Tuples Of Operators Research Articles

Isometric Dilations of Commuting Contractions and Brehmer Positivity

It is well-known that an n-tuple $$(n\ge 3)$$ of commuting contractions does not posses an isometric dilation, in general. Considering a class of n-tuple of commuting contractions satisfying certain positivity assumption, we construct their isometric dilations and consequently establish their von Neumann inequality. The positivity assumption is related to Brehmer positivity and motivated by the study of isometric dilations of operator tuples in Barik et al. (Trans Amer Math Soc 372(2):1429–1450, 2019).

A distance formula for tuples of operators

For a tuple of operators A=(A1,…,Ad), dist(A,CdI) is defined as minz∈Cd⁡‖A−zI‖ and varx(A) as ‖Ax‖2−∑j=1d|〈x|Ajx〉|2. For a tuple A of commuting normal operators, it is known thatdist(A,CdI)2=sup‖x‖=1⁡varx(A). We give an expression for the maximal joint numerical range of a tuple of doubly commuting matrices. Consequently, we obtain that the above distance formula holds for tuples of doubly commuting matrices. We also discuss some general conditions on the tuples of operators for this formula to hold. As a result, we obtain that it holds for tuples of Toeplitz operators as well.

Sectorial Tuples of Operators and Quasilinear Fractional Equations with Multi-Term Linear Part

Sectorial Tuples of Operators and Quasilinear Fractional Equations with Multi-Term Linear Part

Disjoint and simultaneously hypercyclic pseudo-shifts

We characterize disjoint and simultaneously hypercyclic tuples of unilateral pseudo-shift operators on ℓp(N). As a consequence, complementing the results of Bernal and Jung, we give a characterization for simultaneously hypercyclic tuples of unilateral weighted shifts. We also give characterizations for unilateral pseudo-shifts that satisfy the Disjoint and Simultaneous Hypercyclicity Criterions. Contrary to the disjoint hypercyclicity case, tuples of weighted shifts turn out to be simultaneously hypercyclic if and only if they satisfy the Simultaneous Hypercyclicity Criterion.

Homogeneous Hermitian holomorphic vector bundles and operators in the Cowen-Douglas class over the poly-disc

In this article, we obtain two sets of results. The first set of results are for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc.A classification of irreducible hermitian holomorphic vector bundles over D2, homogeneous with respect to Möb×Möb, is obtained assuming that the associated representations are multiplicity-free. Among these the ones that give rise to an operator in the Cowen-Douglas class of D2 of rank 1,2 or 3 are determined.Any hermitian holomorphic vector bundle of rank 2 over Dn, homogeneous with respect to the n-fold direct product of the group Möb is shown to be a tensor product of n hermitian holomorphic vector bundles over D. Among them, n−1 are shown to be the line bundles and one is shown to be a rank 2 bundle. Also, each of the bundles are homogeneous with respect to Möb.The classification of irreducible homogeneous hermitian holomorphic vector bundles over D2 of rank 3 (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of Dn, n>2.It is shown that there is no irreducible n - tuple of operators in the Cowen-Douglas class B2(Dn) that is homogeneous with respect to Aut(Dn), n>1. Also, pairs of operators in B3(D2) homogeneous with respect to Aut(D2) are produced, while it is shown that no n - tuple of operators in B3(Dn) is homogeneous with respect to Aut(Dn), n>2.

Higher dimensional [m,C]-isometric commuting d-tuple of operators

In this paper we recover an [m,C]-isometric operators and (m,C)-isometric commuting tuples of operators on a Hilbert space studied respectively in [11] and [16], we introduce the class of [m,C]-isometries for tuple of commuting operators. This is a generalization of the class of [m,C]-isometric commuting operators on a Hilbert spaces. A commuting tuples of operators S = (S1,..., Sp) ? B(H)p is said to be [m,C]-isometric p-tuple of commuting operators if ?m (S,C):= ?m j=0 (?1)m?j (m j) (? |?|=j j!/?! CS?CS?)=0 for some positive integer m and some conjugation C. We consider a multi-variable generalization of these single variable [m,C]-isometric operators and explore some of their basic properties.

(m,∞)-expansive and (m,∞)-contractive commuting tuple of operators on a Banach space

For a d-tuple of commuting operators S := (S1,..., Sd) ? B[X]d, m ? N and p ? (0,?), we define Q(p) m (S;u) := ? 0?k?m (-1)k (m k) (???Nd0 |?| = k k!/? ||S?u||p). As a natural extension of the concepts of (m,p)-expansive and (m,p)-contractive for tuple of commuting operators, we introduce and study the concepts of (m,?)-expansive tuple and (m,?)-contractive tuple of commuting operators acting on a Banach space. We say that S is (m,?)-expansive d-tuple (resp. (m,?)- contractive d-tuple) of operators if Q(p)m (S;u) ? 0 ? u ? X and p ? ? (resp. Q(p)m (S;u) ? 0 ? u ? X and p ? ?) . These concepts extend the definition of (m,?)-isometric tuple of bounded linear operators acting on Banach spaces was introduced and studied in [13].

$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

Let Ω be an irreducible bounded symmetric domain of rank r in ℂd. Let $$\mathbb{K}$$ be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group $$\mathbb{K}$$ consisting of linear transformations acts naturally on any d-tuple T = (T1, …, Td) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is $$\mathbb{K}$$ -homogeneous. In this paper, we obtain a model for a certain class of $$\mathbb{K}$$ -homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, …, zd on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator $$\sum\nolimits_{i = 1}^d {T_i^\ast {T_i}} $$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.

Disjoint and simultaneous hypercyclic Rolewicz-type operators

We characterize disjoint hypercyclic and supercyclic tuples of unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$, which are a generalization of the unilateral backward shift operator. We show that disjoint hypercyclicity and disjoint supercyclicity are equivalent among a subfamily of these operators and disjoint hypercyclic unilateral Rolewicz-type operators always satisfy the Disjoint Hypercyclicity Criterion. We also characterize simultaneous hypercyclic unilateral Rolewicz-type operators on $c_0(\N)$ and $\ell^p(\N)$, $p \in [1, \infty)$.

Specht's invariant and localization of operator tuples

In this paper we give a Specht-type characterization for unitary equivalence of operator tuples in the Cowen-Douglas class via their point-wise localizations.

Asymptotic relative submajorization of multiple-state boxes

Pairs of states, or “boxes” are the basic objects in the resource theory of asymmetric distinguishability (Wang and Wilde in Phys Rev Res 1(3):033170, 2019. https://doi.org/10.1103/PhysRevResearch.1.033170), where free operations are arbitrary quantum channels that are applied to both states. From this point of view, hypothesis testing is seen as a process by which a standard form of distinguishability is distilled. Motivated by the more general problem of quantum state discrimination, we consider boxes of a fixed finite number of states and study an extension of the relative submajorization preorder to such objects. In this relation, a tuple of positive operators is greater than another if there is a completely positive trace nonincreasing map under which the image of the first tuple satisfies certain semidefinite constraints relative to the other one. This preorder characterizes error probabilities in the case of testing a composite null hypothesis against a simple alternative hypothesis, as well as certain error probabilities in state discrimination. We present a sufficient condition for the existence of catalytic transformations between boxes, and a characterization of an associated asymptotic preorder, both expressed in terms of sandwiched Rényi divergences. This characterization of the asymptotic preorder directly shows that the strong converse exponent for a composite null hypothesis is equal to the maximum of the corresponding exponents for the pairwise simple hypothesis testing tasks.

On the matrix range of random matrices

This note treats a simple minded question: \textit{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.

In search of convexity: diagonals and numerical ranges

We show that the set of all possible constant diagonals of a bounded Hilbert space operator is always convex. This, in particular, answers an open question of J.-C. Bourin ($2003$). Moreover, we show that the joint numerical range of a commuting operator tuple is in general not convex, which fills a gap in the literature. We also prove that the Asplund-Ptak numerical range (which is convex for pairs of operators) is, in general, not convex for tuples of operators.

Automorphisms and the fundamental operators associated with the symmetrized tridisc

The automorphisms of the symmetrized polydisc $${\mathbb {G}}_n$$ are well-known and are given in the coordinates of the polydisc in Edigarian and Zwonek (Arch. Math. 84 (2005) 364–374). We find an explicit formula for the automorphisms of $${\mathbb {G}}_n$$ in its own coordinates. If $$\tau $$ is an automorphism of $${\mathbb {G}}_n$$ , then $$\tau (S_1,\dots ,S_{n-1},P)$$ is a $$\Gamma _n$$ -contraction, where a $$\Gamma _n$$ -contraction is a commuting n-tuple of Hilbert space operators for which the closed symmetrized polydisc $$\Gamma _n$$ is a spectral set. Corresponding to every $$\Gamma _n$$ -contraction $$(S_1,\dots ,S_{n-1},P)$$ , there exist $$n-1$$ unique operators $$A_1,\dots ,A_{n-1}$$ such that $$\begin{aligned} \qquad \qquad \qquad S_i-S_{n-i}^*P=D_PA_iD_P, \quad D_P=(I-P^*P)^{1/2}, \end{aligned}$$ for $$i=1,\dots , n-1$$ . This unique $$(n-1)$$ -tuple $$(A_1,\dots ,A_{n-1})$$ , which is called the fundamental operator tuple or $${\mathcal {F}}_O$$ -tuple of $$(S_1,\dots ,S_{n-1},P)$$ in the literature, plays central role in every section of operator theory on $$\Gamma _n$$ . We find an explicit form of the $${\mathcal {F}}_O$$ -tuple of $$\tau (S_1,\dots ,S_{n-1},P)$$ when $$n=3$$ . We show by an example that a $$\Gamma _n$$ -contraction may not have commuting $${\mathcal {F}}_O$$ -tuple. Also, we obtain a necessary and sufficient condition under which two $$\Gamma _n$$ -contractions are unitarily equivalent.

Joint numerical radius of spherical Aluthge transforms of tuples of Hilbert space operators

Let $\mathbf{T}=(T_1,\ldots,T_d)$ be a $d$-tuple of operators on a complex Hilbert space $\mathcal{H}$. The spherical Aluthge transform of $\mathbf{T}$ is the $d$-tuple given by $\widehat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\ldots,\sqrt{P}V_d\sqrt{P})$ where $P:=\sqrt{T_1^*T_1+\ldots+T_d^*T_d}$ and $(V_1,\ldots,V_d)$ is a joint partial isometry such that $T_k=V_k P$ for all $1 \le k \le d$. In this paper, we prove several inequalities involving the joint numerical radius and the joint operator norm of $\widehat{\mathbf{T}}$. Moreover, a characterization of the joint spectral radius of an operator tuple $\mathbf{T}$ via $n$-th iterated of spherical Aluthge transform is established.

Invariants of Symbols of the Linear Differential Operators

In this paper we classify the symbols of the linear differential operators of order $$k$$ , which act from the module $$C^{\infty}(\xi)$$ to the module $$C^{\infty}(\xi^{t})$$ , where $$\xi\colon E(\xi)\to M$$ is vector bundle over the smooth manifold $$M$$ , bundle $$\xi^{t}$$ is either $$\xi^{*}$$ with fiber $$E^{*}:=\textrm{Hom}(E,\mathbb{C})$$ or $$\xi^{\flat}$$ with fiber $$E^{\flat}:=\textrm{Hom}(E,\Lambda^{n}T^{*})$$ and $$C^{\infty}(\xi)$$ , $$C^{\infty}(\xi^{t})$$ are the modules of their smooth sections. To find invariants of the symbols we associate with every non-degenerated symbol the tuple of linear operators acting on space $$E$$ and reduce our problem to the classification of such tuples with respect to some orthogonal transformations. Using the results of C. Procesi, we find generators for the field of rational invariants of the symbols and in terms of these invariants provide a criterion of equivalence of non-degenerated symbols.

A generalized Fuglede–Putnam theorem for a class of almost normal tuples of operators with finite modulus of Hilbert–Schmidt quasitriangularity

We extend the Fuglede–Putnam theorem modulo the Hilbert– Schmidt class to almost normal m-tuples of operators with finite Hilbert– Schmidt modulus of quasitriangularity.

$$(n_1,\ldots ,n_p)$$-quasi-m-isometric commuting tuple of operators on a Hilbert space

Our aim in this paper is to consider a generalization of the concept of n-quasi-m-isometric operators of a single operator done in Mahmoud Sid Ahmed et al. (Results Math 73:511–531, 2018) and Mechri and Mahmoud Sid Ahmed (Oper Matrices 14(1): 145–157, 2020) to the multi-dimensional operators. We discuss the most interesting results concerning these classes of tuples of operators by extending some results obtained in several recent papers for a single operator.

The Wigner distribution of n arbitrary observables

We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.

Invariant Subspaces for Certain Tuples of Operators with Applications to Reproducing Kernel Correspondences

The techniques developed by Popescu, Muhly-Solel and Good for the study of algebras generated by weighted shifts are applied to generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning dilations and invariant subspaces for commuting tuples of operators. In that paper the authors prove Beurling-Lax-Halmos type results for commuting tuples $$T=(T_1,\ldots ,T_d)$$ of operators that are contractive and pure; that is $$\Phi _T(I)\le I$$ and $$\Phi _T^n(I)\searrow 0$$ where $$\begin{aligned} \Phi _T(a)=\Sigma _i T_iaT_i^*. \end{aligned}$$ Here we generalize some of their results to commuting tuples T satisfying similar conditions but for $$\begin{aligned} \Phi _T(a)=\Sigma _{\alpha \in {\mathbb {F}}^+_d} x_{|\alpha |}T_{\alpha }aT_{\alpha }^* \end{aligned}$$ where $$\{x_k\}$$ is a sequence of non negative numbers satisfying some natural conditions (where $$T_{\alpha }=T_{\alpha (1)}\cdots T_{\alpha (k)}$$ for $$k=|\alpha |$$ ). In fact, we deal with a more general situation where each $$x_k$$ is replaced by a $$d^k\times d^k$$ matrix. We also apply these results to subspaces of certain reproducing kernel correspondences $$E_K$$ (associated with maps-valued kernels K) that are invariant under the multipliers given by the coordinate functions.