thm. 3.1]. Underthis condition, the operator P M defined by P M f = φP θ φf + ψP θ ψf ( f ∈ L ) (2.1)is the orthogonal projection from L onto M := φK θ ⊕ ψK θ .3et now A Mg for g ∈ L be the operator densely defined in M by A M gf = P M gf ( f ∈ L ∞ ∩ M ) , the density of L ∞ ∩ M in M following easily from the density of L ∞ ∩ K θ in K θ , which was given in [31].If this operator is bounded, we also denote by A Mg its unique boundedextension to M . The operator A Mg is bounded, in particular, whenever g ∈ L ∞ . It is easy to see that ( A Mg ) ∗ = A Mg . Theorem 2.2.
Let A Mg be a bounded truncated Toeplitz operator on themultiband space M := φK θ ⊕ ⊥ ψK θ , where θ is inner and φ, ψ ∈ L ∞ areunimodular. Then A Mg is unitarily equivalent to the block truncated Toeplitzoperator W = A θg A θφψg A θψφg A θg ! , (2.2) on K θ ⊕ K θ . Hence A Mg = 0 if and only if each of the four truncated Toeplitzoperators composing W is .Proof. Let M φ denote the operator of multiplication by φ , and similarly forother multiplication operators. We have the factorization A Mg = M φ A θg M φ M φ A θφψg M ψ M ψ A θψφg M φ M ψ A θg M ψ ! = (cid:18) M φ M ψ (cid:19) A θg A θφψg A θψφg A θg ! M φ M ψ ! . (2.3)This has the form U ∗ AU , where U is a unitary operator from M onto K θ ⊕ K θ . Using the fact that M φ maps φK θ bijectively to K θ and P φK θ = M φ P K θ M φ , it is easy to verify that the identity (2.3) holds.There are some simplifications possible here: for A θh = 0 if h ∈ θH or h ∈ θH , so that some of these four blocks may be 0. The basic propertiesof matrix valued truncated Toeplitz operators were studied in [26].Note that A θφψ = 0 if and only if φψ ∈ θH + ¯ θH . If we assumethat φψ = A − ¯ θ + A + θ with A + , A − ∈ H , then A θφψg + = A θA − ¯ θg + and A θψφg + = A θA + ¯ θg + for all g + ∈ H ∞ . Thus we have from (2.2) W = P θ (cid:18) g + A + ¯ θg + A − ¯ θg + g + (cid:19) P θ : K θ → K θ . (2.4)4e also remark that that we can have A − θg + ∈ H ∞ without having g + ∈ θH ∞ . For example, take A − ( z ) = 1 − ¯ z/
2, and θ ( z ) = ( z − / / (1 − z/ g + ( z ) = z . Then A − ( z )¯ θ ( z ) g + ( z ) = 1 − z/
2, but ¯ θ ( z ) g + ( z ) = z (1 − z/ / ( z − / H ∞ .Moreover, if we let A + ∈ H ∞ be chosen to minimize the L ∞ norm of d := A − (¯ θ ) + A + (the Nehari problem) then a result of Sarason [30] saysthat d has constant modulus, and so, dividing A − and A + by a suitableconstant, we may have that A − ¯ θ + A + θ is unimodular.We can also define a natural conjugation C M on L which keeps M in-variant (and is therefore a conjugation on M when restricted to that space).Recall that a conjugation C on a complex Hilbert space H is an antilinearisometric involution, i.e., C = I H and h Cf, Cg i = h g, f i for all f, g ∈ H . The study of conjugations, which generalize complex conjugation, is moti-vated by applications in physics, in connection with the study of complexsymmetric operators [20, 21, 22]. These are the operators A ∈ L ( H ) suchthat CAC = A ∗ for a conjugation C on H . We can define a natural conju-gation C θ on any model space K θ by C θ f = θzf , (2.5)and it is known that any bounded truncated Toeplitz operator is C θ -symmetric[18, Chap. 8].Regarding M , we have the following result. Proposition 2.3.
The antilinear operator C M defined by C M f = θφψzf = φψC θ f (2.6) is a conjugation on L preserving M as an invariant subspace.Proof. If f ∈ M has the form f = φk + ψk with k , k ∈ K θ , then C M f = ψC θ k + φC θ k ∈ M, (2.7)from which the conjugation properties are easily verified. Theorem 2.4. If A Mg is bounded, then it is C M -symmetric. roof. We wish to check the identity A Mg C M = C M ( A Mg ) ∗ . Note that byProposition 2.3, we have U C M U ∗ = (cid:18) C θ C θ (cid:19) , where U is the unitary mapping given in (2.3). Hence U A Mg C M U ∗ = A θg A θφψg A θψφg A θg ! (cid:18) C θ C θ (cid:19) = A θφψg C θ A θg C θ A θg C θ A θψφg C θ ! while U C M ( A Mg ) ∗ U ∗ = (cid:18) C θ C θ (cid:19) A θg A θψφg A θφψg A θg ! = C θ A θφψg C θ A θg C θ A θg C θ A θψφg ! , and these are equal since AC θ = C θ A ∗ for any standard truncated Toeplitzoperator A . Definition 3.1. [3, 24, 32, 33] The operators T : X → e X and S : Y → e Y are said to be (algebraically and topologically) equivalent if and only if T = ESF where
E, F are invertible operators. More generally, T and S are equivalent after extension if and only if there exist (possibly trivial)Banach spaces X , Y , called extension spaces , and invertible bounded linearoperators E : e Y ⊕ Y → e X ⊕ X and F : X ⊕ X → Y ⊕ Y , such that (cid:18) T I X (cid:19) = E (cid:18) S I Y (cid:19) F. (3.1) In this case we say that T ∗ ∼ S . It is shown in [14] that for g ∈ L ∞ the scalar Toeplitz operator A θg isequivalent by extension to the block Toeplitz operator with symbol (cid:18) θ g θ (cid:19) . This result was used in [13] to study spectral properties of A θg and, moregenerally, asymmetric truncated Toeplitz operators.6otivated by the result of Theorem 2.2, we now consider the truncatedToeplitz operator A θG acting on K θ ⊕ K θ , where G = (cid:18) g g g g (cid:19) and linkit with the Toeplitz operator T G acting on ( H ) , where G = θ θ g g θ g g θ . (3.2)Clearly, for p, q, r, s ∈ H , we have ( p, q, r, s ) ∈ ker T G if and only if p, q ∈ K θ and (cid:18) g g g g (cid:19) (cid:18) pq (cid:19) + θ (cid:18) rs (cid:19) ∈ H ⊕ H . So ( p, q ) ∈ ker A θG , and likewisegiven ( p, q ) ∈ ker A θG there exist r, s ∈ H with ( p, q, r, s ) ∈ ker T G .The following theorem shows that the result in [13, Thm. 2.3] can beextended to block truncated Toeplitz operators and in fact we can give theresult more generally for n × n blocks. We shall write P θ for the orthogonalprojection from ( H ) n onto ( K θ ) n , and Q θ for the complementary projectionfrom ( H ) n onto θ ( H ) n . Theorem 3.2.
Let G ∈ ( L ∞ ) n × n and let I n be the n × n identity matrix.The operator A θG = P θ GP θ : K nθ → K nθ is equivalent after extension to T G : ( H ) n → ( H ) n with G = (cid:18) θI n G θI n (cid:19) . (3.3) Proof.
We have, following the proof of [13, Thm. 2.3], A θG ∗ ∼ P θ GP θ + Q θ because (cid:18) A θG I θ ( H ) n (cid:19) = E (cid:18) P θ GP θ + Q θ I { } n (cid:19) F , where F : K nθ ⊕ θ ( H ) n → ( H ) n ⊕{ } n and E : ( H ) n ⊕{ } n → K nθ ⊕ θ ( H ) n are invertible operators, defined in the obvious way. On the other hand, itis clear that, denoting by P + the orthogonal projection from L onto H , P θ GP θ + Q θ ∗ ∼ (cid:18) P θ GP θ + Q θ P + (cid:19) : ( H ) n → ( H ) n . (cid:18) P θ GP θ + Q θ P + (cid:19) = (cid:18) T θI n − P θ GT θI n P θ − P + T θI n (cid:19)| {z } E T G (cid:18) P + T θI n ( P + − T G ) P + (cid:19)| {z } F , where E, F : ( H ) n → ( H ) n are invertible operators with E − = (cid:18) T θI n P + + P θ GQ θ T θI n (cid:19) and F − = (cid:18) P + − T θI n ( P + − T G ) P + (cid:19) . Corollary 3.3.
For g ∈ L ∞ , one has A Mg ∗ ∼ T G with G = θ θ g gφψ θ gφψ g θ . (3.4) Proof.
This is an immediate consequence of Theorems 2.2 and 3.2.We clearly have the following corollary of the above.
Corollary 3.4.
The operators A Mg and W are invertible (resp., Fredholm)if and only if T G is invertible (resp., Fredholm), with G given by (3.4) . More general results will be proved later.
The equivalence after extension proved in Theorem 3.2 implies certain re-lations between the kernels, the ranges, and the invertibility and Fredholmproperties of the two operators A Mg and T G with G given by (3.4) ([3, 32]),and therefore it implies certain relations between the solutions of A Mg f M = h M for a given h M ∈ M (4.1)and those of T G φ + = H + for a given H + ∈ ( H ) . (4.2)8n this section we study these relations, which also allow for a better under-standing of the equivalence after extension obtained in the previous section.First note that the equation (4.2) is equivalent to the Riemann–Hilbertproblem G φ + = φ − + H + , φ ± ∈ ( H ± ) , (4.3)where H +2 = H and H − = H .We start by establishing the equivalence between (4.1) and anotherRiemann–Hilbert problem. Let f θ = P θ φf M , h θ = P θ φh M ,f θ = P θ ψf M , h θ = P θ ψh M , (4.4)so that f M = φf θ + ψf θ and h M = φh θ + ψh θ . Given h M ∈ M wecan write, by (2.1), A Mg f M = h M if and only if P M ( g ( φf θ + ψf θ )) = φh θ + ψh θ , or equivalently, φP θ φ ( gφf + gψf ) + ψP θ ψ ( gφf + gψf ) = φh θ + ψh θ , with f j + = f jθ ∈ K θ . Since φK θ ⊥ ψK θ , this is equivalent to θf = f − ∈ H − , P θ ( gf + gφψf ) = h θ ,θf = f − ∈ H − , P θ ( gφψf + gf ) = h θ , (4.5)which, in its turn, is equivalent to θf = f − , gf + gφψf = h θ + f − − θf ,θf = f − , gφψf + gf = h θ + f − − θf , (4.6)with f j ± ∈ H ± . Equivalently, G F + = F − + H + with F ± = ( f j ± ) ∈ ( H ± ) , H + = (0 , , h θ , h θ ) , (4.7)where G is given by (3.4) and f = f θ = P θ φf M , f = − P + θg ( P θ φf M + φψP θ ψf M ) ,f = f θ = P θ ψf M , f = − P + θg ( φψP θ φf M + P θ ψf M ) . (4.8)We conclude the following: Theorem 4.1. A Mg f M = h M with f M , h M ∈ M if and only if T G F + = H + with F + , H + given by (4.7) and (4.8) . Consequently, if f M ∈ ker A Mg , then F + ∈ ker T G . Theorem 4.2. T G F + = H + with F + = ( ˜ f j + ) , H + = ( h j + ) ∈ ( H ) , if andonly if A Mg f M = h M , where f M = φ ( ˜ f − θh ) + ψ ( ˜ f − θh ) (4.9) and h M = φP θ ( h + − gθ ( h + φh + )) + ψP θ ( h − gθ ( φψh + h )) . (4.10) Proof.
We have T G F + = H + if and only if G F + = F − + H + , with F − ∈ ( H − ) . Now for F + = ( ˜ f j + ) and H + = ( h j + ) ( j = 1 , , ,
4) we have GF + = F − + H + if and only if θ ˜ f = f − + h ,θ ˜ f = f − + h ,g ˜ f + φψg ˜ f + θ ˜ f = f − + h ,gφψ ˜ f + g ˜ f + θ ˜ f = f − + h . (4.11)This in turn is equivalent to the system of equations θ ( ˜ f − θh ) = f − ,θ ( ˜ f − θh ) = f − ,g ( ˜ f − θh ) + φψg ( ˜ f − θh ) + θ ( ˜ f + P + g ( h + φψh ) − P + θh = f − + ( h − θP + θh ) − P θ gθ ( h + φψh ) ,φψg ( ˜ f − θh ) + g ( ˜ f − θh ) + θ ( ˜ f + P + g ( φψh + h ) − P + θh = f − + ( h − θP + θh ) − P θ gθ ( φψh + h ) . (4.12) Taking f = ˜ f − θh , f = ˜ f − θh ,f = ˜ f + P + g ( h + φψh ) − P + θh and f = ˜ f + P + g ( φψh + h ) − P + θh , this gives θf = f − ,θf = f − ,gf + φψgf + θf = f − + h θ ,gφψf + gf + θf = f − + h θ , (4.13)10ith h θ = h − θP + θh | {z } P θ h − P θ gθ ( h + φψh )= P θ ( h − gθ ( h + gφψh )) ,h θ = h − θP + θh − P θ gθ ( φψh + h )= P θ ( h − gθ ( φψh + h )) . By Theorem 4.1 this is equivalent to A Mg f M = h M with f M and h M givenby (4.9) and (4.10). Corollary 4.3. If F + ∈ ker T G , with F + = ( ˜ f j + ) ∈ ( H ) , then f M ∈ ker A Mg with f M = φ ˜ f + ψ ˜ f . Note that, from (4.11), any element of the kernel of T G is determined byits first two components ˜ f and ˜ f , since˜ f = P + θg ( ˜ f + φψ ˜ f )and ˜ f = P + θg ( φψ ˜ f + ˜ f ) . Corollary 4.4.
The map K : ker A Mg → ker T G , K f M = ( f , f , f , f ) with f j + given by (4.8) is an isomorphism. We have ker A Mg = K − ker T G = { φf + ψf : ( f , f , f , f ) ∈ ker T G for some f , f ∈ H } . From Theorems 4.1 and 4.2 we also obtain the following regarding ranges.
Corollary 4.5.
With the same notation as above,(i) ( h , h , h , h ) ∈ ran T G if and only if φP θ ( h − gθ ( h + φψh )) + ψP θ ( h − gθ ( φψh + h )) ∈ ran A Mg ; or equivalently (0 , , P θ ( h − gθ ( h + φψh )) , P θ ( h − gθ ( φψh + h ))) ∈ ran T G . ii) φh θ + ψh θ ∈ ran A Mg if and only if (0 , , h θ , h θ ) ∈ ran T G , or equiva-lently if the set of all ( h , h , h , h ) such that P θ ( h − gθ ( h + φψh )) = h θ and P θ ( h − gθ ( φψh + h )) = h θ is contained in ran T G . Moreover, we obtain a relation between the inverses of A Mg and T G whenthey are invertible. Corollary 4.6. A Mg is invertible if and only if T G is invertible, and in thatcase ( A Mg ) − h M = φP T − G (0 , , P θ φh M , P θ ψh M )+ ψP T − G (0 , , P θ φh M , P θ ψh M ) , i.e., ( A Mg ) − = [ φP , ψP , , T − G U , where P j ( x , x , x , x ) = x j and U : M → ( H +2 ) , is given by U h M = (0 , , P θ φh M , P θ ψh M ) . We can also relate the kernels of A Mg and its adjoint as follows. Theorem 4.7. If g ∈ L ∞ , then ker A Mg ≃ ker( A Mg ) ∗ = ker A Mg .Proof. Since A Mg ∗ ∼ T G , we have that ker A Mg ≃ ker T G and ker( A Mg ) ∗ ≃ ker T ∗G . So it is enough to prove that ker T G , with G given by (3.4), isisomorphic to ker T ∗G = ker T G T . Sinceker T G = { φ + ∈ ( H ) : G φ + = φ − ∈ ( H ) } , we have G φ + = φ − ⇐⇒ zφ + = G − ( zφ − ) ⇐⇒ G − ψ + = ψ − , where ψ + = zφ − ∈ ( H ) and ψ − = zφ + ∈ ( H ) . Since G − = θ θ − g − gφψ θ − gφψ − g θ = − − θ g gφψ θ gφψ g θ
00 0 0 θ | {z } G T −
10 0 − , it is clear that ker T G − ≃ ker T G T . 12 orollary 4.8. If A Mg is Fredholm, then it has index . Consequently, A Mg is invertible if and only if it is Fredholm and injective. Clearly the norm of the multiband TTO A Mg is the same as the norm ofthe block TTO W . One important case that can be analysed is when g isin H ∞ (in the language of multiband signals, this corresponds to a causalconvolution on L (( − b, − a ) ∪ ( a, b ))).The following is an easy generalization of scalar results which apparentlygo back to [30]. Proposition 5.1.
Suppose that the symbol
Φ := (cid:18) g φψgψφg g (cid:19) is in ( H ∞ ) × . Then k A Mg k = k W k = dist( θ Φ , ( H ∞ ) × = k Γ θ Φ k , where the vectorial Hankel operator Γ θ Φ : ( H ) → ( L ⊖ H ) is defined by Γ θ Φ v = P ( L ⊖ H ) θ Φ v .Proof. Since the symbol Φ is analytic, if we write ( u , u ) ∈ H ⊕ H as( k + θℓ , k + θℓ ) with the k j in K θ and ℓ j ∈ H , then we have W ( u , u ) = W ( k , k ) implying that the norm of the truncated Toeplitz operator W isthe same when the domain is H ⊕ H or K θ ⊕ K θ .But W u = θ ( P − ⊕ P − ) θ Φ u = θ Γ θ Φ u , and so k W k = k Γ θ Φ k = dist( θ Φ , H ∞ ( M ( C )) , by the vectorial form of Nehari’s theorem [29, Sec. 2.2].Several results on the spectrum of W can be derived using known resultson the scalar case, particularly in the context of Proposition 2.1. Note thatthe hypotheses of this theorem are satisfied in the original example given by(1.1) and (1.2). 13 heorem 5.2. Suppose that g ∈ H ∞ and that φψ ∈ θH ∞ . Then σ ( A Mg ) = σ ( A θg ) = { λ ∈ C : inf z ( | θ ( z ) | + | g ( z ) − λ | ) = 0 } , where the infimum is taken over D .Proof. Note that W has the form W = A θg A θψφg A θg ! and we claim that W is invertible if and only if A θg is. For the necessity notethat for arbitrary block operator matrices, if we have (cid:18) A B A (cid:19) (cid:18)
P QR S (cid:19) = (cid:18) P QR S (cid:19) (cid:18) A B A (cid:19) = (cid:18) I I (cid:19) , then AP = I and SA = I , so S = SAP = P , and A is invertible withinverse P .The sufficiency follows from the formula W − = ( A θg ) − − ( A θg ) − A θψφg ( A θg ) − ( A θg ) − ! . The spectrum of A θg for g ∈ H ∞ is described in [28, p. 66], and the H ∞ ( C + )case may be found in [13].For the essential spectrum of A θg we may similarly prove the followingresult. Theorem 5.3.
Suppose that g ∈ H ∞ and that φψ ∈ θH ∞ . Then σ e ( A Mg ) = σ e ( A θg ) = { λ ∈ C : lim inf z → ξ ( | θ ( z ) | + | g ( z ) − λ | ) = 0 for some ξ ∈ T } , where z is taken in D .Proof. The method of proof of Theorem 5.2 adapted to inversion modulo thecompact operators (i.e., in the Calkin algebra) shows directly that σ e ( A Mg ) = σ e ( A θg ), and an expression for this is known from results in [5, 13].These results are of particular interest in the case of the restricted shift or truncated shift S M on M , with g ( z ) = z . We thus have14 orollary 5.4. If φψ ∈ θH ∞ , then for the restricted shift S M on M wehave σ ( S M ) = { λ ∈ C : inf z ∈ D ( | θ ( z ) | + | z − λ | ) = 0 } , and σ e ( S M ) = { λ ∈ C : lim inf z → ξ ( | θ ( z ) | + | z − λ | ) = 0 for some ξ ∈ T } = { λ ∈ T : lim inf z → λ ( | θ ( z ) | ) = 0 } . We now use the results of the previous section to study the spectrum of theoperator A Mz and various related properties.Using the result of Corollary 3.3 we associate to the operator A Mz − λ , with λ ∈ C , the matrix symbol G λ = θ θ z − λ ( z − λ ) A + θ θ z − λ ) A − θ z − λ θ , (6.1)where we suppose as in (2.4) that ψφ = A − θ + A + θ with A + ∈ H ∞ and A − ∈ H ∞ . We start by studying the Fredholmness of T G λ , and first weconsider λ T . Let R denote the space of rational functions without poleson T . Then we have: Proposition 6.1.
For any λ ∈ C \ T , G λ admits a meromorphic factorization G λ = M − M − with M ± − ∈ ( H ∞ + R ) × and M ± ∈ ( H ∞ + R ) × , namely M + = θ
00 1 0 θ − ( z − λ ) 00 0 0 − ( z − λ ) ,M − = θ θ z − λ ( z − λ ) A + θ A + ( z − λ )( z − λ ) A − θ z − λ A − ( z − λ ) 0 . (6.2)15 roof. The equality G λ M + = M − , which was obtained by using complexanalytic methods for solving matricial Riemann–Hilbert problems (such asdescribed, for example, in [8, 15]) can be verified directly. Corollary 6.2.
The operator T G λ is Fredholm for all λ ∈ C \ T and, conse-quently, σ e ( A Mz ) ⊂ T .Proof. It follows from the existence of a meromorphic factorization for G λ ,as in Proposition 6.1, that T G λ is Fredholm [9, 11] and, by equivalence afterextension, the same holds for A Mz − λ for all λ ∈ C \ T .To study A Mz − λ with λ ∈ T we shall use the following auxiliary result. Lemma 6.3. If φ + ∈ H then φ + ( z )( z − λ ) → when z → λ ∈ T nontan-gentially in D .Proof. For φ + ∈ H we have | φ + ( w ) | ≤ k φ + k k k w k , where k w is the re-producing kernel function k w ( z ) = 1 / (1 − wz ), with k k w k = 1 / p − | w | .Hence | φ ( z )( z − λ ) | ≤ | z − λ | k φ + k p − | z | , and this tends to 0 if z tends nontangentially to λ ∈ T , as this means that | z − λ | ≤ C (1 − | z | ) for some constant C .We also use the following. We say that an inner function θ has anangular derivative in the sense of Carath´eodory (ADC) if and only if θ hasa nontangential limit θ ( λ ) = lim z → λ n.t. θ ( z ) (6.3)with | θ ( λ ) | = 1 and the difference quotient θ ( z ) − θ ( λ ) z − λ has a nontangentiallimit θ ′ ( λ ) at λ [18, 31]. By Theorem 7.4.1 in [18], θ has an ADC at λ ∈ T if and only if there exists a ∈ T such that θ ( z ) − az − λ ∈ H , which implies, byLemma 6.3, that there exists the limit (6.3) and we have θ ( λ ) = a . Thus wehave: Lemma 6.4. θ has an ADC at λ ∈ T if and only if:(i) lim z → λ n.t. θ ( z ) exists in C (denoted by θ ( λ ) )and(ii) θ ( z ) − θ ( λ ) z − λ ∈ H .
16e denote by T ADC the set of all λ ∈ T where θ has an ADC.The next theorem relates the boundary behaviour of θ with the Fred-holmness of A Mz − λ for λ ∈ T . Theorem 6.5. If A Mz − λ is Fredholm, then θ has an ADC at λ .Proof. (i) If T G λ is Fredholm, then G λ admits a Wiener–Hopf factorizationrelative to L , G λ = G λ − D G − λ + , (6.4)with D = diag( z k j ) j =1 , , , , G ± λ + ∈ ( H ) × , G ± λ − ∈ ( H ) × , where theintegers k j are called the partial indices of G λ . The sum of all partial indicesis the negative of the Fredholm index [27]. Since T G λ must have index 0, thesum of all positive partial indices and the sum of all negative partial indicesmust have the same absolute values.If there exists a negative exponent k j = − n <
0, then there is a non-zerosolution to G λ φ + = z n φ − , ( φ + ∈ ( H ) , φ − ∈ ( H ) ) , (6.5)given by the j th columns of G λ + and G λ − , which means that θφ = z n φ − , (6.6) θφ = z n φ − , (6.7)( z − λ ) φ + θφ = φ − z n − ( z − λ ) A + ¯ z n φ − z }| { θφ := k (6.8)( z − λ ) φ + θφ = φ − z n − ( z − λ ) A − θφ | {z } ¯ z n φ − := k (6.9)with k , k ∈ C . From (6.8), we have θφ ( z − λ ) − θk = − φ = C ∈ C , (6.10)and therefore, if C = 0, θ = ( z − λ ) φ − k C . (6.11)So, by Lemma 6.3, lim z → λ n.t. θ ( z ) exists and is finite unless C = 0. Inthat case, from (6.10) we get ( z − λ ) φ = k so k = 0, and C = 0, k = 0implies that φ = φ = 0, from (6.10). Analogously, we conclude that ifthe nontangential limit does not exist then φ = φ = 0, and since by17ssumption φ + = 0, there can be no negative (nor positive) partial index,i.e., all partial indices would have to be 0. In that case the factorization iscanonical and we must have 4 linearly independent solutions to G λ φ + = φ − , ( φ + ∈ ( H ) , φ − ∈ ( H ) ) , (6.12)given by the columns of the factors. Therefore, θφ = φ − , (6.13) θφ = φ − , (6.14)( z − λ ) φ + θφ = φ − − ( z − λ ) A + φ − z }| { θφ = k ( z − λ ) + k , (6.15)( z − λ ) φ + θφ = φ − − ( z − λ ) A − θφ | {z } φ − = k ( z − λ ) + k , (6.16)where k , . . . , k ∈ C . From (6.15),( z − λ ) θφ − θk ( z − λ ) − θk = − φ = C ( z − λ ) + C (6.17)with C , C ∈ C , and so, if C = 0, θ = ( φ − k )( z − λ ) − k C ( z − λ ) + C , (6.18)which has a finite nontangencial limit when z tends to λ . Thus, if thenontangential limit lim z → λ n.t. θ ( z ) does not exist, we must have C = 0and it then follows from (6.17) that k = 0 and φ = − C ( z − λ ) , φ = θC + k , φ − = C + k θ. (6.19)Analogously,( z − λ ) θφ − θk ( z − λ ) − θk = − φ = C ( z − λ ) + C , (6.20)with C , C ∈ C , hence θ = ( φ − k )( z − λ ) − k C ( z − λ ) + C . (6.21)If lim z → λ n.t. θ ( z ) does not exist, we conclude that C = k = 0 and φ = − C ( z − λ ) , φ = θC + k , φ − = C + k θ. (6.22)18ith C , k ∈ C . Therefore, φ − = ( z − λ )( k + A + φ − ) = ( z − λ )( k + A + ( C + θk )) , (6.23) φ − = ( z − λ )( k + A − φ − ) = ( z − λ )( k + A − ( C + θk )) , (6.24)and we must have, since φ − and φ − are in H , that k + A + (0)( C + θ (0) k ) = 0 ,k + A − (0)( C + θ (0) k ) = 0 , i.e., k + A + (0) θ (0) k + A + (0) C = 0 ,A − (0) θ (0) k + k + A − (0) C = 0 . (6.25)If ∆ := 1 − A + (0) A − (0) θ (0) = 0, then k and k are determined by C and C from (6.25); if ∆ = 0, then A + (0), A − (0) and θ (0) are all nonzeroand k and C are determined from C and k , also by (6.25).Therefore there cannot be 4 linearly independent solutions to (6.12) when θ ( λ ) does not exist.(ii) Assume now that G λ has a Wiener–Hopf factorization (6.4), so that θ ( λ ) exists, and let k j = − n <
0. From (6.10) and (6.17) we get φ = θ ( C ( z − λ ) + C ) + k ( z − λ ) + k z − λ , (6.26)where C = k = 0 if n >
0, as in the proof of (i); then φ − θC − k = θC + k z − λ ∈ H , (6.27)implying that θ ( λ ) C = − k and, from (6.26), C θ − θ ( λ ) z − λ = φ − θC − k . Thus θ − θ ( λ ) z − λ H only if C = 0, which implies that k = 0 by (6.27);analogously, considering φ , we conclude that θ − θ ( λ ) z − λ H only if C = k = 0. In that case, if n >
0, we have C = k = C = k = C =19 = C = k = 0 and we conclude that if any partial index is negative, orpositive, then we must have θ − θ ( λ ) z − λ ∈ H .Assume now that all the partial indices are zero and the factorization(6.4) is canonical. We have C = k = C = k = 0 and, as in (6.19), φ = θC + k , φ = − C ( z − λ ); (6.28)analogously, from (6.22), φ = θC + k , φ = − C ( z − λ ) (6.29)and φ − = C + k θ, φ − = C + θk , (6.30) φ − = ( z − λ )( k + A + φ − ) = ( z − λ )( k + A + C + A + θk ) , (6.31) φ − = ( z − λ )( k + A − φ − ) = ( z − λ )( k + A − C + A − θk ) . (6.32)So we must impose the conditions k + A + (0) C + A + (0) θ (0) k = 0 ,k + A − (0) C + A − (0) θ (0) k = 0 , (6.33)and we conclude as before that this determines two of the unknown constantsin terms of the other two, and so we cannot have four linearly independentsolutions to (6.12) if θ − θ ( λ ) z − λ H . Corollary 6.6. T \ T ADC ⊂ σ e ( A Mz ) ⊂ T . A more precise definition of σ e ( A Mz ) will be given later in this section,after studying the kernel of T G λ and, consequently, the kernel of A Mz − λ . Re-call that, according to Corollary 4.4, we have ker A Mz − λ ≃ ker T G λ and, byCorollary 4.3, f M ∈ ker A Mz − λ if and only if f M = φf + ψf (6.34)where f and f are the first two components of an element of ker T G λ ⊂ ( H ) .We will consider three cases: λ ∈ D , λ ∈ D − = { z ∈ C : | z | > } and λ ∈ T . 20 roposition 6.7. If λ ∈ D , then ker A Mz − λ = { } if and only if ∆ λ := θ ( λ ) − A + (0) A − (0)(1 − θ (0) θ ( λ )) = 0; (6.35) if ∆ λ = 0 , we have dim ker A Mz − λ = 1 if at least one of the conditions θ ( λ ) =0 , A + (0) = 0 or A − (0) = 0 is satisfied; dim ker A Mz − λ = 2 if θ ( λ ) = A + (0) = A − (0) = 0 .Proof. We study ker T G λ by solving the Riemann–Hilbert problem G λ φ + = φ − with φ + ∈ ( H ) , φ − ∈ ( H ) . (6.36)Using the same reasoning as in the proof of Theorem 6.5, we get φ = C θ − θ ( λ ) z − λ , φ − = C − θθ ( λ ) z − λ , (6.37) φ = C θ − θ ( λ ) z − λ , φ − = C − θθ ( λ ) z − λ , (6.38) φ = − C , φ = − C , (6.39) φ − = − θ ( λ ) C + A + (1 − θθ ( λ )) C , (6.40) φ − = − θ ( λ ) C + A − (1 − θθ ( λ )) C , (6.41)and it follows from the last two equations that, since φ − , φ − ∈ H , wemust have − θ ( λ ) C + A + (0)(1 − θ (0) θ ( λ )) C = 0 , − θ ( λ ) C + A − (0)(1 − θ (0) θ ( λ )) C = 0 . (6.42)This system has a nontrivial solution if and only if∆ λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − θ ( λ ) A + (0)(1 − θ (0) θ ( λ )) A − (0) − (1 − θ (0) θ ( λ )) − θ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = θ ( λ ) − A + (0) A − (0)(1 − θ (0) θ ( λ )) = 0 . (6.43)In this case, if θ ( λ ) = 0 we must have also A + (0) , A − (0) = 0 and the system(6.42) is equivalent to C = A + (0) 1 − θ (0) θ ( λ ) θ ( λ ) C ,
21o dim ker T G λ = 1 with a nontrivial solution to G λ φ + = φ − with φ + ∈ ( H ) and φ − ∈ ( H ) (so that φ + ∈ ker T G λ ) given by (6.37) - (6.41) with C = 1and C = A + (0) − θ (0) θ ( λ ) θ ( λ ) .If θ ( λ ) = 0, then A + (0) A − (0) = 0 and we have that:(i) if A + (0) = 0 and A − (0) = 0, then C = 0 and dim ker T G λ = 1, with anontrivial solution to G λ φ + = φ − given by φ = 0 φ − = 0 (6.44) φ = θ − θ ( λ ) z − λ φ − = 1 − θθ ( λ ) z − λ (6.45) φ = 0 φ − = A + (1 − θθ ( λ )) (6.46) φ = − φ − = 0 ; (6.47)(6.48)(ii) if A − (0) = 0 and A + (0) = 0, then C = 0 and dim ker T G λ = 1, with anontrivial solution to G λ φ + = φ − given by φ = θ − θ ( λ ) z − λ φ − = 1 − θθ ( λ ) z − λ (6.49) φ = 0 φ − = 0 (6.50) φ = − φ − = 0 (6.51) φ = 0 φ − = A − (1 − θθ ( λ )) ; (6.52)(6.53)(iii) if A + (0) = A − (0) = 0 then dim ker T G λ = 2 and the nontrivial solutionsto G λ φ + = φ − are given by φ + = C θz − λ − + C θz − λ − (6.54)and φ − = C z − λ A − + C z − λ A + (6.55)where C , C ∈ C .The corresponding results for A Mz − λ follow by equivalence after extension.22 roposition 6.8. If λ ∈ D − , then ker A Mz − λ = { } if and only if f ∆ λ := 1 − A + (0) A − (0)( θ (0) − θ ( λ )) = 0 . If f ∆ λ = 0 then dim ker A Mz − λ = 1 .Proof. For any solution to G λ φ + = φ − we must have( z − λ ) φ + θφ = φ − − ( z − λ ) A + φ − = k , k ∈ C (6.56)( z − λ ) φ + θφ = φ − − ( z − λ ) A − φ − = k , k ∈ C . (6.57)Since θφ = φ − , it follows from (6.56) that( z − λ ) φ − − θk = − φ = C , C ∈ C (6.58)so that ( z − λ ) φ − = θC + k . (6.59)Analogously, since θφ = φ − , it follows from (6.57) that( z − λ ) φ − − θk = − φ = C , C ∈ C (6.60)so that ( z − λ ) φ − = θC + k . (6.61)From (6.59) and (6.61) we get, with θ ( λ ) = θ (1 /λ ), C = − θ ( λ ) k (6.62) C = − θ ( λ ) k (6.63)so φ − = k θ − θ ( λ ) z − λ φ = k − θ ( λ ) θz − λ (6.64) φ − = k θ − θ ( λ ) z − λ φ = k − θ ( λ ) θz − λ (6.65) φ − = k + A + k ( θ − θ ( λ )) φ = θ ( λ ) k (6.66) φ − = k + A − k ( θ − θ ( λ )) φ = θ ( λ ) k . (6.67)Since φ − , φ − ∈ H , we must have (from (6.66) and (6.67)) k + A + (0) k ( θ (0) − θ ( λ )) = 0 A − (0)( θ (0) − θ ( λ )) k + k = 0 (6.68)23nd the system (6.68) admits a nontrivial solution, i.e. ker T G λ = { } , if andonly if f ∆ λ = 1 − A + (0) A − (0)( θ (0) − θ ( λ )) = 0 . In this case we must have A + (0) = 0 , A − (0) = 0 , θ (1 /λ ) = θ (0) and thesystem (6.68) is equivalent to k = − A + (0)( θ (0) − θ ( λ )) k . (6.69)So if f ∆ λ = 0, ker T G λ has dimension 1 and is spanned by φ + = ( φ j + ) j =1 , , , with φ = − A + (0)( θ (0) − θ ( λ )) 1 − θ ( λ ) θz − λ (6.70) φ = 1 − θ ( λ ) θz − λ (6.71) φ = − θ ( λ )) A + (0)( θ (0) − θ ( λ )) (6.72) φ = θ ( λ ) . (6.73)The corresponding results for A Mz − λ follow by equivalence after extension. Proposition 6.9. If λ ∈ T then ker A Mz − λ = { } if and only if λ ∈ T \ T ADC or λ ∈ T ADC and ∆ λ = 0 . If λ ∈ T ADC and ∆ λ = 0 then dim ker A Mz − λ = 1 .Proof. Following the same reasoning as the proof of proposition 6.8 we getthat (6.59) and (6.61) must hold which, taking Lemma 6.3 into account,implies thatlim z → λ n.t. θ ( z ) C = − k lim z → λ n.t. θ ( z ) C = − k . (6.74)If lim z → λ n.t. does not exist, then we must have k = k = C = C = 0and it follows that ker T G λ = { } . If that limit exists in C and we denote itby θ ( λ ), then from (6.59) and (6.61), k = − θ ( λ ) C k = − θ ( λ ) C (6.75) φ = C θ − θ ( λ ) z − λ φ = C θ − θ ( λ ) z − λ (6.76)and we conclude that, if θ − θ ( λ ) z − λ / ∈ H then C = C = 0 + k = k andker T G λ = { } . Thus for ker T G λ = { } we must have λ ∈ T ADC and, in thiscase, following the same reasoning as in the previous two propositions, wesee that ker T G λ = { } if and only if ∆ λ = 0, which is equivalent to f ∆ λ = 0for λ ∈ T ADC since θ ( λ ) = θ ( λ ), and dim ker T G λ = 1.24rom Propositions 6.7–6.9 we conclude therefore; Theorem 6.10. λ ∈ σ p ( A Mz ) ⇐⇒ ( λ ∈ D , ∆ λ = 0) ∨ ( λ ∈ D − , f ∆ λ = 0) ∨ ( λ ∈ T ADC , ∆ λ = 0) , where ∆ λ is defined as in Proposition 6.7. Note that, from the proofs of the previous propositions, and taking (6.34)into account, we also get a description of the eigenspaces ker A Mz − λ . Corollary 6.11. If A + (0) A − (0) = 0 then λ ∈ σ p ( A Mz ) ⇐⇒ λ ∈ D , θ ( λ ) = 0 . Proof.
Let A + (0) A − (0) = 0. If λ ∈ D − then f ∆ λ = 1 = 0 and, if λ ∈ T ADC ,then ∆ λ = θ ( λ ) = 0; so all points of σ p ( A Mz ) must be in the disk D and∆ λ = 0 if and only if θ ( λ ) = 0.Note that the case considered in Theorem 5.2 satisfies the assumptionsof Corollary 6.11, and σ p ( A Mz ) = σ p ( A θz ). Corollary 6.12. If A + (0) A − (0) = 0 then λ ∈ σ p ( A Mz ) ∩ D = ⇒ θ ( λ ) = 0 λ ∈ σ p ( A Mz ) ∩ D − = ⇒ θ (1 /λ ) = θ (0) Remark 6.13.
In particular, we conclude from these corollaries that, if A + (0) A − (0) = 0 , the elements of σ p ( A Mz ) are the zeros of θ in D , while, if A + (0) A − (0) = 0 , the zeros of θ in D cannot be in σ p ( A Mz ) . Remark 6.14.
Since, by Theorem 6.5 and Corollary 6.6, A Mz − λ is notFredholm for λ ∈ T \ T ADC and, from Proposition 6.9 and Theorem 4.7, ker A Mz − λ = ker( A Mz − λ ) ∗ = { } , we conclude that Im A Mz − λ is not closed. Now we return to the question of whether A Mz − λ is Fredholm for λ ∈ T ADC . Fredholmness of A Mz − λ is equivalent to Fredholmness of T G λ in ( H ) ,since the two operators are equivalent after extension. Fredholmness of T G λ ,in its turn, is equivalent to G λ possessing a Wiener–Hopf (or generalized)factorization relative to L [27]. This can take the particular form G λ = G − λ D G + λ (6.77)25ith D = diag( z k j ) j =1 , , , where k j ∈ Z , ( G + λ ) ± ∈ ( H ∞ ) × , ( G − λ ) ± ∈ ( H ∞ ) × , which is called a bounded factorization . The integers k j are calledthe partial indices . They are unique up to permutations and we havedim ker T G λ = X k j < | k j | , dim ker( T G λ ) ∗ = X k j > k j . (6.78)If D is the identity matrix, i.e., k j = 0 for all j = 1 , , , , then the factor-ization takes the form G λ = G − λ G + λ (6.79)which is called a canonical bounded factorization and is unique up to con-stant matrix factors in G ± λ . In this case T G λ is invertible, with inverse( G + λ ) − P + ( G − λ ) − I | ( H ) . Recall that by Corollary 4.6 the invertibility of T G λ is equivalent to theinvertibility of A Mz − λ .To study the Fredholmness and the invertibility of A Mz − λ for λ ∈ T ADC wewill use the relations between Wiener–Hopf factorization and Fredholmnessor invertibility, by explicitly constructing a Wiener–Hopf factorization of G λ .These factorizations, which were obtained by solving a Riemann–Hilbertproblem of the form G λ φ + = φ − , with φ − ∈ ( H ) and φ − ∈ ( H ) , for eachcolumn of the factors, using the same reasoning as before, can be checkeddirectly by multiplication of the matrix factors. Note that they are validalso for λ ∈ D .Let us first assume that ker A Mz − λ = { } , i.e., λ / ∈ σ p ( A Mz ). Proposition 6.15.
Let λ ∈ T ADC and let ∆ λ be defined by (6.35) , with ∆ λ = 0 . Then G λ admits a canonical factorization with respect to L , of theform G λ = G − λ G + λ with ( G + λ ) ± ∈ ( H ) × , ( G − λ ) ± ∈ ( H ) × as follows:(i) If ∆ = 1 − A + (0) A − (0) θ (0) = 0 , ( G + λ ) − = θ + A + (0) A − (0) θ (0)∆ θ − θ ( λ ) z − λ − A + (0)∆ − A − (0)∆ θ + A + (0) A − (0) θ (0)∆ θ − θ ( λ ) z − λ − ( z − λ ) − − ( z − λ ) − , nd G − λ = A + (0) A − (0) θ (0)∆ θ − θ ( λ ) θz − λ − A + (0)∆ θ − A − (0)∆ θ A + (0) A − (0) θ (0)∆ θ − θ ( λ ) θz − λ g − − θ ( λ ) g − A + (1 − θ ( λ ) θ ) g − A − (1 − θ ( λ ) θ ) g − − θ ( λ ) , with g − = − A − (0)∆ ( z − λ )( A + θ − A + (0) θ (0)) ,g − = z − λ ∆ ( A − − A − (0) + A + (0) A − (0) θ (0) A − ( θ − θ (0))) g − = z − λ ∆ ( A + − A + (0) + A + (0) A − (0) θ (0) A + ( θ − θ (0))) ,g − = − A + (0)∆ ( z − λ )( A − θ − A − (0) θ (0)) . Note that in this case det( G + λ ) − = det G − λ = − ∆ λ / ∆ ∈ C \ { } .(ii) If ∆ = 0 , in which case A + (0) , A − (0) , θ (0) = 0 and A + (0) θ (0) =1 A − (0) θ (0) , we have ( G + λ ) − = − A + (0)(1 − θθ (0)) θ − θ ( λ ) z − λ − A + (0) θ (0) 0 θ θ − θ ( λ ) z − λ − θ (0) A + (0)( z − λ ) − − ( z − λ ) 0 0 − , and − λ = − A + (0)( θ − θ (0)) − θ ( λ ) θz − λ − A + (0) θ (0) θ
01 0 θ − θ ( λ ) θz − λ ( A + − A + (0))( z − λ ) − θ ( λ ) ( A + θ − A + (0) θ (0))( z − λ ) A + (1 − θ ( λ ) θ ) − A + (0) A − ( θ − θ (0))( z − λ ) A − (1 − θ ( λ ) θ ) 1 − A + (0) θ (0) A − θ ( z − λ ) − θ ( λ ) with det( G + λ ) − = det G − λ = A + (0)(1 − θ (0) θ ( λ )) = − ∆ λ A − (0) ∈ C \ { } . For λ ∈ T ADC we have θ ( λ ) = 0 so, from Proposition 6.7, dim ker A Mz − λ =1. In this case, as we show next, G λ has a non-canonical factorization whosepartial indices in (6.77) must be − , , ,
1, according to (6.78). We havethe following.
Proposition 6.16.
Let λ ∈ T ADC and let ∆ λ be defined by (6.35) , with ∆ λ = 0 . Then G λ admits a non-canonical factorization with respect to L ,of the form G λ = G − λ diag( z, , , z ) G + λ with ( G + λ ) ± ∈ ( H ) × , ( G − λ ) ± ∈ ( H ) × where:(i) if ∆ = 1 − A + (0) A − (0) θ (0) = 0 , ( G + λ ) − = A + (0) − θ (0) θ ( λ ) θ ( λ ) θ − θ ( λ ) z − λ θ − θ ( λ ) z − λ θ + A + (0) A − (0) θ (0)∆ θ − θ ( λ ) z − λ − A − (0)∆ − A + (0) − θ (0) θ ( λ ) θ ( λ ) − − ( z − λ ) 0 − , and G − λ = A + (0) − θ (0) θ ( λ ) θ ( λ ) 1 − θ ( λ ) θz − λ z − θ ( λ ) θz − λ g − zθ − θ ( λ ) θz − λ z − A − (0)∆ θ z (( A + − A + (0)) − θ ( λ )( A + θ − A + (0) θ (0))) − θ ( λ ) g − z − λz g − A − (1 − θ ( λ ) θ ) g − A − θ z − λz , ith g − = 1 + A + (0) A − (0) θ (0)∆ θ,g − = − A − (0)∆ ( z − λ )( A + θ − A + (0) θ (0)) g − = zA + (0) 1 − θ (0) θ ( λ ) θ ( λ ) (( A − − A − (0)) − θ ( λ )( A − θ − A − (0) θ (0))) ,g − = z − λ ∆ ( A − − A − (0) + A + (0) A − (0) θ (0) A − ( θ − θ (0))) (with det( G + λ ) − = det G − λ = − A − (0) / ∆ = 0 ; note that if ∆ λ = 0 we musthave A − (0) = 0 because θ ( λ ) = 0 );(ii) If ∆ = 0 , in which case A + (0) A − (0) θ (0) = 0 and A + (0) θ (0) =1 A − (0) θ (0) , we have ( G + λ ) − = A + (0) − θ (0) θ ( λ ) θ ( λ ) θ − θ ( λ ) z − λ θ − θ ( λ ) z − λ − A + (0)(1 − θ (0) θ ) 1 θ − θ ( λ ) z − λ θ − A + (0) − θ (0) θ ( λ ) θ ( λ ) − − θ (0) A + (0)( z − λ ) 0 − − ( z − λ ) 0 , and G − λ = A + (0) − θ (0) θ ( λ ) θ ( λ ) 1 − θ ( λ ) θz − λ z − θ ( λ ) θz − λ − A + (0) ( θ − θ (0)) θz − θ ( λ ) θz − λ z g − − θ ( λ ) ( A + − A + (0))( z − λ ) z − λz g − A − (1 − θ ( λ ) θ ) g − z − λz A − θ where g − = z (( A + − A + (0)) − θ ( λ )( A + θ − A + (0) θ (0))) ,g − = A − A + (0) 1 − θ (0) θ ( λ ) θ ( λ ) (1 − θ ( λ ) θ ) − θ ( λ ) ,g − = − ( z − λ )( θ − θ (0)) A + (0) A − . L -factorization for G λ , for λ ∈ T ADC , is not, how-ever, sufficient to conclude that T G λ is Fredholm; the condition( G + λ ) − P + ( G − λ ) − : R → ( H ) is boundedmust also be satisfied [27]. This means that, for all λ ∈ T ADC , the dimensionsof ker A Mz − λ and ker( A Mz − λ ) ∗ are finite, by Propositions 6.7–6.9 and Theorem4.7, but the range of A Mz − λ may not be closed. However, it is clear thatthe factorizations in Proposition 6.15 and 6.16 are bounded (whether or notcanonical) if θ − θ ( λ ) z − λ ∈ H ∞ . (6.80)Thus from Proposition 6.1, Theorem 4.7, Propositions 6.7–6.9 and 6.15,6.16, we have the following. Theorem 6.17. (i) The operator A Mz − λ is Fredholm if λ / ∈ T or λ ∈ T ADC is a regular point of θ (where (6.80) holds); it is not Fredholm if λ ∈ T \ T ADC .(ii) The operator A Mz − λ is invertible if λ ∈ D and ∆ λ = 0 , or λ ∈ D − and f ∆ λ = 0 , or λ ∈ T ADC , (6.80) holds and ∆ λ = 0 . For a large class of inner functions θ including θ ( z ) = e − − z z and all innerfunctions that are relevant in applications, we have that (6.80) if and onlyif λ ∈ T ADC . For those inner functions we have the following.
Corollary 6.18. If θ is such that (6.80) holds for all λ ∈ T ADC , then σ e ( A Mz ) = T \ T ADC σ ( A Mz ) = T \ T ADC ∪ { λ ∈ D ∪ T ADC : ∆ λ = 0 } ∪ { λ ∈ D − : f ∆ λ = 0 } = σ e ( A Mz ) ∪ σ p ( A Mz ) . The residual spectrum is empty and the continuous spectrum coincides with σ e ( A Mz ). Remark 6.19.
Corollary 5.4 says that (under extra assumptions on φ and ψ ), σ e ( A Mz ) consists of all the points on the circle in the spectrum σ ( θ ) of θ (i.e., those λ for which lim inf z → λ ( | θ ( z ) | ) = 0 ). At these points we cannothave (6.80) , unless θ ( λ ) = 0 , in which case the inner function θ has noangular derivative at λ ; indeed then | θ ( λ ) | = 1 (cf. [18, Thm. 7.24]).However, if λ ∈ T \ σ ( θ ) then θ has an analytic continuation to a neigh-bourhood of λ and there clearly is an ADC there.Thus, Corollaries 5.4 and 6.18 are giving the same conclusion underdifferent assumptions. emark 6.20. For general inner functions θ , there remains the question ofwhether the range of A Mz − λ is closed if λ ∈ T ADC is a singular point, sincein that case θ − θ ( λ ) z − λ is not bounded [31]. However, even in that case, we candescribe the kernels of A Mz − λ and its adjoint by using Propositions 6.15 and6.16. Remark 6.21.
For λ / ∈ σ ( A Mz ) , the resolvent operator A Mz − λ can be explicitlyobtained by using Corollary 4.6 and the canonical factorization of G λ givenin Proposition 6.15, since T − G λ = ( G + λ ) − P + ( G − λ ) − [27]. Acknowledgements
This work was partially supported by FCT/Portugal throughUID/MAT/04459/2020. The second author is grateful to the EPSRC forfinancial support.
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