scholarly journals Jordan {g,h}-derivations on triangular algebras

2020 ◽  
Vol 18 (1) ◽  
pp. 894-901
Author(s):  
Liang Kong ◽  
Jianhua Zhang
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Necessary Condition ◽  
Nest Algebra ◽  
Triangular Algebras ◽  
Sufficient And Necessary Condition ◽  
Complex Separable Hilbert Space

Abstract In this article, we give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on \tau ({\mathscr{N}}) is a {g,h}-derivation if and only if \dim {0}_{+}\ne 1 or \dim {H}_{-}^{\perp }\ne 1 , where {\mathscr{N}} is a non-trivial nest on a complex separable Hilbert space H and \tau ({\mathscr{N}}) is the associated nest algebra.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

scholarly journals Characterizations of Double Commutant Property on B H

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Chaoqun Chen ◽  
Fangyan Lu ◽  
Cuimei Cui ◽  
Ling Wang
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Necessary Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Double Commutant ◽  
Sufficient And Necessary Condition

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

scholarly journals Perturbation of the spectra of complex symmetric operators

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 191-199
Author(s):  
Qinggang Bu ◽  
Cun Wang
Keyword(s):  
Hilbert Space ◽  
Symmetric Operator ◽  
Separable Hilbert Space ◽  
Symmetric Matrix ◽  
Compact Perturbation ◽  
Complex Symmetric Operator ◽  
Single Valued Extension Property ◽  
Complex Symmetric ◽  
Symmetric Operators ◽  
Complex Separable Hilbert Space

An operator T on a complex Hilbert space H is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for H. This paper focuses on the perturbation theory for the spectra of complex symmetric operators. We prove that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and having the single-valued extension property. Also it is proved that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and satisfying generalized Weyl?s theorem.

scholarly journals Preservers for the Tsallis Entropy of Convex Combinations of Density Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xiaochun Fang ◽  
Yihui Lao
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Tsallis Entropy ◽  
Unitary Equivalence ◽  
Density Operators ◽  
Complex Separable Hilbert Space

Let H be a complex separable Hilbert space; we first characterize the unitary equivalence of two density operators by use of Tsallis entropy and then obtain the form of a surjective map on density operators preserving Tsallis entropy of convex combinations.

scholarly journals A New Exceptional Family of Elements and Solvability of General Order Complementarity Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Na Huang ◽  
Changfeng Ma
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Complementarity Problems ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Exceptional Family Of Elements ◽  
Exceptional Family ◽  
General Order ◽  
Sufficient And Necessary Condition

By using the concept of exceptional family, we propose a sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in Németh, 2010 (Theorem 3.1). Then we study some sufficient conditions for the nonexistence of exceptional family for GOCP in Hilbert space. Moreover, we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone general order complementarity problems.

Obstructions for semigroups of partial isometries to be self-adjoint

2016 ◽  
Vol 161 (1) ◽  
pp. 107-116
Author(s):  
JANEZ BERNIK ◽  
ALEXEY I. POPOV
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Finite Multiplicity ◽  
Partial Isometries ◽  
Von Neumann ◽  
Infinite Multiplicity ◽  
Complex Separable Hilbert Space

AbstractIn this paper we study the following question: given a semigroup ${\mathcal S}$ of partial isometries acting on a complex separable Hilbert space, when does the selfadjoint semigroup ${\mathcal T}$ generated by ${\mathcal S}$ again consist of partial isometries? It has been shown by Bernik, Marcoux, Popov and Radjavi that the answer is positive if the von Neumann algebra generated by the initial and final projections corresponding to the members of ${\mathcal S}$ is abelian and has finite multiplicity. In this paper we study the remaining case of when this von Neumann algebra has infinite multiplicity and show that, in a sense, the answer in this case is generically negative.

Representation of Hilbert space operators by (nJ)-matrices

1957 ◽  
Vol 53 (2) ◽  
pp. 304-311 ◽  
Author(s):  
D. R. Smart
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Separable Hilbert Space ◽  
Jacobi Matrix ◽  
Hermitian Matrix ◽  
Infinite Matrix ◽  
Closed Linear Operator ◽  
Hilbert Space Operators ◽  
Complex Separable Hilbert Space

Introduction. Let be the complex separable Hilbert space. We say that the closed linear operator T, with domain dense in. , is represented by the infinite matrix H if T is the operator T˜1(H) defined† by H (with respect to some complete orthonormal set). We define an (nJ)-matrix as a Hermitian matrix H = [hij]i, j ≥ 1 for which hij = 0 when i − j > n and hij ╪ 0 when i − j = n. (Thus a Jacobi matrix is a (1J)-matrix.) If, in addition, hij = 0 when 0 < i − j < n, we call H an (nJ ┴)-matrix.

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

Sign in / Sign up

Export Citation Format

Share Document