Let A&#92;subsetC(K) be a unital closed subalgebra of the algebra of all continuous functions on a compact set K in &#92;mathbb{C}^{n}. We define the notion of an A-isometry and show that, under a suitable regularity condition needed to apply Aleksandrov\u27s work on the inner function problem, every A-isometry T&#92;in L(&#92;mathcal{H})^{n} is reflexive. This result applies to commuting isometries, spherical isometries, and more general, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain

Eschmeier, Jörg

Let A&#92;subsetC(K) be a unital closed subalgebra of the algebra of all continuous functions on a compact set K in &#92;mathbb{C}^{n}. We define the notion of an A-isometry and show that, under a suitable regularity condition needed to apply Aleksandrov's work on the inner function problem, every A-isometry T&#92;in L(&#92;mathcal{H})^{n} is reflexive. This result applies to commuting isometries, spherical isometries, and more general, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain

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On the reflexivity of multivariable isometries

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Universita¨t des SaarlandesUN IV E R S IT A SSAR A V I E NS ISFachrichtung 6.1 – MathematikPreprint Nr. 125On the reflexivity of multivariable isometriesJo¨rg EschmeierSaarbru¨cken 2005Fachrichtung 6.1 – Mathematik Preprint No. 125Universita¨t des Saarlandes submitted: January 31, 2005On the reflexivity of multivariable isometriesJo¨rg EschmeierSaarland UniversityDepartment of MathematicsPostfach 15 11 50D–66041 Saarbru¨ckenGermanyeschmei@math.uni-sb.deEdited byFR 6.1 – MathematikUniversita¨t des SaarlandesPostfach 15 11 5066041 Saarbru¨ckenGermanyFax: + 49 681 302 4443e-Mail: preprint@math.uni-sb.deWWW: http://www.math.uni-sb.de/On the reflexivity of multivariable isometriesJo¨rg EschmeierAbstractLet A ⊂ C(K) be a unital closed subalgebra of the algebra of allcontinuous functions on a compact set K in Cn. We define the notionof an A–isometry and show that, under a suitable regularity conditionneeded to apply Aleksandrov’s work on the inner function problem,every A–isometry T ∈ L(H)n is reflexive. This result applies to com-muting isometries, spherical isometries, and more general, to all sub-normal tuples with normal spectrum contained in the Bergman-Shilovboundary of a strictly pseudoconvex or bounded symmetric domain.1. INTRODUCTIONLet S ⊂ L(H) be an arbitrary family of bounded linear operators on acomplex Hilbert space H. We denote by Lat(S) the set of all closed linearsubspaces of H that are invariant under every operator S ∈ S. The setAlg Lat(S) = {C ∈ L(H); Lat(C) ⊃ Lat(S)}is a subalgebra of L(H) which contains S and is closed in the weak operatortopology (WOT). The family S is called reflexive ifAlg Lat(S) = WS ,where the right-hand side denotes the smallest WOT-closed subalgebra ofL(H) containing S and the identity operator 1H.Sarason [17] proved in 1966 that analytic Toeplitz operators on the Hardyspace H2(D) over the unit disc D and commuting families of normal op-erators are reflexive. In 1971 it was shown by Deddens [8] that all isome-tries are reflexive. These results were generalized in a paper [15] of Olinand Thomson from 1980 which showed that every subnormal operator isreflexive. Whether the corresponding result holds for subnormal tuples2000 Mathematics Subject Classification. Primary 47A15; Secondary 47A13, 47B20,47L45.1T = (T1, . . . , Tn) ∈ L(H)n of Hilbert space operators, is one of the ma-jor open problems in this area of multivariable invariant subspace theory.The result of Deddens on the reflexivity of single isometries was extendedby Li and McCarthy [13] to finite families of commuting isometries andby Bercovici [4] to arbitrary families of commuting isometries. In a recentpaper of Didas [10] the result of Bercovici is used to show that every sphericalisometry, that is, each commuting system T = (T1, . . . , Tn) ∈ L(H)n withn∑i=1T ∗i Ti = 1His reflexive.An n-tuple T = (T1, . . . , Tn) ∈ L(H)n consists of commuting isometries ifand only if T is subnormal and the joint spectrum of its minimal normalextension is contained in the Shilov boundary of the polydisc algebra A(Dn).Analogously, the class of spherical isometries consists precisely of those sub-normal n-tuples T = (T1, . . . , Tn) with the property that the joint spectrumof their minimal normal extension is contained in the Shilov boundary of theball algebra A(Bn). In this note we replace the polydisc and ball algebra bysuitable closed subalgebras A ⊂ C(K) over compact sets K in Cn to definethe notion of an A-isometry and prove that, under a regularity conditionneeded to apply Aleksandrov’s work [1] on the existence of inner functions,every A-isometry T ∈ L(H)n is reflexive.The main tools are the above cited result of Bercovici [4] on the reflexivityof commuting families of isometries and the work of Aleksandrov [1] on theexistence of inner functions for uniform algebras.2. REFLEXIVITY OF A-ISOMETRIESRecall that a commuting tuple T = (T1, . . . , Tn) ∈ L(H)n on a complexHilbert space H is called subnormal if it can be extended to a commutingtuple N = (N1, . . . , Nn) ∈ L(K)n of normal operators on a larger Hilbertspace K ⊃ H. A normal extension N of T as above is called minimal if Kis the only reducing subspace for N that contains H. All minimal normalextensions of a subnormal tuple T are unitarily equivalent and, for everynormal extension N of T , there is a reducing subspace K0 for N whichcontains H such that N |K0 is a minimal normal extension of T .Let T ∈ L(H)n be subnormal with minimal normal extension N ∈ L(K)n.The normal spectrum of T defined as σn(T ) = σ(N), where σ(N) denotesthe Taylor spectrum of N , is independent of the choice of N . A result of2Putinar [16] shows that σn(T ) ⊂ σ(T ). Let µ be a scalar spectral measure ofN . Denote by Ψ : L∞(µ) → L(H) the L∞-functional calculus of the normaltuple N . The restriction algebraR(T ) = {f ∈ L∞(µ); Ψ(f)H ⊂ H}is a weak∗ closed subalgebra of L∞(µ) and the induced algebra homomor-phismγT : R(T ) → L(H), f 7→ Ψ(f)|His isometric and weak∗ continuous if L(H) is equipped with its weak∗ topol-ogy as the dual space of the trace class operators (see Proposition 1.1 in[7]). Since all scalar spectral measures of T , that is, measures µ arising asa scalar spectral measure of some minimal normal extension N of T , aremutually absolutely continuous, both the restriction algebra R(T ) and thealgebra homomorphism γT are independent of the choice of N .Let K ⊂ Cn be a compact subset, and let A ⊂ C(K) be a unital closedsubalgebra containing the restrictions of the polynomials. We write S(A)for the Shilov boundary of A, that is, the smallest closed set S ⊂ K suchthat‖f‖∞,K = ‖f‖∞,S (f ∈ A).Denote by M+(K) the set of all positive regular Borel measures on K. Forµ ∈ M+(K), the triple (A,K, µ) is called regular in the sense of Aleksandrov[1] if, for each function ϕ ∈ C(K) with ϕ > 0, there exists a sequence (fk) inA with |fk| < ϕ for all k and limk→∞ |fk| = ϕ µ-almost everywhere. If thetriple (A,K, µ) is regular, then the support of µ is necessarily contained inS(A). For this and other properties of regular triples, the reader is referredto [1], [2].Fix a unital closed subalgebra A ⊂ C(K) containing the polynomials. Asubnormal tuple T ∈ L(H)n will be called A-subnormal if σn(T ) ⊂ K andA ⊂ R(T ). By an A-isometry we mean an A-subnormal tuple with σn(T ) ⊂S(A). An A-isometry T ∈ (H)n is said to be regular if the triple (A,K, µ) isregular for some, or equivalently, every scalar spectral measure µ of T . Hereµ is regarded as a measure on K via trivial extension.With these notations our main result can be formulated as follows.Theorem 1 Let A ⊂ C(K) be a unital closed subalgebra over somecompact set K in Cn containing the polynomials. Then every regular A-isometry is reflexive.In the following we fix A and K as in Theorem 1. For any given measureµ ∈ M+(K), we shall denote by H∞A (µ) the weak∗ closure of A in L∞(µ)and by H2A(µ) the norm-closure of A in L2(µ).3The proof of Theorem 1 will be based on the result of Bercovici [4] citedabove and on the following observation of Aleksandrov (Corollary 2.9 in [1])on the existence of µ-inner functions.Proposition 2 (Aleksandrov) Let µ ∈ M+(K) be a measure withµ({z}) = 0 for every z ∈ K such that (A,K, µ) is regular. Then the weak∗closure of the set {f ∈ H∞A (µ); |f | = 1 µ-almost everywhere} contains theclosed unit ball of A. 2As usual we call a measure µ ∈ M+(K) continuous if µ({z}) = 0 for everyz ∈ K and discrete if there is a countable set D ⊂ K with µ(K \ D) = 0.For an arbitrary measure µ ∈ M+(K), the set D = {z ∈ K; µ({z}) > 0} iscountable, and the measures µc, µd ∈ M+(K) defined byµc(A) = µ(A ∩ (K \D)), µd(A) = µ(A ∩D)yield the unique representation of µ as a sum µ = µc + µd of a continuousmeasure µc and a discrete measure µd. The following observation of Alek-sandrov (Proposition 1.5 in [1]) allows us to reduce the proof of Theorem 1to the case of A-isometries with continuous scalar spectral measure.Lemma 3 Suppose that the triple (A,K, µ) is regular. Then the orthog-onal decomposition H2A(µ) = H2A(µc)⊕ L2(µd) holds.Proof. Since this result is stated without proof in [1], we indicate the mainideas.For µ ∈ M+(K), define µc and µd as the continuous and discrete part of µ asin the section leading to the lemma. Denote by κc and κd the characteristicfunctions of K \D and D, respectively. Then the mapσ : L2(µc)⊕ L2(µd) → L2(µ), (f, g) 7→ fκc + gκdis easily seen to be a unitary operator between Hilbert spaces with inversegiven by ρ : L2(µ) → L2(µc)⊕ L2(µd),ρ([f ]L2(µ))=([f ]L2(µc), [f ]L2(µd)).To prove the equality ρ(H2A(µ))= H2A(µc) ⊕ L2(µd), it suffices to showthe inclusion σ(L2(µd))⊂ H2A(µ). Since µd is discrete, it is enough to provethat the characteristic function κ{w} of every point w ∈ D belongs to H2A(µ).Using the density of C(K) in L2(µ), one obtains a sequence (fk) in C(K)converging to κ{w} in L2(µ) such that 0 < fk < 2 for all k. The hypothesis4that (A,K, µ) is regular allows us to choose (Theorem 37 in [1]) functionsgk ∈ A with |gk| ≤ fk andµ({|gk| 6= fk}) < µ({w})/(k + 1) (k ∈ N).Obviously, one can achieve that gk(w) = fk(w) holds for all k. In view ofthe estimate‖gk − κ{w}‖2L2(µ) = |gk(w) − 1|2µ({w}) +∫K\{w}|gk|2dµ≤ |fk(w) − 1|2µ({w}) + ‖fk − κ{w}‖2L2(µ) + 4µ({|gk| 6= fk})it is clear that (gk) converges to κ{w} in L2(µ). 2Each subnormal tuple T ∈ L(H)n admits a unique orthogonal decompositionT = T0 ⊕ T1 ∈ L(H0 ⊕H1)nsuch that T1 ∈ L(H1)n is normal and T0 ∈ L(H0)n is pure, that is, possessesno non-zero reducing subspace M for which T0|M is normal.Lemma 4 Let T = T0 ⊕ T1 ∈ L(H0 ⊕ H1)n be the decomposition ofa given subnormal tuple T ∈ L(H)n into its pure part T0 ∈ L(H0)n andnormal part T1 ∈ L(H1)n. If T is a regular A-isometry, then so is T0.Proof. Let N ∈ L(K)n be a minimal normal extension of T . Denote by Ethe projection-valued spectral measure of N . The spaceK0 =∨(N∗kH0; k ∈ Nn)reduces N and N0 = N |K0 is a minimal normal extension of T0. Henceσn(T0) = σ(N0) ⊂ σ(N) ⊂ S(A).Set K1 = K 	 K0. Choose separating vectors x0 ∈ K0 for N0 and x1 ∈ K1for N1 = N |K1. Since the von Neumann algebras generated by N , N0 andN1 satisfy the relationW ∗(N) ⊂ W ∗(N0)⊕W∗(N1),the vector x = x0 + x1 is a separating vector for N . Because of〈E(·)x, x〉 = 〈E(·)x0, x0〉+ 〈E(·)x1, x1〉the scalar spectral measure µ0 = 〈E(·)x0, x0〉 of N0 is absolutely continuouswith respect to the scalar spectral measure µ = 〈E(·)x, x〉 of N . Hence(A,K, µ0) remains regular.5The projection-valued spectral measure E0 of N0 acts asE0(A) = E(A)|K0 ∈ L(K0),where A runs through all Borel subsets of σ(N0). Hence the L∞-functionalcalculi of N0 and N are related byΨN0(f) = Ψ(f)|K0 (f ∈ L∞(µ)).It follows that ΨN0(f)H0 ⊂ H for every function f ∈ A Since H1 is areducing subspace for N , we find that〈ΨN0(f)x, y〉 = 〈x,Ψ(f)∗y〉 = 0for all f ∈ A and x ∈ H0, y ∈ H1. Therefore A ⊂ R(T0), and the proof iscomplete. 2Let A ⊂ L(H) be a weak∗ closed subalgebra. The set of all weak∗ continuouslinear functionals onA can be identified isometrically with the quotient spaceQA = C1(H)/⊥A, where C1(H) denotes the Banach space of all trace classoperators on H. For x, y ∈ H, we write x ⊗ y ∈ QA for the equivalenceclass of the rank-one operator H → H, ξ 7→ 〈ξ, y〉x. The associated weak∗continuous functional on A acts as x⊗ y(A) = 〈Ax, y〉.Recall that A is said to possess the factorisation property (A1) if everyelement L ∈ QA is of the form L = x⊗ y with suitable vectors x, y ∈ H.To reduce Theorem 1 to the pure case, we shall use the following well knownresult (cf. for instance Lemma 4.4.1 in [9]).Lemma 5 Suppose that Ai ⊂ L(Hi) (i = 1, 2) are reflexive subalge-bras with property (A1). Then each weak∗ closed unital subalgebra B ⊂L(H1 ⊕ H2) which is contained in A1 ⊕ A2 is reflexive and has property(A1). Furthermore, the weak∗ and weak operator topology coincide on B.2Now we have gathered all tools necessary for the proof of our main result.Proof of Theorem 1. Let T ∈ L(H)n be a regular A-isometry. Denoteby T = T0 ⊕ T1 ∈ L(H0 ⊕ H1)n the decomposition of T into its pure partT0 ∈ L(H0)n and normal part T1 ∈ L(H1)n. Let us observe first that itsuffices to show that T0 is reflexive and that Alg Lat(T0) = WT0 has property(A1). Indeed, since the normal part T1 satisfies both of these properties ([17]and Proposition 2.05 in [5]) and since the weak∗ closed subalgebra AT ofL(H) generated by T is contained in WT0 ⊕WT1 , an application of Lemma5 will complete the proof of Theorem 1.6Since by Lemma 4 the pure part T0 is a regular A-isometry again, it sufficesto assume from the very beginning that T is pure and to show that T isreflexive and that AlgLat(T) = WT has property (A1). Thus let us fix apure regular A-isometry T ∈ L(H)n. Choose a minimal normal extensionN ∈ L(K)n of T and denote by E its projection-valued spectral measure.Exactly as in the one-variable case (Proposition V.17.14 in [6]) one can showthat there is a separating vector x ∈ H for N . Let µ = 〈E(·)x, x〉 be theassociated scalar spectral measure of T , and let Ψ : L∞(µ) → L(K) be theL∞-functional calculus of N . Since‖Ψ(f)x‖2 = 〈Ψ(|f |2)x, x〉 = ‖f‖2L2(µ)for all f ∈ L∞(µ), the map A → H, f 7→ Ψ(f)x, extends to a uniqueisometry j : H2A(µ) →H. The relationj(gf) = Ψ(gf)x = Ψ(g)j(f)obviously holds for all functions f, g ∈ A. A continuity argument impliesthat j(gf) = Ψ(g)j(f) holds for all g ∈ H∞A (µ) and f ∈ H2A(µ). In particu-lar, we find thatj ◦Mzi = Ti ◦ j (1 ≤ i ≤ n),where Mzi : H2A(µ), f 7→ zif , is the operator of multiplication with the i-thcoordinate function.Using the decomposition H2A(µ) = H2A(µc)⊕ L2(µd) from Lemma 3, we de-duce that T |j(L2(µd))is unitarily equivalent to the normal tuple Mz|L2(µd)via j. Our hypothesis that T is pure therefore implies that µ = µc is con-tinuous.Let p ∈ C[z] be a polynomial with ‖p‖∞,K ≤ 1. By Proposition 2 there is anet (θi)i∈I inI = {f ∈ H∞A (µ); |f | = 1 µ-almost everywhere}such that p is the weak∗ limit of the net (θi)i∈I in H∞A (µ). It follows thatp(T ) is the weak∗ limit of the net(γT (θi))i∈Iin L(H). Thus we have shownthatAlg Lat(T ) ⊂ Alg Lat(γT (I)).Since γT (I) consists of commuting isometries, the cited result of Bercovici(Theorem 2.3 in [4]) implies that the operator algebra on the right-hand sidehas property (A1). But then Proposition 2.5 in [12] shows that T is reflexiveand that Alg Lat(T ) = WT has property (A1). Thus the proof of Theorem1 is complete. 27As a consequence of the above proof we obtain the following slightly strongerresult.Corollary 6 Let T ∈ L(H)n be a regular A-isometry. Then the weak∗and weak operator topology coincide on the operator algebraAlg Lat(T ) = WT = AT .Furthermore, this algebra has property (A1) and is super-reflexive, that is,each of its unital weak∗ closed subalgebras is reflexive.Proof. The super-reflexivity of AT is a consequence of Proposition 2.5 in[12]. The remaining assertions follow from the above proof of Theorem 1together with Lemma 5. 2By specializing to the cyclic case we obtain the following result.Corollary 7 Suppose that µ ∈ M+(K) is a measure such that (A,K, µ) isa regular triple. Then the tuple Mz = (Mz1 , . . . ,Mzn) ∈ L(H2A(µ))n consist-ing of the multiplication operators with the coordinate functions is reflexive,and all the remaining assertions of Corollary 6 hold.Proof. Let T = Mz ∈ L(H2A(µ))n. By Stone-Weierstraß the minimal nor-mal extension of T is the corresponding multiplication tuple Mz ∈ L(L2(µ))n.Hence µ is a scalar spectral measure for T and T is a regular A-isometry.2Let D ⊂ Cn be a bounded open set. Define K = D and A = A(D) asthe Banach algebra of all continuous functions on D that are analytic onD. If D = Bn is the open Euclidean unit ball, then S(A) = ∂Bn is theunit sphere and the triple (A(Bn), Bn, µ) is regular for every measure µ ∈M+(Bn) with support contained in S(A) (Proposition 2 in [1]). Hence everyA(Bn)-isometry is regular. By a result of Athavale [3] the A(Bn)-isometriesare precisely the spherical isometries, that is, the commuting tuples T =(T1, . . . , Tn) ∈ L(H)n satisfying the identityn∑i=1T ∗i Ti = 1H.Thus in this case, Theorem 1 specializes to a recent result of Didas [10].8If D = Dn is the open unit polydisc, then S(A) = Tn is the n-torus andagain each A(Dn)-isometry is regular. The A(Dn)-isometries are preciselythe commuting n-tuples T ∈ L(H)n of isometries and Theorem 1 reduces tothe cited result of Li and McCarthy [13] on the reflexivity of finite commutingfamilies of isometries.More generally, in [2] (Theorem 3) Aleksandrov described conditions whichensure that the triple (A,K, µ) is regular for every measure µ ∈ M +(K)with support contained in the Shilov boundary S(A) of A. Under theseconditions every A-isometry is regular, and hence reflexive. It is well known(see [11]) that these conditions of Aleksandrov hold for A(D) when D isa bounded symmetric and circled domain in Cn or when D is a relativelycompact strictly pseudoconvex open set in Cn, or in a Stein submanifold Xof Cn.References[1] A.B. Aleksandrov, Inner functions on compact spaces, Funct. Anal.Appl. 18 (1984), 87–98.[2] A.B. Aleksandrov, Inner functions: Results, methods, problems,Proc. Int. Cong. Math., Berkeley 1986, Vol.1, 699–707.[3] A. Athavale, On the intertwining of joint isometries, J. OperatorTheory 23 (1990), 339–350.[4] H. Bercovici, A factorization theorem with applications to invari-ant subspaces and the reflexivity of isometries, Math. Res. Lett. 1(1994), 511–518.[5] H. Bercovici, C. Foias and C. Pearcy, Dual algebras with applica-tions to invariant subspaces and dilation theory, CBMS RegionalConf. Ser. in Math. 56, Amer. Math. Soc., Providence, RI, 1985.[6] J.B. Conway, The theory of subnormal operators, Math. SurveysMonogr. 36, Amer. Math. Soc., Providence, RI, 1991.[7] J.B. Conway, Towards a functional calculus of subnormal tuples:The minimal normal extension, Trans. Am. Math. Soc. 326 (1991),543–567.[8] J.A. Deddens, Every isometry is reflexive, Proc. Amer. Math. Soc.28 (1971), 509–512.[9] M. Didas, Dual algebras generated by von Neumann n-tuples overstrictly pseudoconvex sets, Dissertationes math. 425 (2004).9[10] M. Didas, Spherical isometries are reflexive, Preprint, Univ. desSaarlandes, 2004.[11] M. Didas and J. Eschmeier, Subnormal tuples on strictly pseudo-convex and bounded symmetric domains, Preprint, Univ. des Saar-landes, 2004.[12] D. Hadwin and E. Nordgren, Subalgebras of reflexive algebras, J.Operator Theory 7 (1982), 3–23.[13] W.S. Li and J. McCarthy, unpublished manuscript.[14] V. Mu¨ller and M. Ptak, Spherical isometries are hyporeflexive,Rocky Mountain J. Math. 29 (1999), 677–683.[15] R. Olin and J. Thomson, Algebras of subnormal operators, J. Funct.Anal. 37 (1980), 271–301.[16] M. Putinar, Spectral inclusion for subnormal n-tuples, Proc. Amer.math. Soc. 90 (1984), 405–406.[17] D. Sarason, Invariant subspaces and unstarred operator algebras,Pacific J. Math. 17 (1966), 511–517.Jo¨rg Eschmeier, Fachrichtung Mathematik, Universita¨t des Saarlandes,Postfach 151150, D–66041 Saarbru¨cken, GermanyE-mail address: eschmei@math.uni-sb.de10

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On the reflexivity of multivariable isometries

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