aa r X i v : . [ m a t h . F A ] J u l THE FAILURE OF RATIONAL DILATION ON THETETRABLOCK
SOURAV PAL
Abstract.
We show by a counter example that rational dilationfails on the tetrablock, a polynomially convex and non-convex do-main in C defined as E = { ( x , x , x ) ∈ C : 1 − zx − wx + zwx = 0 whenever | z | ≤ , | w | ≤ } . A commuting triple of operators ( T , T , T ) for which the closedtetrablock E is a spectral set, is called an E -contraction. For an E -contraction ( T , T , T ), the two operator equations T − T ∗ T = D T X D T and T − T ∗ T = D T X D T , D T = ( I − T ∗ T ) , have unique solutions A , A on D T = RanD T and they are calledthe fundamental operators of ( T , T , T ). For a particular class of E -contractions, we prove it necessary for the existence of ratio-nal dilation that the corresponding fundamental operators A , A satisfy A A = A A and A ∗ A − A A ∗ = A ∗ A − A A ∗ . (0.1)Then we construct an E -contraction from that particular classwhich fails to satisfy (0.1). We produce a concrete functionalmodel for pure E -isometries, a class of E -contractions analogousto the pure isometries in one variable. The fundamental operatorsplay the main role in this model. Introduction
Let X be a compact subset of C n and let R ( X ) denote the algebra ofall rational functions on X , that is, all quotients p/q of polynomials p, q for which q has no zeros in X . The norm of an element f in R ( X )is defined as k f k ∞ ,X = sup {| f ( ξ ) | : ξ ∈ X } . Mathematics Subject Classification.
Key words and phrases.
Tetrablock, Spectral set, Complete spectral set, Rationaldilation, Functional model.This work was supported in part by the Center for Advanced Studies in Mathe-matics at Ben-Gurion University of the Negev, Israel when the author was visitingthe Mathematics Department of BGU during 2012 - 2014. At present the author’swork is supported by INSPIRE Faculty Award of DST, India.
Also for each k ≥
1, let R k ( X ) denote the algebra of all k × k matricesover R ( X ). Obviously each element in R k ( X ) is a k × k matrix ofrational functions F = ( f i,j ) and we can define a norm on R k ( X ) inthe canonical way k F k = sup {k F ( ξ ) k : ξ ∈ X } , thereby making R k ( X ) into a non-commutative normed algebra. Let T = ( T , · · · , T n ) be an n -tuple of commuting operators on a Hilbertspace H . The set X is said to be a spectral set for T if the Taylor jointspectrum σ ( T ) of T is a subset of X and k f ( T ) k ≤ k f k ∞ ,X , for every f ∈ R ( X ) . (1.1)Here f ( T ) can be interpreted as p ( T ) q ( T ) − when f = p/q . Moreover, X is said to be a complete spectral set if k F ( T ) k ≤ k F k for every F in R k ( X ), k = 1 , , · · · .Let A ( X ) be the algebra of continuous complex-valued functions on X which separates the points of X . A boundary for A ( X ) is a closedsubset F of X such that every function in A ( X ) attains its maximummodulus on F . It follows from the theory of uniform algebras that if bX is the intersection of all the boundaries of X then bX is a boundaryfor A ( X ) (see Theorem 9.1 of [6]). This smallest boundary bX is calledthe ˇS ilov boundary relative to the algebra A ( X ) .A commuting n -tuple of operators T that has X as a spectral set, issaid to have a rational dilation or normal bX - dilation if there exists aHilbert space K , an isometry V : H → K and an n -tuple of commutingnormal operators N = ( N , · · · , N n ) on K with σ ( N ) ⊆ bX such that f ( T ) = V ∗ f ( N ) V, for every f ∈ R ( X ) . (1.2)One of the important discoveries in operator theory is Sz.-Nagy’sunitary dilation for a contraction, [20], which opened a new horizonby announcing the success of rational dilation on the closed unit discof C . Since then one of the main aims of operator theory has beento determine the success or failure of rational dilation on the closureof a bounded domain in C n . It is evident from the definitions that if X is a complete spectral set for T then X is a spectral set for T . Acelebrated theorem of Arveson states that T has a normal bX -dilationif and only if X is a complete spectral set for T (Theorem 1.2.2 andits corollary, [8]). Therefore, the success or failure of rational dilationis equivalent to asking whether the fact that X is a spectral set for T automatically turns X into a complete spectral set for T . History HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 3 witnessed an affirmative answer to this question given by Agler when X is an annulus [3] and by Ando when X = D [7]. Agler, Harlandand Raphael have produced, by machine computation, an example of atriply connected domain in C where the answer is negative [4]. Dritscheland M c Cullough also gave a negative answer to that question when X isan arbitrary triply connected domain [12]. Parrott showed by a counterexample [18] that rational dilation fails on the closed tridisc D . Alsorecently we have success of rational dilation on the closed symmetrizedbidisc Γ [5, 10, 16], where Γ is defined asΓ = { ( z + z , z z ) : | z | ≤ , | z | ≤ } . (1.3)In this article, we show that rational dilation fails when X is theclosure of the tetrablock E , a polynomially convex, non-convex andinhomogeneous domain in C , defined as E = { ( x , x , x ) ∈ C : 1 − zx − wx + zwx = 0 whenever | z | ≤ , | w | ≤ } . This domain has attracted the attention of a number of mathematicians[1, 2, 22, 13, 14, 23, 9, 11, 17] because of its relevance to µ -synthesisand H ∞ control theory. The following result from [1] (Theorem 2.4,part-(9)) characterizes points in E and E and provides a geometricdescription of the tetrablock. Theorem 1.1.
A point ( x , x , x ) ∈ C is in E if and only if | x | ≤ and there exist β , β ∈ C such that | β | + | β | ≤ and x = β +¯ β x , x = β + ¯ β x . It is evident from the above result that the tetrablock lives insidethe tridisc D . The distinguished boundary (which is same as the ˇS ilov boundary) of the tetrablock was determined in [1] (see Theorem 7.1 of[1]) to be the set b E = { ( x , x , x ) ∈ C : x = ¯ x x , | x | ≤ , | x | = 1 } = { ( x , x , x ) ∈ E : | x | = 1 } . In [9], Bhattacharyya introduced the study of commuting operatortriples that have E as a spectral set. There such a triple was called a tetrablock contraction . As a notation is always convenient, we shall callsuch a triple an E -contraction. So we are led to the following definition: Definition 1.2.
A triple of commuting operators ( T , T , T ) on aHilbert space H for which E is a spectral set is called an E - contraction .Since the tetrablock lives inside the tridisc, an E -contraction consistsof commuting contractions. Evidently ( T ∗ , T ∗ , T ∗ ) is an E -contraction SOURAV PAL when ( T , T , T ) is an E -contraction. We briefly recall from the lit-erature the special classes of E -contractions which are analogous touniteries, isometries and co-isometries in one variable operator theory. Definition 1.3.
Let T , T , T be commuting operators on a Hilbertspace H . We say that ( T , T , T ) is(i) an E - unitary if T , T , T are normal operators and the jointspectrum σ T ( T , T , T ) is contained in b E ;(ii) an E - isometry if there exists a Hilbert space K containing H and an E -unitary ( ˜ T , ˜ T , ˜ T ) on K such that H is a commoninvariant subspace of T , T , T and that T i = ˜ T i | H for i = 1 , , E - co-isometry if ( T ∗ , T ∗ , T ∗ ) is an E -isometry.Moreover, an E -isometry ( T , T , T ) is said to be pure if T is a pureisometry, i.e, if T ∗ n → n → ∞ . We accumulate someresults from the literature in section 2 and they will be used in sequel.It is clear that a rational dilation of an E -contraction ( T , T , T ) isnothing but an E -unitary dilation of ( T , T , T ), that is, an E -unitary N = ( N , N , N ) that dilates T by satisfying (1.2). Similarly an E -isometric dilation of T = ( T , T , T ) is an E -isometry V = ( V , V , V )that satisfies (1.2). In Theorem 3.5 in [9], it was shown that for every E -contraction ( T , T , T ) there were two unique operators A , A in L ( D T ) such that T − T ∗ T = D T A D T , T − T ∗ T = D T A D T . Here D T = ( I − T ∗ T ) and D T = Ran D T and L ( H ), for a Hilbertspace H , always denotes the algebra of bounded operators on H . Anexplicit E -isometric dilation was constructed for a particular class of E -contractions in [9] (see Theorem 6.1 in [9]) and A , A played thefundamental role in that explicit construction of dilation. For theirpivotal role in the dilation, A and A were called the fundamentaloperators of ( T , T , T ).In section 4, we produce a set of necessary conditions for the existenceof rational dilation for a class of E -contractions. Indeed, in Proposition4.5, we show that if ( T , T , T ) is an E -contraction on H ⊕H for someHilbert space H , satisfying(i) Ker ( D T ) = H ⊕ { } and D T = { } ⊕ H (ii) T ( D T ) = { } and T Ker ( D T ) ⊆ D T HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 5 and if A , A are the fundamental operators of ( T , T , T ), then for theexistence of an E -isometric dilation of ( T ∗ , T ∗ , T ∗ ) it is necessary that[ A , A ] = 0 and [ A ∗ , A ] = [ A ∗ , A ] . (1.4)Here [ S , S ] = S S − S S , for any two operators S , S . In section5, we construct an example of an E -contraction that satisfies the hy-potheses of Proposition 4.5 but fails to satisfy (1.4). This concludesthe failure of rational dilation on the tetrablock.The proof of Proposition 4.5 depends heavily upon a functionalmodel for pure E -isometries which we provide in Theorem 3.3. There isan Wold type decomposition for an E -isometry (see Theorem 2.3) thatsplits an E -isometry into two parts of which one is an E -unitary andthe other is a pure E -isometry. Theorem 2.2 describes the structure ofan E -unitary. Therefore, a concrete model for pure E -isometries gives acomplete description of an E -isometry. In Theorem 3.3, we show that apure E -isometry ( ˆ T , ˆ T , ˆ T ) can be modelled as a commuting triple ofToeplitz operators ( T A ∗ + A z , T A ∗ + A z , T z ) on the vectorial Hardy space H ( D ˆ T ∗ ), where A and A are the fundamental operators of the E -co-isometry ( ˆ T ∗ , ˆ T ∗ , ˆ T ∗ ). The converse is also true, that is, every suchtriple of commuting contractions ( T A + Bz , T B ∗ + A ∗ z , T z ) on a vectorialHardy space is a pure E -isometry.2. Preliminary results
We begin with a lemma that simplifies the definition of E -contraction. Lemma 2.1.
A commuting triple of bounded operators ( T , T , T ) isan E -contraction if and only if k f ( T , T , T ) k ≤ k f k ∞ , E for any holo-morphic polynomial f in three variables. This actually follows from the fact that E is polynomially convex.For a proof to this lemma see Lemma 3.3 of [9]. The following theoremgives a set of characterization for E -unitaries (Theorem 5.4 of [9]). Theorem 2.2.
Let N = ( N , N , N ) be a commuting triple of boundedoperators. Then the following are equivalent. (1) N is an E -unitary, (2) N is a unitary and N is an E -contraction, (3) N is a unitary, N is a contraction and N = N ∗ N . Here is a structure theorem for the E -isometries. Theorem 2.3.
Let V = ( V , V , V ) be a commuting triple of boundedoperators. Then the following are equivalent. SOURAV PAL (1) V is an E -isometry. (2) V is an isometry and V is an E -contraction. (3) V is an isometry, V is a contraction and V = V ∗ V . (4) (Wold decomposition) H has a decomposition H = H ⊕H intoreducing subspaces of V , V , V such that ( V | H , V | H , V | H ) isan E -unitary and ( V | H , V | H , V | H ) is a pure E -isometry. See Theorem 5.6 and Theorem 5.7 of [9] for a proof.3.
A functional model for pure E -isometries Let us recall that the numerical radius of an operator T on a Hilbertspace H is defined by ω ( T ) = sup {|h T x, x i| : k x k H = 1 } . It is well known that r ( T ) ≤ ω ( T ) ≤ k T k and 12 k T k ≤ ω ( T ) ≤ k T k , (3.1)where r ( T ) is the spectral radius of T . We state a basic lemma onnumerical radius whose proof is a routine exercise. We shall use thislemma in sequel. Lemma 3.1.
The numerical radius of an operator T is not greater thanone if and only if Re βT ≤ I for all complex numbers β of modulus . We recall from section 1, the existence-uniqueness theorem ([9], The-orem 3.5) for the fundamental operators of an E -contraction. Theorem 3.2.
Let ( T , T , T ) be an E -contraction. Then there aretwo unique operators A , A in L ( D T ) such that T − T ∗ T = D T A D T and T − T ∗ T = D T A D T . (3.2) Moreover, ω ( A + zA ) ≤ for all z ∈ D . As we mentioned in Section 1 that these two unique operators A , A are called the fundamental operators of ( T , T , T ). The following the-orem gives a concrete model for pure E -isometries in terms of Toeplitzoperators on a vectorial Hardy space. Theorem 3.3.
Let ( ˆ T , ˆ T , ˆ T ) be a pure E -isometry acting on a Hilbertspace H and let A , A denote the corresponding fundamental operators.Then there exists a unitary U : H → H ( D ˆ T ∗ ) such that ˆ T = U ∗ T ϕ U, ˆ T = U ∗ T ψ U and ˆ T = U ∗ T z U, where ϕ ( z ) = G ∗ + G z, ψ ( z ) = G ∗ + G z, z ∈ D and G = U A U ∗ and G = U A U ∗ . Moreover, A , A satisfy HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 7 (1) [ A , A ] = 0 ;(2) [ A ∗ , A ] = [ A ∗ , A ] ; and (3) k A ∗ + A z k ≤ for all z ∈ D .Conversely, if A and A are two bounded operators on a Hilbert space E satisfying the above three conditions, then ( T A ∗ + A z , T A ∗ + A z , T z ) on H ( E ) is a pure E -isometry.Proof. Suppose that ( ˆ T , ˆ T , ˆ T ) is a pure E -isometry. Then ˆ T is apure isometry and it can be identified with the Toeplitz operator T z on H ( D ˆ T ∗ ). Therefore, there is a unitary U from H onto H ( D ˆ T ∗ ) suchthat ˆ T = U ∗ T z U . Since for i = 1 , , ˆ T i is a commutant of ˆ T , thereare two multipliers ϕ, ψ in H ∞ ( L ( D ˆ T ∗ )) such that ˆ T = U ∗ T ϕ U andˆ T = U ∗ T ψ U . Claim.
If ( V , V , V ) on a Hilbert space H is an E -isometry then V = V ∗ V . Proof of Claim.
Let ( V , V , V ) be the restriction of an E -unitary( N , N , N ) to the common invariant subspace H . By part-(3) ofTheorem 2.2 we have that N is a unitary and N = N ∗ N . Therefore, N ∗ = N ∗ N and hence N ∗ = N N ∗ by an application of Fuglede’s the-orem, [15], which states that if a normal operator N commutes with abounded operator T then it commutes with T ∗ too. Also since N is aunitary we have that N = N ∗ N . Now H is an invariant subspace for N and thus H is invariant under N ∗ N . So V = N | H = N ∗ N | H .Again H is invariant under N . Therefore, N ∗ ( N ( H )) ⊆ H . So wehave that P H N ∗ | N ( H ) = N ∗ | N ( H ) . Again V ∗ = P H N ∗ | H . There-fore, N ∗ N | H = V ∗ V . So, we have that V = V ∗ V .We apply this claim and part-(3) of Theorem 2.3 to the E -isometry( T ϕ , T ψ , T z ) to get T ϕ = T ∗ ψ T z and T ψ = T ∗ ϕ T z and by these two relationswe have that ϕ ( z ) = G ∗ + G z and ψ ( z ) = G ∗ + G z for some G , G ∈ L ( D ˆ T ∗ ) . By the commutativity of ϕ ( z ) and ψ ( z ) we obtain[ G , G ] = 0 and [ G ∗ , G ] = [ G ∗ , G ] . We now compute the fundamental operators of the E -co-isometry ( T ∗ ϕ , T ∗ ψ , T ∗ z )that is of ( T ∗ G ∗ + G z , T ∗ G ∗ + G z , T ∗ z ). Clearly I − T z T ∗ z is the projection ontothe space D T ∗ z . Now T ∗ G ∗ + G z − T G ∗ + G z T ∗ z = T G + G ∗ ¯ z − T G ∗ + G z T ¯ z = T G = ( I − T z T ∗ z ) G ( I − T z T ∗ z ) . SOURAV PAL
Similarly, T ∗ G ∗ + G z − T G ∗ + G z T ∗ z = ( I − T z T ∗ z ) G ( I − T z T ∗ z ) . Therefore, G , G are the fundamental operators of ( T ∗ ϕ , T ∗ ψ , T ∗ z ). Thefundamental operators of ( ˆ T ∗ , ˆ T ∗ , ˆ T ∗ ) are A , A . Thereforeˆ T ∗ − ˆ T ˆ T ∗ = D ˆ T ∗ A D ˆ T ∗ that is U ∗ ( T ∗ ϕ − T ψ T ∗ z ) U = U ∗ D T ∗ z ( U A U ∗ ) D ∗ T z U or equivalently T ∗ ϕ − T ψ T ∗ z = D T ∗ z ( U A U ∗ ) D ∗ T z . Similarly, T ∗ ψ − T ϕ T ∗ z = D T ∗ z ( U A U ∗ ) D ∗ T z . Therefore, by the uniqueness of fundamental operators (see Theorem3.2) we have that G = U A U ∗ and G = U A U ∗ . From [ G , G ] = 0 and [ G ∗ , G ] = [ G ∗ , G ] it trivially follows that[ A , A ] = 0 and [ A ∗ , A ] = [ A ∗ , A ]. Also since ( T ϕ , T ψ , T z ) is an E -contraction, we have that k T ϕ k ≤ k ϕ ( z ) k = k G ∗ + G z k ≤ z ∈ D . Therefore, k A ∗ + A z k = k U ∗ ( G ∗ + G ) U k ≤ z ∈ D .For the converse, we first prove that the triple of multiplication op-erators ( M A ∗ + A z , M A ∗ + A z , M z ) on L ( E ) is an E -unitary when A , A satisfy the given conditions. It is evident that ( M A ∗ + A z , M A ∗ + A z , M z )is a commuting triple of normal operators when [ A , A ] = 0 and[ A ∗ , A ] = [ A ∗ , A ]. Also M A ∗ + A z = M ∗ A ∗ + A z M z and M z on L ( E ) isunitary. Therefore, by part-(3) of Theorem 2.2, ( M A ∗ + A z , M A ∗ + A z , M z )becomes an E -unitary if we prove that k M A ∗ + A z k ≤ z ∈ T .We have that ω ( A + zA ) ≤ z ∈ T , which is same as sayingthat ω ( z A + z A ) ≤ z , z of unit modulus.Thus by Lemma 3.1,( z A + z A ) + ( z A + z A ) ∗ ≤ I, that is ( z A + ¯ z A ∗ ) + ( z A + ¯ z A ∗ ) ∗ ≤ I. Therefore, ¯ z ( A ∗ + zA ) + z ( A ∗ + zA ) ∗ ≤ I for all z, z ∈ T . This issame as saying thatRe z ( A ∗ + zA ) ≤ I, for all z, z ∈ T . HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 9
Therefore, by Lemma 3.1 again ω ( A ∗ + A z ) ≤ z in T . Since M A ∗ + A z is a normal operator we have that k M A ∗ + A z k = ω ( M A ∗ + A z )and thus k M A ∗ + A z k for all z ∈ T . Therefore, ( M A ∗ + A z , M A ∗ + A z , M z )on L ( E ) is an E -unitary and hence ( T A ∗ + A z , T A ∗ + A z , T z ), being therestriction of ( M A ∗ + A z , M A ∗ + A z , M z ) to the common invariant sub-space H ( E ), is an E -isometry. Also T z on H ( E ) is a pure isometry.Thus we conclude that ( T A ∗ + A z , T A ∗ + A z , T z ) is a pure E -isometry.4. A necessary condition for the existence of dilation
Let us recall from section 1 the definitions of the E -isometric and E -unitary dilations of an E -contraction. In fact they can be definedin a simpler way by involving polynomials only. This is because thepolynomials are dense in the rational functions. Definition 4.1.
Let ( T , T , T ) be a E -contraction on H . A com-muting tuple ( Q , Q , V ) on K is said to be an E -isometric dilation of( T , T , T ) if H ⊆ K , ( Q , Q , V ) is an E -isometry and P H ( Q m Q m V n ) | H = T m T m T n , for all non-negative integers m , m , n. Here P H : K → H is the orthogonal projection of K onto H . Moreover,the dilation is called minimal if K = span { Q m Q m V n h : h ∈ H and m , m , n ∈ N ∪ { }} . Definition 4.2.
A commuting tuple ( R , R , U ) on K is said to be an E -unitary dilation of ( T , T , T ) if H ⊆ K , ( R , R , U ) is an E -unitaryand P H ( R m R m U n ) | H = T m T m T n , for all non-negative integers m , m , n. Moreover, the dilation is called minimal if K = span { R m R m U n h : h ∈ H and m , m , n ∈ Z } . Here R m i i = R ∗ i − m i for i = 1 , U n = U ∗− n when m i and n arenegative integers. Proposition 4.3.
If a E -contraction ( T , T , T ) defined on H has a E -isometric dilation, then it has a minimal E -isometric dilation.Proof. Let ( Q , Q , V ) on K ⊇ H be a E -isometric dilation of ( T , T , T ).Let K be the space defined as K = span { Q m Q m V n h : h ∈ H and m , m , n ∈ N ∪ { }} . Clearly K is invariant under Q m , Q m and V n , for any non-negativeinteger m , m and n . Therefore if we denote the restrictions of Q , Q and V to the common invariant subspace K by Q , Q and V re-spectively, we get Q m k = Q m k, Q m k = Q m k, and V n k = V n k, for any k ∈ K . Hence K = span { Q m Q m V n h : h ∈ H and m , m , n ∈ N ∪ { }} . Therefore for any non-negative integers m , m and n we have that P H ( Q m Q m V n ) h = P H ( Q m Q m V n ) h, for all h ∈ H . Now ( Q , Q , V ) is an E -contraction by being the restriction of an E -contraction ( Q , Q , V ) to a common invariant subspace K . Also V , being the restriction of an isometry to an invariant subspace, isalso an isometry. Therefore by Theorem 2.3 - part(2), ( Q , Q , V ) isan E -isometry. Hence ( Q , Q , V ) is a minimal E -isometric dilationof ( T , T , T ). Proposition 4.4.
Let ( Q , Q , V ) on K be an E -isometric dilation ofan E -contraction ( T , T , T ) on H . If ( Q , Q , V ) is minimal, then ( Q ∗ , Q ∗ , V ∗ ) is an E -co-isometric extension of ( T ∗ , T ∗ , T ∗ ) .Proof. We first prove that T P H = P H Q , T P H = P H Q and T P H = P H V . Clearly K = span { Q m Q m V n h : h ∈ H and m , m , n ∈ N ∪ { }} . Now for h ∈ H we have that T P H ( Q m Q m V n h ) = T ( T m T m T n h ) = T m +11 T m T n h = P H ( Q m +11 Q m V n h )= P H Q ( Q m Q m V n h ) . Thus we have that T P H = P H Q and similarly we can prove that T P H = P H Q and T P H = P H V . Also for h ∈ H and k ∈ K we havethat h T ∗ h, k i = h P H T ∗ h, k i = h T ∗ h, P H k i = h h, T P H k i = h h, P H Q k i = h Q ∗ h, k i . Hence T ∗ = Q ∗ | H and similarly T ∗ = Q ∗ | H and T ∗ = V ∗ | H . Therefore,( Q ∗ , Q ∗ , V ∗ ) is an E -co-isometric extension of ( T ∗ , T ∗ , T ∗ ). Proposition 4.5.
Let H be a Hilbert space and let ( T , T , T ) be an E -contraction on H = H ⊕ H with fundamental operators A , A . Let (i) Ker ( D T ) = H ⊕ { } and D T = { } ⊕ H ;(ii) T ( D T ) = { } and T Ker ( D T ) ⊆ D T .If ( T ∗ , T ∗ , T ∗ ) has an E -isometric dilation then HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 11 (1) A A = A A , (2) A ∗ A − A A ∗ = A ∗ A − A A ∗ .Proof. Let ( Q , Q , V ) on a Hilbert space K be a minimal E -isometricdilation of ( T ∗ , T ∗ , T ∗ ) (such a minimal E -isometric dilation exists byProposition 4.3) so that ( Q ∗ , Q ∗ , V ∗ ) is an E -co-isometric extension of( T , T , T ) by Proposition 4.4. Since ( Q , Q , V ) on K is an E -isometry,by part-(4) of Theorem 2.3, K has decomposition K = K ⊕ K intoreducing subspaces K , K of Q , Q , V such that ( Q | K , Q | K , V | K ) =( Q , Q , U ) is an E -unitary and ( Q | K , Q | K , V | K ) = ( Q , Q , V )is a pure E -isometry. Since ( Q , Q , V ) on K is a pure E -isometry,by Theorem 3.3, K can be identified with H ( E ), where E = D V ∗ and Q , Q , V can be identified with T ϕ , T ψ , T z respectively on H ( E ),where ϕ ( z ) = A + Bz and ψ ( z ) = B ∗ + A ∗ z, z ∈ D . Here A ∗ , B are the fundamental operators of ( Q ∗ , Q ∗ , V ∗ ). Again H ( E ) can beidentified with l ( E ) and T ϕ , T ψ , T z on H ( E ) can be identified with themultiplication operators M ϕ , M ψ , M z on l ( E ) respectively. So withoutloss of generality we can assume that K = l ( E ) and Q = M ϕ , Q = M ψ and V = M z on l ( E ). The block matrices of M ϕ , M ψ , M z are givenby M ϕ = A . . .B A . . . B A . . .. . . . . . . . . . . . , M ψ = B ∗ . . .A ∗ B ∗ . . . A ∗ B ∗ . . .. . . . . . . . . . . . and M z = . . .I . . . I . . .. . . . . . . . . . . . . From now onward we shall consider H as a subspace of K and T , T , T on H as the restrictions of Q ∗ , Q ∗ , V ∗ respectively to H . Claim 1 . D T ⊆ E ⊕ { } ⊕ { } ⊕ · · · ⊆ l ( E ) . Proof of claim.
Let h = h ⊕ h ∈ D T ⊆ H , where h ∈ K and h = ( c , c , c , . . . ) T ∈ l ( E ). Here ( c , c , c , . . . ) T denotes thetranspose of the vector ( c , c , c , . . . ). Since T ( D T ) = { } , we havethat T h = V ∗ h = V ∗ ( h ⊕ h ) = U ∗ h ⊕ M ∗ z h = U ∗ h ⊕ ( c , c , · · · ) T = 0which implies that h = 0 and c = c = · · · = 0. This completes theproof of Claim 1 . Claim 2 . Ker ( D T ) ⊆ { } ⊕ E ⊕ { } ⊕ { } ⊕ · · · ⊆ l ( E ) . Proof of claim.
For h = h ⊕ h ∈ Ker ( D T ) ⊆ H , where h ∈ K and h = ( c , c , c , . . . ) T ∈ l ( E ), we have that D T h = ( I − T ∗ T ) h = P H ( I − V V ∗ ) h = P H ( h ⊕ h − h ⊕ M z M ∗ z h ) = 0which implies that P H ( h ⊕ h ) = P H ( h ⊕ M z M ∗ z h ). Therefore, h ⊕ ( c , c , · · · ) T = P H ( h ⊕ (0 , c , c , · · · ) T )which further implies that k h ⊕ (0 , c , c , · · · ) T k ≥ k h ⊕ ( c , c , c , · · · ) T k .Thus c = 0. Again T ( Ker ( D T )) ⊆ D T . Therefore, for h ⊕ (0 , c , c , . . . ) T ∈ Ker ( D T ), we have that T ( h ⊕ (0 , c , c , . . . ) T ) = U ∗ h ⊕ M ∗ z (0 , c , c , · · · ) T = U ∗ h ⊕ ( c , c , · · · ) T ∈ D T . Then by Claim 1, h = 0 and c = c = · · · = 0. Hence Claim 2 isestablished.Now since H = D T ⊕ Ker ( D T ), we can conclude that H ⊆ E ⊕ E ⊕ { } ⊕ { } ⊕ · · · ⊆ l ( E ) = K . Therefore, ( M ∗ ϕ , M ∗ ψ , M ∗ z ) on l ( E )is an E -co-isometric extension of ( T , T , T ). We now compute thefundamental operators of ( M ∗ ϕ , M ∗ ψ , M ∗ z ). M ∗ ϕ − M ψ M ∗ z = A ∗ B ∗ · · · A ∗ B ∗ · · · A ∗ · · · ... ... ... . . . − B ∗ . . .A ∗ B ∗ · · · A ∗ B ∗ · · · ... ... ... . . . I · · · I · · · · · · ... ... ... . . . = A ∗ B ∗ · · · A ∗ B ∗ · · · A ∗ · · · ... ... ... . . . − B ∗ · · · A ∗ B ∗ · · · A ∗ · · · ... ... ... . . . = A ∗ · · · · · · · · · ... ... ... . . . . Similarly M ∗ ψ − M ϕ M ∗ z = B · · · · · · · · · ... ... ... . . . . HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 13
Also D M ∗ z = I − M z M ∗ z = I · · · · · · · · · ... ... ... . . . . Therefore, D M ∗ z = E ⊕ { } ⊕ { } · · · and D M ∗ z = D M ∗ z = I d on E ⊕ { } ⊕ { } · · · . If we setˆ A = A ∗ . . . . . . . . .. . . . . . . . . . . . , ˆ A = B . . . . . . . . .. . . . . . . . . . . . , (4.1)then M ∗ ϕ − M ψ M ∗ z = D M ∗ z ˆ A D M ∗ z and M ∗ ψ − M ϕ M ∗ z = D M ∗ z ˆ A D M ∗ z . Therefore, ˆ A , ˆ A are the fundamental operators of ( M ∗ ϕ , M ∗ ψ , M ∗ z ).Let us denote ( M ∗ ϕ , M ∗ ψ , M ∗ z ) by ( R , R , W ). Therefore, R − R ∗ W = D W ˆ A D W (4.2) R − R ∗ W = D W ˆ A D W . (4.3) Claim 3. ˆ A i D W | D T ⊆ D T and ˆ A i ∗ D W | D T ⊆ D T for i = 1 , Proof of claim.
Clearly D W = D M ∗ z = I d on D W . Let h = ( c , , , · · · ) T ∈D T . Then ˆ A D W h = ( A ∗ c , , , · · · ) T = M ∗ ϕ h = R h . Since R | H = S , R h ∈ H . Therefore ( A ∗ c , , , · · · ) T ∈ D T and ˆ A D W | D T ⊆ D T .Similarly we can prove that ˆ A D W | D T ⊆ D T .We compute the adjoint of T . Let ( c , c , , · · · ) T and ( d , d , , · · · ) T be two arbitrary elements in H where ( c , , , · · · ) T , ( d , , , · · · ) T ∈D T and (0 , c , , · · · ) T , (0 , d , , · · · ) T ∈ Ker ( D T ). Now h T ∗ ( c , c , , · · · ) T , ( d , d , , · · · ) T i = h ( c , c , , · · · ) T , T ( d , d , , · · · ) T i = h ( c , c , , · · · ) T , W ( d , d , , · · · ) T i = h ( c , c , , · · · ) T , ( d , , , · · · ) T i = h c , d i E = h (0 , c , , · · · ) T , ( d , d , , · · · ) T i . Therefore T ∗ ( c , c , , · · · ) T = (0 , c , , · · · ) T . Now h = ( c , , , · · · ) T ∈ D T implies that T ∗ h = (0 , c , , · · · ) T ∈ H and M ∗ ψ (0 , c , , · · · ) T = R (0 , c , , · · · ) T = ( Ac , , , · · · ) T ∈ H . Inparticular, ( Ac , , , · · · ) T ∈ D T . Therefore ˆ A ∗ D W h = ( Ac , , , · · · ) T ∈D T and ˆ A ∗ D W | D T ⊆ D T . Similarly we can prove that ˆ A ∗ D W | D T ⊆D T . Hence Claim 3 is proved.
Claim 4. ˆ A i | D T = A i and ˆ A i ∗ | D T = A ∗ i for i = 1 , Proof of Claim.
It is obvious that D T ⊆ D W = E ⊕ { } ⊕ { } ⊕ · · · .Now since W | H = T and D W is projection onto D W , we have that D W | H = D W | H = D W | D T = D T . Therefore, D T is a projection onto D T and D T = D T . From (4.2) we have that P H ( R − R ∗ W ) | H = P H ( D W ˆ A D W ) | H . (4.4)Since ( R , R , W ) is an E -co-isometric extension of ( T , T , T ), the LHSof (4.4) is equal to T − T ∗ T . Again since A , A are the fundamentaloperators of ( T , T , T ), we have that T − T ∗ T = D T A D T , A ∈ L ( D T ) . (4.5)It is clear that T − T ∗ T is 0 on the ortho-complement of D T , that ison Ker ( D T ). Therefore, T − T ∗ T = P D T ( R − R ∗ W ) | D T = P D T ( D W ˆ A D W ) | D T . (4.6)Again since D W | D T = D T = I d on D T , the RHS of (4.6) is equal to( D W ˆ A D W ) | D T and hence T − T ∗ T = ( R − R ∗ W ) | D T = ( D W ˆ A D W ) | D T = D T ˆ A D T . (4.7)The last identity follows from the fact ( Claim 3 ) that ˆ A D W | D T ⊆ D T .By the uniqueness of A we get that ˆ A | D T = A . Also since D T isinvariant under ˆ A ∗ by Claim 3 , we have that ˆ A ∗ | D T = A ∗ . Similarlywe can prove that ˆ A | D T = A and ˆ A ∗ | D T = A ∗ . Thus the proof to Claim 4 is complete.
HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 15
Now since ( M ϕ , M ψ , M z ) on l ( E ) is an E -isometry, M ϕ and M ψ commute, that is A . . .B A . . . B A . . .. . . . . . . . . . . . B ∗ . . .A ∗ B ∗ . . . A ∗ B ∗ . . .. . . . . . . . . . . . = B ∗ . . .A ∗ B ∗ . . . A ∗ B ∗ . . .. . . . . . . . . . . . A . . .B A . . . B A . . .. . . . . . . . . . . . which implies that AB ∗ . . .BB ∗ + AA ∗ AB ∗ . . .BA ∗ BB ∗ + AA ∗ AB ∗ . . .. . . . . . . . . . . . = B ∗ A . . .A ∗ A + B ∗ B B ∗ A . . .A ∗ B A ∗ A + B ∗ B B ∗ A . . .. . . . . . . . . . . . . Comparing both sides we obtain the following,(1) A ∗ B = BA ∗ (2) A ∗ A − AA ∗ = BB ∗ − B ∗ B .Therefore from (4.1) we have that(1) ˆ A ˆ A = ˆ A ˆ A (2) ˆ A ∗ ˆ A − ˆ A ˆ A ∗ = ˆ A ∗ ˆ A − ˆ A ˆ A ∗ .Taking restriction of the above two operator identities to the subspace D T we get(1) A A = A A (2) A ∗ A − A A ∗ = A ∗ A − A A ∗ .The proof is now complete.5. A counter example
Let H = l ( E ) ⊕ l ( E ) , E = C and let H = H ⊕ H . Let T , T , T on H ⊕ H be the block operator matrices T = (cid:20) J (cid:21) , T = (cid:20) (cid:21) and T = (cid:20) Y (cid:21) where J = (cid:20) F
00 0 (cid:21) and Y = (cid:20) VI (cid:21) on H = l ( E ) ⊕ l ( E ) . Here V = M z and I = I d on l ( E ) and F on l ( E ) is defined as F : l ( E ) → l ( E )( c , c , c , · · · ) T ( F c , , , · · · ) T , where we choose F = (cid:18) (cid:19) so that F is a non-normal contraction such that F = 0. Clearly F = 0 and F ∗ F = F F ∗ . Since F V = 0,
J Y = 0 and thus theproduct of any two of T , T , T is equal to 0. Now we unfold theoperators T , T , T and write their block matrices with respect to thedecomposition H = l ( E ) ⊕ l ( E ) ⊕ l ( E ) ⊕ l ( E ): T = F
00 0 0 0 , T = and T = V I . We shall prove later that ( T , T , T ) is an E -contraction and let usassume it for now. Here D T = I − T ∗ T = I I I
00 0 0 I − I V ∗
00 0 0 00 0 0 0 V I = I
00 0 0 I = D T . Clearly D T = { } ⊕ { } ⊕ l ( E ) ⊕ l ( E ) = { } ⊕ H and Ker ( D T ) = l ( E ) ⊕ l ( E ) ⊕ { } ⊕ { } = H ⊕ { } . Also for a vector k =( h , h , , T ∈ Ker ( D T ) and for a vector k = (0 , , h , h ) T ∈ D T , T k = V I ( h , h , , T = (0 , , V h , h ) T ∈ D T HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 17 and T k = V I (0 , , h , h ) T = (0 , , , T . Thus ( T , T , T ) satisfies all the conditions of Proposition 4.5. Wenow compute the fundamental operators A , A of ( T , T , T ). T − T ∗ T = T = F
00 0 0 0 = D T A D T = I
00 0 0 I A I
00 0 0 I . Since D T = { } ⊕ H and A ∈ L ( D T ) we can set A = 0 ⊕ (cid:20) F
00 0 (cid:21) on { } ⊕ H (= D T )so that T − T ∗ T = D T A D T . Again T ∗ T = 0 as X ∗ V = 0 and therefore T − T ∗ T = 0. This showsthat the fundamental operator A , for which T − T ∗ T = D T A D T holds, has to be equal to 0. Clearly A ∗ A − A A ∗ = 0 ⊕ (cid:20) F ∗ F − F F ∗
00 0 (cid:21) = 0 as F ∗ F = F F ∗ but A ∗ A − A A ∗ = 0. This violets the conclusion of Proposition 4.5and it is guaranteed that the E -contraction ( T ∗ , T ∗ , T ∗ ) does not havean E -isometric dilation. Since every E -unitary dilation is necessarily an E -isometric dilation, ( T ∗ , T ∗ , T ∗ ) does not have an E -unitary dilation.Now we prove that ( T , T , T ) is an E -contraction. By Lemma 2.1,it suffices to show that k p ( T , T , T ) k ≤ k p k ∞ , E , for any polynomial p ( x , x , x ) in the co-ordinates of E . Let p ( x , x , x ) = a + X i =1 a i x i + q ( x , x , x ) , where q is a polynomial containing only terms of second or higherdegree. Now p ( T , T , T ) = a I + a T + a T = (cid:20) a I a Y a I + a J (cid:21) Since Y is a contraction and k J k = 14 , it is obvious that (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) a I a Y a I + a J (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . We divide the rest of the proof into two cases.
Case 1.
When | a | ≤ | a | .We show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . Let (cid:18) ǫδ (cid:19) be a unit vector in C such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Without loss of generality we can choose ǫ, δ ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = | a ǫ | + (cid:12)(cid:12)(cid:12)(cid:12) | a ǫ | + (cid:18) | a | + | a | (cid:19) δ (cid:12)(cid:12)(cid:12)(cid:12) and if we replace (cid:18) ǫδ (cid:19) by (cid:18) | ǫ || δ | (cid:19) we see that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) | ǫ || δ | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 19
So, assuming ǫ, δ ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = | a ǫ | + (cid:26) | a ǫ | + (cid:18) | a | + | a | (cid:19) δ (cid:27) = | a ǫ | + | a ǫ | + (cid:26) | a | + | a a | | a | (cid:27) δ + 2 | a | (cid:18) | a | + | a | (cid:19) ǫδ = (cid:8) ( | a | + | a | ) ǫ + | a | δ + 2 | a a | ǫδ (cid:9) + (cid:26) | a |
16 + | a a | (cid:27) δ + | a a | ǫδ . (5.1)Again (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19) (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = | a ǫ | + { ( | a | + | a | ) ǫ + | a | δ } = | a | ǫ + {| a | + | a | + 2 | a a |} ǫ + 2 | a | ( | a | + | a | ) ǫδ + | a | δ = (cid:8) ( | a | + | a | ) ǫ + | a | δ + 2 | a a | ǫδ (cid:9) + ( | a | ǫ + 2 | a a | ǫδ ) + 2 | a a | ǫ . (5.2)We now compare (5.1) and (5.2). If ǫ ≥ δ then( | a | ǫ + 2 | a a | ǫδ ) + 2 | a a | ǫ ≥ (cid:18) | a |
16 + | a a | (cid:19) δ + | a a | ǫδ Therefore, it is evident from (5.1) and (5.2) that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19) (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . If ǫ < δ we consider the unit vector (cid:18) δǫ (cid:19) and it suffices if we show that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19) (cid:18) δǫ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . A computation similar to (5.2) gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19) (cid:18) δǫ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = | a | δ + {| a | + | a | + 2 | a a |} δ + 2 | a | ( | a | + | a | ) ǫδ + | a | ǫ = {| a | ( ǫ + δ ) + 2 | a a | ǫδ } + {| a | + | a | + 2 | a a |} δ + 2 | a a | ǫδ = {| a | + 2 | a a | ǫδ } + {| a | + | a | + 2 | a a |} δ + 2 | a a | ǫδ . (5.3)In the last equality we used the fact that | ǫ | + | δ | = 1. Again from(5.1) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = {| a | ( ǫ + δ ) + 2 | a a | ǫδ } + (cid:26) | a | ǫ + | a a | ǫδ (cid:27) + (cid:26) | a |
16 + | a a | (cid:27) δ ≤ {| a | ( ǫ + δ ) + 2 | a a | ǫδ } + (cid:26) | a | ǫ + | a a | ǫδ (cid:27) + (cid:26) | a |
16 + | a | (cid:27) δ = {| a | + 2 | a a | ǫδ } + (cid:26) | a | δ + | a | ǫ + | a a | ǫδ (cid:27) (5.4)The last inequality follows from the fact that | a | ≤ | a | . Since ǫ < δ we can conclude from (5.3) and (5.4) that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | ! (cid:18) ǫδ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19) (cid:18) δǫ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . Therefore, k p ( T , T , T ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . A classical result of Caratheodory and Fej´er states thatinf k b + b z + r ( z ) k ∞ , D = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) b b b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , where the infimum is taken over all polynomials r ( z ) in one variablewhich contain only terms of degree two or higher. For an elegant proofto this result, see Sarason’s seminal paper [19], where the result isderived as a consequence of the classical commutant lifting theorem of HE FAILURE OF RATIONAL DILATION ON THE TETRABLOCK 21
Sz.-Nagy and Foias (see [21]). Using this fact we have that k p ( T , T , T ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = inf k| a | + ( | a | + | a | ) z + r ( z ) k ∞ , D ≤ inf k| a | + | a | x + | a | x + r ( x , x , x ) k ∞ , Λ (5.5) ≤ inf k| a | + | a | + | a | x + | a | x + r ( x , x , x ) k ∞ , Λ (5.6)= inf k| a | + | a | x + | a | x + | a | x + r ( x , x , x ) k ∞ , Λ ≤ k a + a x + a x + a x + q ( x , x , x ) k ∞ , Λ (5.7) ≤ k a + a x + a x + a x + q ( x , x , x ) k ∞ , E = k p ( x , x , x ) k ∞ , E . Here Λ = { ( x, , x ) : x ∈ D } ⊆ E (by choosing β = 0 , β = 1in Theorem 1.1) and r ( z ) and r ( x , x , x ) range over polynomials ofdegree two or higher. The inequality (5.5) was obtained by putting x = x = z and x = 1 which makes the set of polynomials | a | + | a | x + | a | x + r ( z , z , z ), a subset of the set of polynomials | a | +( | a | + | a | ) z + r ( z ). The infimum taken over a subset is always biggerthan or equal to the infimum taken over the set itself. We obtained theinequality (5.6) by applying a similar argument because we can extractthe polynomial | a | x from the set r ( x , x , x ) and | a | x = | a | when x = 1. The inequality (5.7) was obtained by choosing r ( x , x , x ) inparticular to be equal to( a − | a | + a − | a | ) x + ( a − | a | ) x x + ( a − | a | ) x x + q ( x , x , x ) . Case 2.
When | a | > | a | .It is obvious from Case 1 that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | a | | a | | a | + | a | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . Therefore, k p ( T , T , T ) k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) | a | | a | + | a | | a | (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = inf k| a | + ( | a | + | a | ) z + r ( z ) k ∞ , D ≤ inf k| a | + | a | x + | a | x + r ( x , x , x ) k ∞ , Λ ≤ inf k| a | + | a | + | a | x + | a | x + r ( x , x , x ) k ∞ , Λ = inf k| a | + | a | x + | a | x + | a | x + r ( x , x , x ) k ∞ , Λ ≤ k a + a x + a x + a x + q ( x , x , x ) k ∞ , Λ (5.8) ≤ k a + a x + a x + a x + q ( x , x , x ) k ∞ , E = k p ( x , x , x ) k ∞ , E . Here all notations used are as same as they were in Case 1 and weobtained the inequality (5.8) by choosing r ( x , x , x ) in particular tobe equal to( a − | a | + a − | a | ) x + ( a − | a | ) x x + ( a − | a | ) x x + q ( x , x , x ) . Acknowledgement.
The author is indebted to the referee for his/hercareful and rigorous reading of this article and for pointing out aninaccuracy in the first version of the paper. The author greatly appre-ciates the warm and generous hospitality provided by Indian StatisticalInstitute, Delhi, India during the course of the work.
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Department of Mathematics, Indian Institute of Tech-nology Bombay, Powai, Mumbai - 400076, India.
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