Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 318-325
Author(s):  
Nadia Mesbah ◽  
Hadia Messaoudene ◽  
Asma Alharbi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Generalized Derivation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Similarity Orbit

Abstract Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

Generalized Derivation and Double Operator Integrals

2005 ◽  
Vol 12 (4) ◽  
pp. 717-726
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Generalized Derivation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional

Abstract Let 𝐻 be a separable infinite dimensional complex Hilbert space, and let 𝔹(𝐻) denote the algebra of all bounded linear operators on 𝐻. Let 𝐴, 𝐵 be operators in 𝔹(𝐻). We define the generalized derivation δ 𝐴, 𝐵 : 𝔹(𝐻) ↦ 𝔹(𝐻) by δ 𝐴, 𝐵(𝑋) = 𝐴𝑋 – 𝑋𝐵. In this paper we consider the question posed by Turnsek [Publ. Math. Debrecen 63: 293–304, 2003], when ? We prove that this holds in the case where 𝐴 and 𝐵 satisfy the Fuglede–Putnam theorem. Finally, we apply the obtained results to double operator integrals.

Generalized D-symmetric Operators II

2011 ◽  
Vol 54 (1) ◽  
pp. 21-27 ◽  
Author(s):  
S. Bouali ◽  
M. Ech-chad
Keyword(s):  
Hilbert Space ◽  
Generalized Derivation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Symmetric Operators

AbstractLet H be a separable, infinite-dimensional, complex Hilbert space and let A, B ∈ ℒ(H), where ℒ(H) is the algebra of all bounded linear operators on H. Let δAB : ℒ(H) → ℒ(H) denote the generalized derivation δAB(X) = AX – XB. This note will initiate a study on the class of pairs (A, B) such that .

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

Operators with Compact Self-Commutator

1974 ◽  
Vol 26 (1) ◽  
pp. 115-120 ◽  
Author(s):  
Carl Pearcy ◽  
Norberto Salinas
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Open Problems ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Calkin Algebra ◽  
The Ideal

Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.Some open problems in the theory of extensions of C*-algebras (cf. [1]) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

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 A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

Operators of the form PAQ-QAP

1968 ◽  
Vol 20 ◽  
pp. 1353-1361 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

In this note the Hilbert spaces under consideration are complex, and the operators referred to are bounded, linear operators. If is a Hilbert space, then the algebra of all operators on is denoted by .It is known (1) that if is any Hilbert space, then the class of commutators on , i.e., the class of all operators that can be written in the form PQ — QP for some , can be exactly described. A similar problem is that of characterizing all operators on that can be written in the form for some .

scholarly journals Two characterisations of additive *-automorphisms of B(H)

1996 ◽  
Vol 53 (3) ◽  
pp. 391-400 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Preserver Problems ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this paper we give two necessary and sufficient conditions for an additive bijection of B(H) to be a *-automorphism. Both of the results in the paper are related to the so-called preserver problems.

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