arXiv subjects

Orr Shalit

Publications and source records attributed to Orr Shalit.

At least 19 recordsLinked to original sources

arXiv2026

On the spectral radius of operator tuples

In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $\rho_{\mathcal{E}}$ on $d$-tuples of operators. The main goal of this paper is to elucidate how this spectral radius depends on the operator space structure. Let $V = (\mathbb{C}^d, \|\cdot\|_V)$ be a normed space and let $\mathcal{E}$ be a quantization of $V$. We show that for a commuting operator tuple $X$, the spectral radius depends only on the underlying normed space; more precisely, \[ \rho_{\mathcal{E}}(X) = \max\{ \|\lambda\|_V : \lambda \in \sigma(X)\}, \] where $\sigma(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $\rho_{\min(V)}(X) \neq \rho_{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $\rho_{\mathcal{E}_1}(X) = \rho_{\mathcal{E}_2}(X)$ for all tuples $X$ implies $\mathcal{E}_1 = \mathcal{E}_2$. We present two proofs of this result; a key ingredient in one of them is a characterization, of independent interest, of $\rho_{\mathcal{E}}(A)$ in terms of the invertibility domain of the linear pencil associated with $A$. Finally, we prove that if two operator spaces give rise to the same spectral radius function, then the algebras of locally uniformly bounded NC functions on the corresponding NC unit balls coincide.

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arXiv2025

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random $N{\times}N$ unitary matrices converges to $\sqrt2$ as $N \to \infty$ almost surely. Under the assumption that this is the case, we prove that the commuting dilation constant of an arbitrary pair of contractions is strictly smaller than $2$. Our experiments are based on a simple algorithm that we introduce for the purpose of computing dilation constants between tuples of matrices.

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arXiv2025

A spectral radius for matrices over an operator space

With every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$, we associate a spectral radius function $\rho_{\mathcal{E}}$ on $d$-tuples of operators. For a $d$-tuple $X = (X_1, \ldots, X_d) \in M_n(\mathbb{C}^d)$ of matrices we show that $\rho_{\mathcal{E}}(X)<1$ if and only if $X$ is jointly similar to a tuple in the open unit ball of $M_n(\mathcal{E})$, that is, there is an invertible matrix $S$ such that $\|S^{-1}X S\|_{M_n(\mathcal{E})}<1$, where $S^{-1} X S =(S^{-1} X_1 S, \ldots, S^{-1} X_d S)$. When $\mathcal{E}$ is the row operator space, for example, our spectral radius coincides with the joint spectral radius considered by Bunce, Popescu, and others, and we recover the condition for a tuple of matrices to be simultaneously similar to a strict row contraction. When $\mathcal{E}$ is the minimal operator space $\min(\ell^\infty_d)$, our spectral radius $\rho_{\mathcal{E}}$ is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that $\rho_{\mathcal{E}}(X)<1$ if and only if $X$ is simultaneously similar to a tuple of strict contractions. We show that for a nc rational function $f$ with descriptor realization $(A,b,c)$, the spectral radius $\rho_{\mathcal{E}}(A)<1$ if and only the domain of $f$ contains a neighborhood of the noncommutative closed unit ball of the operator space dual $\mathcal{E}^*$ of $\mathcal{E}$.

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arXiv2024

Weak-* and completely isometric structure of noncommutative function algebras

We study operator algebraic and function theoretic aspects of algebras of bounded nc functions on subvarieties of the nc domain determined by all levels of the unit ball of an operator space (nc operator balls). Our main result is the following classification theorem: under very mild assumptions on the varieties, two such algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are completely isometrically and weak-* isomorphic if and only if there is a nc biholomorphism between the varieties. For matrix spanning homogeneous varieties in injective operator balls, we further sharpen this equivalence, showing that there exists a linear isomorphism between the respective balls that maps one variety onto the other; in general, we show, the homogeneity condition cannot be dropped. We highlight some difficulties and open problems, contrasting with the well studied case of row ball.

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arXiv2024

Dilation distance and the stability of ergodic commutation relations

We revisit and generalize the notion of dilation distance ${\rm d_{D}}(u,v)$ between unitary tuples and study its relation to the natural Haagerup-Rørdam distance ${\rm d_{HR}}(u,v) = \inf\{\|π(u) - ρ(v)\|\}$, where the infimum is taken over all pairs of faithful representations $π\colon C^*(u) \to B(\mathcal{H})$, $ρ\colon C^*(v) \to B(\mathcal{H})$. We show that ${\rm d_{HR}}(u,v)\leq 10\operatorname{d_{rD}}(u,v)^{1/2}$, where ${\rm d_{rD}}(u,v)$ is a relaxed dilation distance, improving and extending earlier results. For an antisymmetric matrix $Θ$, we show via a concrete dilation construction that a tuple of unitaries $u$ that almost commutes according to $Θ$ (i.e., $\|u_\ell u_k - e^{i θ_{k,\ell}} u_k u_\ell\|$ is small) can be nearly dilated to a tuple of unitaries $v$ that commutes according to $Θ$ (i.e., $v_\ell v_k - e^{i θ_{k,\ell}} v_k v_\ell = 0$). We show that the dilation can be "reversed" by a second application of the dilation construction, which leads to a rotated version of the original tuple. Thus, a gauge invariant almost $Θ$-commuting unitary tuple can be approximated (in some faithful representation) by a $Θ$-commuting unitary tuple. Moreover, when $Θ$ is ergodic, a $Θ$-commuting tuple is shown to be {\em almost} gauge invariant, and it follows from the results above that these can be approximated in norm by $Θ$-commuting tuples. In particular, we obtain the following counterpart of Lin's theorem on almost commuting unitaries: if $q \in \mathbb{T}$ is {\em not} a root of unity, then for every $\varepsilon >0$ there exists $δ> 0$ such that for every pair of unitaries $u_1,u_2 \in B(\mathcal{H})$ for which $\|u_1 u_2 - qu_2 u_1\| < δ$, there exists two $q$-commuting unitaries $v_1, v_2 \in B(\mathcal{H} \otimes \ell^2)$ such that $\|v_i - u_i \otimes 1\| < \varepsilon$ ($i=1,2$).

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arXiv2023

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

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arXiv2022

Tensor algebras of subproduct systems and noncommutative function theory

We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly continuous noncommutative functions on a noncommutative homogeneous variety or, equivalently, when it is residually finite dimensional: this happens precisely when the closed homogeneous ideal associated to the subproduct system satisfies a Nullstellensatz with respect to the algebra of uniformly continuous noncommutative functions on the noncommutative closed unit ball. We show that - in contrast to the finite dimensional case - in the case of infinite dimensional fibers this Nullstellensatz may fail. Finally, we also resolve the isomorphism problem for tensor algebras of subproduct systems: two such tensor algebras are (isometrically) isomorphic if and only if their subproduct systems are isomorphic in an appropriate sense.

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arXiv2022

Bounded perturbations of the Heisenberg commutation relation via dilation theory

We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows.

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arXiv2021

von Neumann's inequality for row contractive matrix tuples

We prove that for all $n\in \mathbb{N}$, there exists a constant $C_{n}$ such that for all $d \in \mathbb{N}$, for every row contraction $T$ consisting of $d$ commuting $n \times n$ matrices and every polynomial $p$, the following inequality holds: \[ \|p(T)\| \le C_{n} \sup_{z \in \mathbb{B}_d} |p(z)| . \] We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason's problem cannot be solved contractively in $H^\infty(\mathbb{B}_d)$ for $d \ge 2$. Second, we prove that the multiplier algebra $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ of the weighted Dirichlet space $\mathcal{D}_a(\mathbb{B}_d)$ on the ball is not topologically subhomogeneous when $d \ge 2$ and $a \in (0,d)$. In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra $A(\mathcal{D}_a(\mathbb{B}_d))$ of $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball $\mathfrak{C}\mathfrak{B}_d$ that is levelwise uniformly continuous but not globally uniformly continuous.

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arXiv2021

Dilations of unitary tuples

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $*$-isomorphic copy of $cv$. This gives rise to a metric \[ d_D(u,v)=\log\max\{c(u,v),c(v,u)\} \] on the set of equivalence classes of $*$-isomorphic tuples of unitaries. We also consider the metric \[ d_{HR}(u,v)=\inf\left\{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u\textrm{ and } v'\sim v\right\}, \] and we show the inequality \[ d_{HR}(u,v)\leq K d_D(u,v)^{1/2}. \] Let $u_Θ$ be the universal unitary tuple $(u_1,\ldots,u_d)$ satisfying $u_\ell u_k=e^{iθ_{k,\ell}} u_k u_\ell$, where $Θ=(θ_{k,\ell})$ is a real antisymmetric matrix. We find that $c(u_Θ, u_{Θ'})\leq e^{\frac{1}{4}\|Θ-Θ'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $Θ\mapsto U(Θ)\in B(H)^d$ such that $U(Θ)\sim u_Θ$ and \[ \|U(Θ)-U({Θ'})\|\leq K\|Θ-Θ'\|^{1/2}. \] Of special interest are: the universal $d$-tuple of noncommuting unitaries ${\mathrm u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds \[ 2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}. \] From this, we recover Passer's upper bound for the universal unitaries $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.

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arXiv2020

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are "close" to one another if and only if their multiplier algebras are "close", and that this happens if and only if the underlying point-sets are "almost congruent", meaning that one of the sets is very close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.

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arXiv2020

On the matrix range of random matrices

This note treats a simple minded question: what does a typical random matrix range look like? We study the relationship between various modes of convergence for tuples of operators, on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric, on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C*-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.

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arXiv2020

Dilations of $q$-commuting unitaries

Let $q = e^{i θ} \in \mathbb{T}$ (where $θ\in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the optimal constant $c = c_θ$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that \[ c_θ= \frac{4}{\|u_θ+u_θ^*+v_θ+v_θ^*\|}, \] where $u_θ, v_θ$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q'$-commuting unitaries. The techniques that we develop allow us to give new and simple "dilation theoretic" proofs of well known results regarding the continuity of the field of rotations algebras. In particular, for the so-called "Almost Mathieu Operator" $h_θ= u_θ+u_θ^*+v_θ+v_θ^*$, we recover the fact that the norm $\|h_θ\|$ is a Lipshitz continuous function of $θ$, as well as the result that the spectrum $σ(h_θ)$ is a $\frac{1}{2}$-Hölder continuous function in $θ$ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every selfadjoint $*$-polynomial $p(u_θ,v_θ)$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

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arXiv2020

CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond

These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond. While exisiting work on the two- and d-parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and-only-if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS- (respectively, the Arveson-Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong - but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?

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arXiv2020

Dilation theory: a guided tour

Dilation theory is a paradigm for studying operators by way of exhibiting an operator as a compression of another operator which is in some sense well behaved. For example, every contraction can be dilated to (i.e., is a compression of) a unitary operator, and on this simple fact a penetrating theory of non-normal operators has been developed. In the first part of this survey, I will leisurely review key classical results on dilation theory for a single operator or for several commuting operators, and sample applications of dilation theory in operator theory and in function theory. Then, in the second part, I will give a rapid account of a plethora of variants of dilation theory and their applications. In particular, I will discuss dilation theory of completely positive maps and semigroups, as well as the operator algebraic approach to dilation theory. In the last part, I will present relatively new dilation problems in the noncommutative setting which are related to the study of matrix convex sets and operator systems, and are motivated by applications in control theory. These problems include dilating tuples of noncommuting operators to tuples of commuting normal operators with a specified joint spectrum. I will also describe the recently studied problem of determining the optimal constant $c = c_{θ,θ'}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{iθ} UV$ can be dilated to a pair of $cU', cV'$, where $U',V'$ are unitaries that satisfy the commutation relation $V'U' = e^{iθ'} U'V'$. The solution of this problem gives rise to a new and surprising application of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.

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arXiv2018

Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem

Given a noncommutative (nc) variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we consider the algebra $H^\infty(\mathfrak{V})$ of bounded nc holomorphic functions on $\mathfrak{V}$. We investigate the problem of when two algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are isomorphic. We prove that these algebras are weak-$*$ continuously isomorphic if and only if there is an nc biholomorphism $G : \widetilde{\mathfrak{W}} \to \widetilde{\mathfrak{V}}$ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form $f \mapsto f \circ G$, where $G$ is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras $H^\infty(\mathfrak{B}_d)$ studied by Davidson--Pitts and by Popescu. In particular, we find that $\operatorname{Aut}(H^\infty(\mathfrak{B}_d))$ is a proper subgroup of $\operatorname{Aut}(\widetilde{\mathfrak{B}}_d)$. When $d<\infty$ and the varieties are homogeneous, we remove the weak-$*$ continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a bi-Lipschitz nc biholomorphism between the similarity envelopes of the nc varieties. We provide two proofs. In the noncommutative setting, our main tool is the noncommutative spectral radius, about which we prove several new results. In the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case.

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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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arXiv2017

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

We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given a nc variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we identify the algebra of bounded analytic functions on $\mathfrak{V}$ --- denoted $H^\infty(\mathfrak{V})$ --- as the multiplier algebra $\operatorname{Mult} \mathcal{H}_{\mathfrak{V}}$ of a certain reproducing kernel Hilbert space $\mathcal{H}_{\mathfrak{V}}$ consisting of nc functions on $\mathfrak{V}$. We find that every such algebra $H^\infty(\mathfrak{V})$ is completely isometrically isomorphic to the quotient $H^\infty(\mathfrak{B}_d)/ \mathcal{J}_{\mathfrak{V}}$ of the algebra of bounded nc holomorphic functions on the ball by the ideal $\mathcal{J}_{\mathfrak{V}}$ of bounded nc holomorphic functions which vanish on $\mathfrak{V}$. We investigate the problem of when two algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are isometrically isomorphic. If the variety $\mathfrak{W}$ is the image of $\mathfrak{V}$ under a nc analytic automorphism of $\mathfrak{B}_d$, then $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are (completely) isometrically isometric. 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 isometrically isomorphic, then there must be nc holomorphic maps between the varieties. 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. We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of $\mathfrak{B}_d$.

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