Isometric Operators Research Articles

Impossibility of masking a set of quantum states of nonzero measure

We study the quantum information masking based on isometric linear operators that distribute the information encoded in pure states to the correlations in bipartite states. It is shown that a isometric linear operator can not mask any nonzero measure set of pure states. We present a geometric characterization of the maskable sets, and show that any maskable set must be on a spherical circle in certain Euclidean spaces. Detailed examples and potential applications in such as secret sharing and quantum cryptography are analyzed.

Composition operators on Hardy spaces of the homogenous rooted trees

In Muthukumar and Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017), the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space \(\mathbb {T}_{p}\) defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator \(C_\phi \) to be bounded on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces. Also, we discuss the compactness of \(C_\phi \) on \({\mathbb {T}}_{p}\) and finally prove there are no compact composition operators on \({\mathbb {T}}_{p,0}\) spaces.

Operators on positive semidefinite inner product spaces

Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form 〈⋅,⋅〉. If a linear operator A:U→U is bounded (i.e., ‖Au‖⩽c‖u‖ for some c∈R and all u∈U), then the subspace U0:={u∈U|〈u,u〉=0} is invariant, and so A defines the linear operators A0:U0→U0 and A1:U/U0→U/U0.Let A be an indecomposable bounded operator on U such that 0≠U0≠U. Let λ be an eigenvalue of A0. We prove that the algebraic multiplicity of λ in A1 is not less than the geometric multiplicity of λ in A0, and the geometric multiplicity of λ in A1 is not less than the number of Jordan blocks Jt(λ) of each fixed size t×t in the Jordan canonical form of A0.We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U.For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.

Isometric Composition Operators on Lipschitz Spaces

Given pointed metric spaces X and Y, we characterize the basepoint-preserving Lipschitz maps $$\phi $$ from Y to X inducing an isometric composition operator $$C_\phi $$ between the Lipschitz spaces $$\mathrm {Lip}_0(X)$$ and $$\mathrm {Lip}_0(Y)$$, whenever X enjoys the peak property. This gives an answer to a question posed by Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].

Classes of operators related to m-isometric operators

Isometries played a pivotal role in the development of operator theory, in particular with the theory of contractions and polar decompositions and has been widely studied due to its fundamental importance in the theory of stochastic processes, the intrinsic problem of modeling the general contractive operator via its isometric dilation and many other areas in applied mathematics. In this paper we present some properties of n-quasi-(m;C)-isometric operators. We show that a power of a n-quasi-(m;C)-isometric operator is again a n-quasi-(m;C)-isometric operator and some products and tens

Classes of operators related to isometries

In this paper, we introduce the classes of analytic extensions of isometric operators and F-quasi-isometric operators. We show that every analytic extension of isometric operator and F-quasi-isometric operator have scalar extensions. Some spectral properties are also presented.

Spectral properties for some extensions of isometric operators

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-isometric operators. We also prove that if T, \(S^{*}\) are 2-isometric and invertible isometric operators respectively and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\). Moreover, we show that every Riesz projection E with respect to a non-zero isolated spectral point \(\lambda \) of a 2-isometric operator T is self-adjoint and satisfies \({\mathcal {R}}(E) = {\mathcal {N}}(T-\lambda ) = {\mathcal {N}}(T- \lambda )^{*}\). Further, we show that quasi-2-isometric operator satisfies Bishop’s property (\(\beta \)). Finally, we prove Weyl type theorems for \(f(d_{TS})\), where \( d_{TS}\) denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and \(S^{*}\) and \(f \in H(\sigma (d_{TS}))\), the set of analytic functions which are defined on an open neighborhood of \(\sigma (d_{TS})\).

On n-quasi-[m,C]-isometric operators

For positive integers m and n, an operator T in B ( H ) is said to be an n-quasi-[m,C]-isometric operator if there exists some conjugation C such that . In this paper, some basic structural properties of n-quasi-[m,C]-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi-[m,C]-isometric operator is again an n-quasi-[m,C]-isometric operator. Moreover, we show that the class of n-quasi-[m,C]-isometric operators is norm closed. Finally, we examine the stability of n-quasi-[m,C]-isometric operator under perturbation by nilpotent operators commuting with T.

Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form

Let V be a vector space over a field F with scalar product given by a nondegenerate sesquilinear form whose matrix is diagonal in some basis. If F=C, then we give canonical matrices of isometric and selfadjoint operators on V using known classifications of isometric and selfadjoint operators on a complex vector space with nondegenerate Hermitian form. If F is a field of characteristic different from 2, then we give canonical matrices of isometric, selfadjoint, and skewadjoint operators on V up to classification of symmetric and Hermitian forms over finite extensions of F.

Operators from the Hardy space to the α‐Bloch space

AbstractLet be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space into the α‐Bloch space on . Also, we show the multiplication operator from into is bounded. Finally, we show that there exist no isometric composition operators.

Power bounded m-left invertible operators

A Hilbert space operator is left m-invertible by if S is m-isometric if and S is -isometric for some conjugation C of if If a power bounded operator S is left invertible by a power bounded operator T, then S (also, ) is similar to an isometry. Translated to m-isometric and -isometric operators S this implies that S is 1-isometric, equivalently isometric, and (respectively) -isometric.

On the dynamics of the $d$-tuples of $m$-isometries

A commuting $d$-tuple $T=(T_{1}, \ldots , T_{d})$ of bounded linear operators on a Hilbert space $\mathcal {H}$ is called a spherical $m$-isometry if $\sum _{j=0}^{m}(-1)^{j}\binom {m}{j}Q_{T}^{j}(I)=0$, where $I$ denotes the identity operator and $Q_{T}(A)=\sum _{i=1}^{d}T_{i}^{*}AT_{i}$ for every bounded linear operator $A$ on $\mathcal {H}$. Also, $T$ is called a toral $m$-isometry if $\sum _{p\in \mathbb {N}^{d},\, 0\leq p\leq n}(-1)^{\vert p\vert }\binom {n}{p}{T^{\ast }}^{p}T^{p}=0 $ for all $n\in \mathbb {N}^{d}$ with $\vert n\vert = m$. The present paper mainly focuses on the convex-cyclicity of the $d$-tuples of operators on a separable infinite-dimensional Hilbert space $\mathcal {H}$. In particular, we prove that spherical $m$-isometries are not convex-cyclic. Also, we show that toral and spherical $m$-isometric operators are never supercyclic.

Dominated and pointwise ergodic theorems with “weighted” averages for bounded Lamperti representations of amenable groups

Group representations by bounded Lamperti operators in the spaces Lα(1≤α<∞) form a wide class of representations, including representations by bounded positive operators and (when α≠2) representations by isometric operators. The Dominated and the Pointwise Ergodic Theorems (DET and PET) for Cesàro averages for the bounded Lamperti representations of amenable σ-compact locally compact groups in Lα(1<α<∞) were proved by A. Tempelman in Proc. Amer. Math. Soc. 143 (2015) 4989–5004. By using a completely different, functional-analytical method, developed by A. Shulman in his PhD thesis in 1988, we generalize this result to “weighted” averages of such representations and discuss various conditions on the “weights” under which the DET and the PET hold. We conclude with applications of the general results to the bounded Lamperti representations of groups of polynomial growth and of the groups Rm and Zm.

The properties of [∞,C]-isometric operators

In this paper we introduce the class of [?,C]-isometric operators and study various properties of this class. In particular, we show that if T is an [?,C]-isometric operator and Q is a quasi-nilpotent operator, then T + Q is an [?,C]-isometric operator under suitable conditions. Also, we show that the class of [?,C]-isometric operators is norm closed. Finally, we examine properties of products of [?,C]-isometric operators.

Some properties of (m,C)-isometric operators

In this paper, we study if T is an (m,C)-isometric operator and CT+C commutes with T, then T+ is an (m,C)-isometric operator. We also give local spectral properties and spectral relations of (m;C)-isometric operators, such as property (?), decomposability, the single-valued extension property and Dunford?s boundedness. We also investigate perturbation of (m,C)-isometric operators by nilpotent operators and by algebraic operators and give some properties.

Wold–Słociński decompositions for commuting isometric triples

Extending two remarkable results by von Neumann–Halmos–Wold (for isometric operators) and Słociński (for pairs of commuting isometries) we discuss the possibility to decompose a given commuting triple V = ( V 1 , V 2 , V 3 ) of isometric operators, acting on a Hilbert space H , into the direct sum between commuting triples consisting of unitary operators and/or unilateral shifts. We prove that such a decomposition exists if and only if the pairs ( V 1 V 2 , V 3 ) , ( V 2 V 3 , V 1 ) and ( V 1 V 3 , V 2 ) have decompositions of Wold–Słociński type. If only two of these pairs are supposed to have such a decomposition then the Wold–Słociński decomposition associated to V has seven summands. Several structure results, of geometric type, for these summands are also presented. Examples and counterexamples are used for illustrative purposes. Certain results are presented in full generality, i.e., for commuting isometric n -tuples.

On n-quasi-(m,C)-isometric operators

ABSTRACT In this paper, we introduce the class of n-quasi--isometric operator on a Hilbert space which is a variation of -isometric operators Chō M, Ko E, Lee JE. [On -isometric operators. Complex Anal Oper Theory. 2016;10:1679–1694]. An operator is said to be n-quasi--isometric if for some positive integers n and m. We study this class of such operators and give some of their basic properties.

Invariant subspaces and operator model theory on noncommutative varieties

Let $${{\mathcal {Q}}}\subset {{\mathbb {C}}}\left $$ be an arbitrary set of polynomials in noncommutative indeterminates such that $$q(0)=0$$ for all $$q\in {{\mathcal {Q}}}$$ . The noncommutative variety $$\begin{aligned} {{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}}):=\left\{ { X}=(X_1,\ldots , X_n)\in \mathbf{D}_f^m({{\mathcal {H}}}): \ q({ X})=0 \ \text { for all } q\in {{\mathcal {Q}}}\right\} , \end{aligned}$$ where $$\mathbf{D}_f^m({{\mathcal {H}}})$$ is a noncommutative regular domain in $$B({{\mathcal {H}}})^n$$ and $$B({{\mathcal {H}}})$$ is the algebra of bounded linear operators on a Hilbert space $${{\mathcal {H}}}$$ , admits a universal model $$B^{(m)}=(B_1^{(m)},\ldots , B_n^{(m)})$$ such that $$q(B^{(m)})=0$$ , $$q\in {{\mathcal {Q}}}$$ , acting on a model space which is a subspace of the full Fock space with n generators. In this paper, we obtain a Beurling type characterization of the joint invariant subspaces under the operators $$B_1^{(m)},\ldots , B_n^{(m)}$$ , in terms of partially isometric multi-analytic operators acting on model spaces. More generaly, a Beurling-Lax-Halmos type representation is obtained and used to parameterize the wandering subspaces of the joint invariant subspaces under $$B_1^{(m)}\otimes I_{{\mathcal {E}}},\ldots , B_n^{(m)}\otimes I_{{\mathcal {E}}}$$ , and to characterize when they are generating for the corresponding invariant subspaces. Similar results are obtained for any pure n-tuple $$(X_1,\ldots , X_n)$$ in the noncommutative variety $${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$$ . We characterize the elements in the noncommutative variety $${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$$ which admit characteristic functions, develop an operator model theory for the completely non-coisometric elements, and prove that the characteristic function is a complete unitary invariant for this class of elements. This extends the classical Sz.-Nagy–Foias functional model for completely non-unitary contractions, based on characteristic functions. Our results apply, in particular, when $${{\mathcal {Q}}}$$ consists of the noncommutative polynomials $$Z_iZ_j-Z_jZ_i$$ , $$i,j=1,\ldots , n$$ . In this case, the model space is a symmetric weighted Fock space, which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in $${{\mathbb {C}}}^n$$ , and the universal model is the n-tuple $$(M_{\lambda _1},\ldots , M_{\lambda _n})$$ of multipliers by the coordinate functions.

Triangular n-isometric operators

In this paper, we introduce the class of triangular n-isometric operators and study various properties. We show that every triangular n-isometric operator is subscalar of order 2n; in particular, every isometric operator is subscalar of order two. Consequently, if the spectrum of a triangular n-isometric operator T has a nonempty interior in , then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for triangular n-isometric operators. Some spectral properties of this class of operators are also presented.

On $$\varvec{(m,C)}$$ ( m , C ) -Isometric Commuting Tuples of Operators on a Hilbert Space

Inspired by recent works on (m, C)-isometric and [m, C]-isometric operators on Hilbert spaces studied respectively in Chō et al. (Complex Anal. Oper. Theory 10:1679–1694, 2016; Filomat 31:7, 2017), in this paper we introduce the class of (m, C)-isometries for tuple of commuting operators. This is a generalization of the class of (m, C)-isometric operators. A commuting tuples of operators $$\mathbf{\large T}=(T_1,\ldots ,T_d)\in {\mathcal {B}}^{(d)}({\mathcal {H}})$$ is said to be (m, C)-isometric tuple if $$\begin{aligned} {{\mathcal {Q}}}_{m}(\mathbf{T}):=\sum _{0\le k\le m}(-1)^{m-k}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( \sum _{|\beta |=k}\frac{k!}{\beta !}\mathbf{\large T}^{*\beta }C\mathbf{\large T}^{\beta }C\right) =0 \end{aligned}$$ for some positive integer m and some conjugation C. We consider a multivariable generalization of these single variable (m, C)-isometric operators and explore some of their basic properties.