A parametrization for the symbols of a Hankel type operator

First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004 A PARAMETRIZATION FOR THE SYMBOLS OF A HANKEL TYPE OPERATOR SERGIO BERMUDO, STEFANIA. A. M. MARCANTOGNINI, AND MAR´ IA D. MOR´ AN Abstract. Hankel operators and their symbols, as generalized by V. Pt´ak and P. Vrbov´a, are considered. In this more general framework, a linear operator Xfrom a Hilbert space H1to a Hilbert space H2is said to be a Hankel operator for given contractions T1on H1and T2on H2if, and only if, XT ∗ 1=T2Xand Xsatisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations V1of T1and V2of T2.A Hankel symbol of Xis a dilation Zof X, with a certain norm constraint, such that ZV ∗ 1=V2Z. The boundedness condition imposed to Xhas revealed to be essential, indeed necessary and sufficient, for X to admit Hankel symbols. As for a description of the symbols of X, this work provides a parametric labeling of all of them by means of Schur like formula. As a by-product, a new proof of the existence of Hankel symbols is obtained. The proof is established by associating to X,T1and T2a suitable isometry Vso that there is a bijective correspondence between the symbols of Xand the family of all minimal unitary extensions of V. 1. Introduction Under the commutant perspective, the classical Hankel operators can be characterized by the intertwining relation they satisfy. From the same point of view, other intertwining operators may be thought as abstract or generalized Hankel operators. In this more general setting, which was the one adopted by Pt´ak and Vrbov´a [7], [8], [9], a Hankel operator is a linear map Xfrom a 2000 Mathematics Subject Classification. 47B35. Revised August 31, 2004. The first author was partially supported by grant BFM2001-3735 of Ministerio de Ciencia y Tecnolog´ıa and Junta de Andaluc´ıa (ref. FQM-260). The third author was partially supported by grant G-97000668 FONACIT, Venezuela. 71 72 S. BERMUDO, S. A. M. MARCANTOGNINI, AND M. D. MOR ´ AN separable Hilbert space H1into a separable Hilbert space H2such that XT ∗ 1= T2X, where T1on H1and T2on H2are given contraction operators and T∗ 1 denotes the adjoint of T1. If we consider the contractions T1:= P S∗|H2and T2:= P−S|H2 − , with Sthe shift operator on L2,H2the Hardy space, H2 −its orthogonal complement in L2, and Pand P−the corresponding orthogonal projections, then a classical Hankel operator can be regarded as a linear map X:H2→H2 −such that XT ∗ 1=T2X. In the classical case, the celebrated Nehari’s Theorem states that a Hankel operator Xis bounded if and only if there exists an L∞function Φ, a symbol of X, such that Xf =P−Φffor all f∈H2. Hence, the symbols are either multiplication operators induced by L∞functions with prescribed antianalytic part or, from another point of view, operators that commute with Sand have the same fixed component from H2into H2 −. Since the unitary operators V1=S∗and V2=Sare the corresponding minimal isometric dilations of the contractions T1and T2defined above, we can conclude that the symbols of the given Hankel operator Xare the intertwining dilations Zof X, namely, those linear operators Z:L2→L2such that P−Z|H2=Xand ZV ∗ 1=V2Z. In the abstract framework the operators that play the role of symbols might be the solutions Zof the commutant dilation problem ZV ∗ 1=V2Z, where V1 and V2are the minimal isometric dilations of T1and T2, respectively. The investigations carried on by Pt´ak and Vrbov´a revealed that the problem is solvable whenever Xsatisfies certain boundedness condition that depends on the unitary parts of the Wold-von Neumann decompositions of V1and V2. Since the Wold-von Neumann decomposition is trivial in the classical case, for Sbeing unitary, the result includes the classical situation. In this survey, we deal with the problem of describing the symbols Zof any abstract Hankel operator X, for given contractions T1and T2. We show that there is a bijective correspondence between the symbols of Xand the minimal unitary extensions of a Hilbert space isometry Vdetermined by X,T1and T2. Since any Hilbert space isometry has at least one minimal unitary extension, our approach provides a direct proof of the existence of the symbols for the generalized Hankel operator. The Arov-Grossman functional model [1] yields a complete description of the minimal unitary extensions of V, as it associates to each minimal unitary extension Uof Va function θUin a suitable Schur class of operator valued functions, and to each function θin the Schur class, an operator model Uθwhich gives rise to a minimal unitary extension of V, in such a way that the outlined correspondence is bijective. This method along with the Arov-Grossman model gives, in turn, a bijective correspondence between the symbols of Xand the Schur class. As a consequence, the connection between the symbols and the Schur functions can be realized as a parametric description. We also present uniqueness criteria and a Schur like formula. A PARAMETRIZATION FOR THE SYMBOLS OF A HANKEL TYPE OPERATOR 73 We point out that the line of investigations initiated by Pt´ak and Vrbov´a has been pursued mainly by Mancera and Pa´ul [5], [6]. It is worth to mention that abstract Hankel operators can be treated as bilinear forms defined in the even more general framework of the algebraic scattering systems as by Cotlar and Sadosky (see, for instance, [3], [4] and further references given therein.) The construction of the isometry V, which plays a key role in the proof of our main result, is in fact inspired by the Cotlar-Sadosky algebraic scattering systems methods. 2. Preliminaries Throughout this survey, all Hilbert spaces are assumed to be complex and separable. If Gis a closed linear subspace of a Hilbert space K, then PK Gstands for the orthogonal projection from Konto G. We denote by L(H,K) the space of all bounded linear operators from the Hilbert space Hinto the Hilbert space K. The space L(H,H) is denoted by L(H). By 1 we denote either the scalar unit or the identity operator, depending on the context. If T∈ L(H,K) and kTk ≤ β, then Dβ T= (β2−T∗T)1 2and Dβ T=Dβ TH. For β= 1, we use the standard notation DTand DTfor the defect operator and the defect space of T, respectively. For an isometric operator V∈ L(K), we denote by Rthe closed linear subspace of Kthat reduces Vto its unitary part in the Wold-von Neumann decomposition. In particular, R=T∞ n=0 VnKand PK R= lim n→∞ VnV∗n. In what follows, T1∈ L(H1) and T2∈ L(H2) are two given contractions with minimal isometric dilations V1∈ L(K1) and V2∈ L(K2), respectively. As defined by Pt´ak and Vrbov´a [7], [8], [9], an operator X∈ L(H1,H2) is said to be a Hankel operator for T1and T2if and only if XT ∗ 1=T2Xand, for some β≥0, (1) |hXh1, h2i| ≤ βkPK1 R1h1kkPK2 R2h2k,for all h1∈ H1and h2∈ H2, where Rjis the subspace of Kjwhich reduces the minimal isometric dilation Vjof Tjto the unitary part Rjof Vj(j= 1,2). We define kXkP V = inf β, where βvaries over all numbers satisfying (1). Given a Hankel operator Xfor T1and T2, we say that Z∈ L(K1,K2) is a Hankel symbol of Xif and only if (i) ZV ∗ 1=V2Z, (ii) PK2 H2Z|H1=X, and (iii) kZk=kXkP V . As already remarked in the introduction, the relation XT ∗ 1=T2Xalone is not sufficient to guarantee the existence of symbols. This difficulty is overtaken by means of the boundedness condition (1), since it turns out to be necessary and sufficient to ensure that there exist intertwining dilations Zof X((i) and (ii)), which altogether satisfy (iii). The reader is referred to [7], [8] and [9], as the original sources. If Xis a fixed Hankel operator for T1and T2, then Douglas’ Lemma (cf. [8, Proposition 1.4]) yields a unique bounded linear operator e Xfrom E1:= PK1 R1H1 74 S. BERMUDO, S. A. M. MARCANTOGNINI, AND M. D. MOR ´ AN into E2:= PK2 R2H2such that (2) X= (PK2 R2|H2)∗e XP K1 R1|H1and ke Xk=kXkP V . As the minimal unitary extensions of an isometry, on one hand, and the so called Schur functions, on the other, play key roles in the description of the symbols of a given Hankel operator, we conclude this section with a few words about these objects. If Vis an isometric operator on a Hilbert space Hwith domain D(V) and range R(V), both closed linear subspaces of H, then a minimal unitary extension of Vis a unitary operator Uacting on a Hilbert space Fthat contains Has closed linear subspace such that U|D(V)=Vand F=Wn∈ZUnH. Two minimal unitary extensions of V, namely U∈ L(F) and U0∈ L(F0), are to be interpreted as indistinguishable whenever there exists an isometric isomorphism ϕ:F → F0such that ϕ|H= 1 and ϕU =U0ϕ. As for the existence of minimal unitary extensions of any given isometry V, we remark that if UT, acting boundedly on the Hilbert space FT, is the minimal unitary dilation of the contraction T:= V P H D(V), then UTis a minimal unitary extension of V. The defect spaces of the isometry Vare N=H ª D(V) and M=H ª R(V). If either N={0}or M={0}, then Vhas a unique (up to isometric isomorphisms) minimal unitary extension. If Nand Mare Hilbert spaces, then the Schur class S(N,M) is the family of all analytic functions θ:D→ L(N,M) such that sup z∈D kθ(z)k ≤ 1. The Schur class S(N,M) features the Arov-Grossman functional model. The following theorem can be found in [1]. Theorem 2.1. Let Vbe an isometric operator on a Hilbert space Hwith domain D(V), range R(V)and defect spaces Nand M. The map that to each minimal unitary extension U∈ L(F)associates the function θU(z) := PF MU(1 −zP F FªHU)−1|N, z ∈D, establishes a bijection between the family U(V)of all minimal unitary extensions of Vand the Schur class S(N,M). 3. Description of the Hankel symbols of a given Hankel operator We now turn our attention to the problem of describing the Hankel symbols of a given Hankel operator X. We have the following theorem whose proof can be found in [2]. Theorem 3.1. Let T1∈ L(H1)and T2∈ L(H2)be two contractions with minimal isometric dilations V1∈ L(K1)and V2∈ L(K2), respectively. For j= 1,2, let Rjbe the subspace of Kjwhich reduces Vjto its unitary part. Given X, a Hankel operator for T1and T2, with kXkP V = 1, let e X∈ L(E1,E2) be the contraction operator uniquely determined by Xas in (2). Then there is 2.as P 76 S. BERMUDO, S. A. M. MARCANTOGNINI, AND M. D. MOR ´ AN space H1⊕H2, with the standard inner product, consider the 2×2block matrix operators f T1:= µT10 0 1¶,f T2:= µ1 0 0T2¶ and E:= µPK1 H1PK1 R1|H1X∗ X P K2 H2PK2 R2|H2¶. Then Xhas a unique Hankel symbol if and only if either (a) kernel (f T1E)⊆kernel E or (b) kernel (f T2E)⊆kernel E. Corollary 3.5. If either T1PK1 H1|E1or T2PK2 H2|E2is injective, any Hankel operator Xfor T1and T2has a unique Hankel symbol, say ZX. Acknowledgements. The authors would like to thank the referee for some comments that improved the exposition. References [1] Arov, D.Z.; Grossman, L.Z. Scattering matrices in the theory of extensions of isometric operators, Soviet Math. Dokl. 27 (1983), 518–522. [2] Bermudo, S.; Marcantognini, S. A. M.; Mor´an, M. D. Operators of Hankel type, Preprint. [3] Cotlar, M.; Sadosky, C. Prolongements des formes de Hankel g´en´eralis´ees et formes de Toeplitz, C. R. Acad. Sci Paris S´er. I Math. 305 (1987), 167–170. [4] Cotlar, M.; Sadosky, C. Integral representations of bounded Hankel forms defined in scattering systems with a multiparametric evolution group, Operator Theory: Adv. Appl., Vol. 35 (1988), 357–375. [5] Mancera, C. H.; Pa´ul, P. J. On Pt´ak’s generalization of Hankel operators, Czech. Math. J. 51 (2001), 323–342. [6] Mancera, C. H.; Pa´ul, P. J. Compact and finite rank operators satisfying a Hankel type equation T2X=XT ∗ 1, Integral Equations and Operator Theory 39 (2001), 475–495. [7] Pt´ak, V. Factorization of Toeplitz and Hankel operators, Math. Bohem. 122 (1997), No. 2, 131–140. [8] Pt´ak, V.; Vrbov´a, P. Operators of Toeplitz and Hankel type, Acta Sci. Math. (Szeged), 52 (1988), No. 1-2, 117–140. [9] Pt´ak, V.; Vrbov´a, P. Lifting intertwining relations, Integral Equations Operator Theory 11 (1988), No. 1, 128–147. A PARAMETRIZATION FOR THE SYMBOLS OF A HANKEL TYPE OPERATOR 77 Departamento de Econom´ ıa y Empresa, Universidad Pablo de Olavide, Carretera de Utrera, Km. 1, 41013 Seville, Spain E-mail address:[email protected] Departamento de Matem´ aticas, IVIC, Apartado Postal 21827, Caracas 1020A, Venezuela E-mail address:[email protected] Facultad de Ciencias, Escuela de Matem´ aticas, Universidad Central de Venezuela, Apartado Postal 20513, Caracas 1020A, Venezuela E-mail address:[email protected]