Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

Shalit, Orr

doi:10.1007/978-3-0348-0692-3_60-1

Cited by 27 publications

(38 citation statements)

References 104 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let ψ ∈ M d ∩ C(B d ) be such that ψ / ∈ A d (the existence of such a multiplier was noted in [77,Section 5.2] by invoking [30]). Then ψ is, in particular, in the ball algebra A(B d ).…”

Section: Corollary 92mentioning

confidence: 99%

“…The isomorphism problem in the commutative case. We start by recalling that the Drury-Arveson space H 2 d is the reproducing kernel Hilbert space (in the usual, commutative function-theoretic sense) on the unit ball B d ⊆ C d , with reproducing kernel k w (z) = k(z, w) = 1 1− z,w (see [77]). Let M d denote the multiplier algebra (in the usual, commutative function-theoretic sense) of H 2 d .…”

Section: Connection To the Commutative Casementioning

confidence: 99%

“…Let d ∈ N ∪ {∞}, and let H 2 d denote the Drury-Arveson space, and let M d denote the multiplier algebra of the Drury-Arveson space (see the survey [77]). We note that M d is closed in the weak-operator topology (WOT) on B(H 2 d ) and, furthermore, it is obtained as the WOT-closure of the algebra generated by the shifts M z i : f → z i f (i = 1, .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Salomon

Shalit

Shamovich

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given an nc variety V in the nc unit ball B d , we identify the algebra of bounded analytic functions on V -denoted H ∞ (V) -as the multiplier algebra Mult H V of a certain reproducing kernel Hilbert space H V consisting of nc functions on V. We find that every such algebra H ∞ (V) is completely isometrically isomorphic to the quotient H ∞ (B d )/J V of the algebra of bounded nc holomorphic functions on the ball by the ideal J V of bounded nc holomorphic functions which vanish on V. In order to demonstrate this isomorphism, we prove that the space H V is an nc complete Pick space (a fact recently proved -by other methods -by Ball, Marx and Vinnikov).We investigate the problem of when two algebras H ∞ (V) and H ∞ (W) are (completely) isometrically isomorphic. If the variety W is the image of V under an nc analytic automorphism of B d , then H ∞ (V) and H ∞ (W) are completely isometrically isomorphic. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case d = ∞ we need to assume that the isomorphism is also weak- * continuous).We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of B d and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case d = ∞ without further assumptions.Along the way, we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases.Analogues of the Nevanlinna-Pick interpolation on the noncommutative ball first appeared in [21] and [12]. More general noncommutative versions of the classical interpolation and realization results appeared recently in the works of Agler and M c Carthy [5] and Ball, Marx and Vinnikov [14,15], who also introduced a generalization of reproducing kernel Hilbert spaces to the free setting.Our first goal in this work is to show that the full Fock space is a noncommutative reproducing kernel Hilbert space (nc RKHS), and its algebra of multipliers is, on the one hand, the algebra of bounded functions on the noncommutative ball (such that the multiplier norm and the supremum norm coincide), and, on the other hand, that this algebra coincides with the WOT closed algebra considered by Arias-Popescu and Davidson-Pitts. With this identification in hand, our second goal is to show that several results of [14] in the case of the noncommutative ball follow from established operator algebraic techniques and results, in particular, the complete Pick property of the noncommutative kernel of the full Fock space.W...

show abstract

Section: Corollary 92mentioning

confidence: 99%

Section: Connection To the Commutative Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Salomon

Shalit

Shamovich

2018

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Of course, if X is chosen such that 2 (X, β) is backshift-invariant, then B X = B 2 (X,β) and (i) and (ii) hold, see [3]. More in general, doing standard calculations it is not hard to see that B X satisfies (i) if and only if (7) n, n + e i + e j , n + e i ∈ X =⇒ n + e j ∈ X, for i, j = 1, . .…”

Section: Drury Type Inequalitymentioning

confidence: 99%

“…This function space was first introduced by Drury in [3], then further developed in [1]. See also [7]. It naturally arises as the right space to consider when trying to generalize to tuples of commuting operators a notable result by Von Neumann, saying that for any linear contraction A on a Hilbert space and any complex polinomial Q, it holds…”

Section: Introductionmentioning

confidence: 99%

On a class of shift-invariant subspaces of the Drury-Arveson space

Arcozzi

Levi

2018

Concrete Operators

View full text Add to dashboard Cite

In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in the complement of a set Y ⊂ N d with the property that Y +ej ⊂ Y for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitely calculated. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury's inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.Following Drury, we will relate A to an operator acting on a Hilbert space of holomorphic functions of several variables on the unit ball. We write B d for the open unit ball {z = (z 1 , . . . , z d ) ∈ C d : |z| < 1}, where |z| 2 := d j=1 |z j | 2 . Assuming that multiplication by z j defines a bounded linear operator (and it does on the spaces we are dealing with), on such a space we can consider a very natural d-tuple of operators, namely the d-shift

show abstract

A Gleason Solution Model for Row Contractions

Ramanantoanina

2019

Interpolation and Realization Theory With Applications to Control Theory

View full text Add to dashboard Cite

In the de Branges-Rovnyak functional model for contractions on Hilbert space, any completely non-coisometric (CNC) contraction is represented as the adjoint of the restriction of the backward shift to a de Branges-Rovnyak space, H (b), associated to a contractive analytic operator-valued function, b, on the open unit disk. We extend this model to a large class of CNC contractions of several copies of a Hilbert space into itself (including all CNC row contractions with commuting component operators). Namely, we completely characterize the set of all CNC row contractions, T , which are unitarily equivalent to an extremal Gleason solution for a de Branges-Rovnyak space, H (b T ), contractively contained in a vector-valued Drury-Arveson space of analytic functions on the open unit ball in several complex dimensions. Here, a Gleason solution is the appropriate several-variable analogue of the adjoint of the restricted backward shift and the characteristic function, b T , belongs to the several-variable Schur class of contractive multipliers between vector-valued Drury-Arveson spaces. The characteristic function, b T , is a unitary invariant, and we further characterize a natural sub-class of CNC row contractions for which it is a complete unitary invariant. 1 2 R.T.W. MARTIN AND A. RAMANANTOANINARecall that the shift, S : H 2 (D) → H 2 (D) is the canonical isometry of multiplication by z and its adjoint, the backward shift, S * , acts as the difference quotient:The shift plays a central role in the classical theory of Hardy spaces [29,42,25]. If T is any CNC contraction, there is a (essentially unique) contractive, operator-valued analytic function, bT , on the unit disk, bT (z) ∈ L (H, K) (i.e. a member of the operatorvalued Schur class), so that T is unitarily equivalent to X where X * := S * | H (b T ) is the restriction of the backward shift of the vector-valued Hardy space H 2 (D) ⊗ K to the de Branges-Rovnyak space H (bT ) [7,11]. (Any de Branges-Rovnyak space H (b) associated to a contractive, operator-valued analytic function, b, on the disk is always contractively contained in vector-valued Hardy space and is always co-invariant for the shift [48].) This provides a natural model for CNC contractions as adjoints of restrictions of the backward shift to de Branges-Rovnyak reproducing kernel Hilbert spaces, and this is the model we extend to several variables in this paper. A canonical several-variable extension of the Hardy space of the disk is the Drury-Arveson space, H 2 d , the unique RKHS of analytic functions on the open unit ball B d := (C d )1 corresponding to the several-variable Szegö kernel. (If Y is a Banach space, let (Y )1 denote the open unit ball and let [Y ]1 denote the closed unit ball.) The Schur classes of contractive, operator-valued functions on the disk are promoted to the multi-variable Schur classes, S d (J, K), of contractive, operator-valued multipliers between vector-valued Drury-Arveson spaces H 2 d ⊗ J and H 2 d ⊗ K (see Subsection 2.1), and the appropriate analogue of the adjoint o...

show abstract

Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients

Cited by 27 publications

References 104 publications

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

On a class of shift-invariant subspaces of the Drury-Arveson space

A Gleason Solution Model for Row Contractions

Contact Info

Product

Resources

About