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.

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.

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.

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.

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.

On Quasi-Class A Operators

2013 ◽  
Vol 59 (1) ◽  
pp. 163-172
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Class A ◽  
Generalized Weyl’S Theorem

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). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

Essentially Convexoid Operators

1981 ◽  
Vol 33 (2) ◽  
pp. 257-274
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Convex Hull ◽  
Numerical Range ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Quotient Mapping ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Essentially Normal

Let H be a separable complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Let π be the quotient mapping from B(H) onto the Calkin algebra B(H)/K(H), where K(H) denotes all compact operators on B(H). An operator T ∈ B(H) is said to be convexoid[14] if the closure of its numerical range W(T) coincides with the convex hull co σ(T) of its spectrum σ(T). T ∈ B(H) is said to be essentially normal, essentially G1, or essentially convexoid if π(T) is normal, G1 or convexoid in B(H)/K(H) respectively.

scholarly journals A Formula for the Numerical Range of Elementary Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
M. Barraa
Keyword(s):  
Hilbert Space ◽  
Numerical Range ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Unitary Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Elementary Operators

Let BH be the algebra of bounded linear operators on a complex Hilbert space H. For k-tuples of elements of BH, A=A1,…,Ak and B=B1,…,Bk, let RA,B denote the elementary operator on BH defined by RA,BX=∑i=1kAiXBi. In this paper, we prove the following formula for the numerical range of RA,B: VRA,B,BBH=[∪U∈UHW(∑i=1kUAiU*Bi)-]-, where UH is the set of unitary operators.

scholarly journals Inequalities for the Schatten P-norm

1985 ◽  
Vol 26 (2) ◽  
pp. 141-143 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
The Ideal

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 K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1<p<∞, si(A)p <∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).

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