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Orr Shalit

Publications and source records attributed to Orr Shalit.

At least 37 records · Page 2Linked to original sources
arXiv2018

Dilations, inclusions of matrix convex sets, and completely positive maps

A matrix convex set is a set of the form $\mathcal{S} = \cup_{n\geq 1}\mathcal{S}_n$ (where each $\mathcal{S}_n$ is a set of $d$-tuples of $n \times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $\mathcal{S} = \cup_{n \geq 1} \mathcal{S}_n,$ and $\mathcal{T} = \cup_{n \geq 1} \mathcal{T}_n$, we find geometric conditions on $\mathcal{S}$ or on $\mathcal{T}$, such that $\mathcal{S}_1 \subseteq \mathcal{T}_1$ implies that $\mathcal{S} \subseteq C\mathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $\mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $\mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=\sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the "number of variables" $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.

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arXiv2017

Minimal and maximal matrix convex sets

To every convex body $K \subseteq \mathbb{R}^d$, one may associate a minimal matrix convex set $\mathcal{W}^{\textrm{min}}(K)$, and a maximal matrix convex set $\mathcal{W}^{\textrm{max}}(K)$, which have $K$ as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies $K,L \subseteq \mathbb{R}^d$ does $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$ hold? For a convex body $K$, we aim to find the optimal constant $θ(K)$ such that $\mathcal{W}^{\textrm{max}}(K) \subseteq θ(K) \cdot \mathcal{W}^{\textrm{min}}(K)$; we achieve this goal for all the $\ell^p$ unit balls, as well as for other sets. For example, if $\overline{\mathbb{B}}_{p,d}$ is the closed unit ball in $\mathbb{R}^d$ with the $\ell^p$ norm, then \[ θ(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p - 1/2|}. \] This constant is sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$. In particular, we obtain that a convex body $K$ satisfies $\mathcal{W}^{\textrm{max}}(K) = \mathcal{W}^{\textrm{min}}(K)$ if and only if $K$ is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every $d$-tuple of self-adjoint operators of norm less than or equal to $1$, can be dilated to a commuting family of self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit constructions of these (and other) dilations.

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arXiv2016

Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel \[ k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} . \] We prove that if an ideal $I \triangleleft \mathbb{C}[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the "approximate stable division property", then the closure of $I$ in $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$.

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arXiv2015

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = \sum a_n n^{-s-\bar u}$, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be "the same", and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury-Arveson space $H^2_d$ in $d$ variables, where $d$ can be any number in $\{1,2,\ldots, \infty\}$, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of $H^2_d$. Thus, a family of multiplier algebras of Dirichlet series are exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic to $H^2_d$ and when its multiplier algebra is isometrically isomorphic to $Mult(H^2_d)$.

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arXiv2015

Corrigendum to: Essential normality, essential norms and hyperrigidity

In our paper "Essential normality, essential norms and hyperrigidity" we claimed that the restriction of the identity representation of a certain operator system (constructed from a polynomial ideal) has the unique extension property, however the justification we gave was insufficient. In this note we provide the required justification under some additional assumptions. Fortunately, homogeneous ideals that are "sufficiently non-trivial" are covered by these assumptions. This affects the section of our paper relating essential normality and hyperrigidity. We show here that Proposition 4.11 and Theorem 4.12 hold under the additional assumptions. We do not know if they hold in the generality considered in our paper.

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arXiv2015

Spectral sets and distinguished varieties in the symmetrized bidisc

We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $Γ$ as a spectral set, there is a one dimensional complex algebraic variety $Λ$ in $Γ$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $Λ$. The variety $Λ$ is shown to have a particular determinantal representation, related to the so-called "fundamental operator" of the pair (S,P). When (S,P) is a strict $Γ$-contraction, then $Λ$ is a distinguished variety in the symmetrized bidisc, i.e., a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.

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arXiv2014

The isomorphism problem for complete Pick algebras: a survey

Complete Pick algebras - these are, roughly, the multiplier algebras in which Pick's interpolation theorem holds true - have been the focus of much research in the last twenty years or so. All (irreducible) complete Pick algebras may be realized concretely as the algebras obtained by restricting multipliers on Drury-Arveson space to a subvariety of the unit ball; to be precise: every irreducible complete Pick algebra has the form $M_V = \{f|_V : f \in M_d\}$, where $M_d$ denotes the multiplier algebra of the Drury-Arveson space $H^2_d$, and $V$ is the joint zero set of some functions in $M_d$. In recent years several works were devoted to the classification of complete Pick algebras in terms of the complex geometry of the varieties with which they are associated. The purpose of this survey is to give an account of this research in a comprehensive and unified way. We describe the array of tools and methods that were developed for this program, and take the opportunity to clarify, improve, and correct some parts of the literature.

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arXiv2014

Operator algebras for analytic varieties

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions $\mathcal M_V$ of the multiplier algebra $\mathcal M$ of Drury-Arveson space to a holomorphic subvariety $V$ of the unit ball $\mathbb{B}_d$. We find that $\mathcal M_V$ is completely isometrically isomorphic to $\mathcal M_W$ if and only if $W$ is the image of $V$ under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when $d<\infty$, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When $V$ and $W$ are each a finite union of irreducible varieties and a discrete variety in $\mathbb{B}_d$ with $d<\infty$, then an isomorphism between $\mathcal M_V$ and $\mathcal M_W$ determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-$*$ continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold---particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences.

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arXiv2014

Multipliers of embedded discs

We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of $\ell^2$ which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of $\ell^2$ which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety $V$ in $\mathbb B_2$ such that the multiplier algebra is not all of $H^\infty(V)$. We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.

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arXiv2014

Essential normality, essential norms and hyperrigidity

Let $S = (S_1, \ldots, S_d)$ denote the compression of the $d$-shift to the complement of a homogeneous ideal $I$ of $\mathbb{C}[z_1, \ldots, z_d]$. Arveson conjectured that $S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary. The unital norm closed algebra $\mathcal{B}_I$ generated by $S_1,\ldots,S_d$ modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on $\overline{V} := \overline{\mathcal{Z}(I) \cap \mathbb{B}_d}$, where $\mathcal{Z}(I)$ is the variety corresponding to $I$. Consequently, the essential norm of an element in $\mathcal{B}_I$ is equal to the sup norm of its Gelfand transform, and the C*-envelope of $\mathcal{B}_I$ is identified as the algebra of continuous functions on $\overline{V} \cap \partial \mathbb{B}_d$, which means it is a complete invariant of the topology of the variety determined by $I$ in the ball. Motivated by this determination of the C*-envelope of $\mathcal{B}_I$, we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple $S$ is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arveson's notion of a boundary representation. We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal $I$ can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the $d$-shift.

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arXiv2013

On the isomorphism question for complete Pick multiplier algebras

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $\mv = \{f\big|_V : f \in \cM_d\}$, where $d$ is some integer or $\infty$, $\cM_d$ is the multiplier algebra of the Drury-Arveson space $H^2_d$, and $V$ is a subvariety of the unit ball. For finite $d$ it is known that, under mild assumptions, every isomorphism between two such algebras $\mv$ and $\mw$ is induced by a biholomorphism between $W$ and $V$. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where $V$ is the proper image of a finite Riemann surface. The second deals with the case where $V$ is a disjoint union of varieties.

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arXiv2012

Essential normality and the decomposability of algebraic varieties

We consider the Arveson-Douglas conjecture on the essential normality of homogeneous submodules corresponding to algebraic subvarieties of the unit ball. We prove that the property of essential normality is preserved by isomorphisms between varieties, and we establish a similar result for maps between varieties that are not necessarily invertible. We also relate the decomposability of an algebraic variety to the problem of establishing the essential normality of the corresponding submodule. These results are applied to prove that the Arveson-Douglas conjecture holds for submodules corresponding to varieties that decompose into linear subspaces, and varieties that decompose into components with mutually disjoint linear spans.

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arXiv2011

Dilation theory in finite dimensions: the possible, the impossible and the unknown

This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These results can be used to give very elementary proofs of sharpened versions of some von Neumann type inequalities, as well as some other striking consequences about polynomials and matrices. Exploring the limits of the finite dimensional approach sheds light on the difference between those techniques and phenomena in operator theory that are inherently infinite dimensional, and those that are not.

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arXiv2011

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.

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arXiv2011

Unitary N-dilations for tuples of commuting matrices

We show that whenever a contractive $k$-tuple $T$ on a finite dimensional space $H$ has a unitary dilation, then for any fixed degree $N$ there is a unitary $k$-tuple $U$ on a finite dimensional space so that $q(T) = P_H q(U) |_H$ for all polynomials $q$ of degree at most $N$.

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arXiv2011

Stable polynomial division and essential normality of graded Hilbert modules

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1, ..., f_k\}$ such that every $h \in M$ can be written as $h = \sum_i a_i f_i$ for some polynomials $a_i$ such that $\sum \|a_i f_i\| \leq C\|h\|$. We show that certain classes of modules have this property, and that the stable decomposition $h = \sum a_i f_i$ may be obtained by carefully applying techniques from computational algebra. We show that when the algebra of polynomials in $d$ variables is given the natural $\ell^1$ norm, then every ideal is linearly equivalent to an ideal that has the stable division property. We then show that a module $M$ that has the stable division property (with respect to the appropriate norm) is $p$-essentially normal for $p > \dim(M)$, as conjectured by Douglas. This result is used to give a new, unified proof that certain classes of graded submodules are essentially normal. Finally, we reduce the problem of determining whether all graded submodules of the $d$-shift Hilbert module are essentially normal, to the problem of determining whether all ideals generated by quadratic scalar valued polynomials are essentially normal.

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