scholarly journals A Weierstrass theorem for a complex separable Hilbert space

1975 ◽  
Vol 15 (1) ◽  
pp. 18-22 ◽  
Author(s):  
Milton Chaika ◽  
S.J Perlman
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Weierstrass Theorem ◽  
Complex Separable Hilbert Space

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.

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.

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 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.

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

Dense Subalgebras of Left Hilbert Algebras

1982 ◽  
Vol 34 (6) ◽  
pp. 1245-1250 ◽  
Author(s):  
A. van Daele
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Quantum Field ◽  
Von Neumann ◽  
Hilbert Algebras

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

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