Operators That Are Orthogonal to the Range of a Derivation (original) (raw)

On the range of a generalized derivation

Journal of Mathematical Sciences, 2000

A generalized derivation 5A,B : f~(9-O' , ]L(gT.), is defined by the formula ~A,B(X) : AX -XB, where A, B E L (gf.) and IL (J'O is the Banach algebra of bounded linear operators in a Hilbert space J-C. Sufficient conditions under which R ( SA,B ) A kerSA,B : {0} and R ( SA,B ) fq ker 5A*,B* = {0} a,'e given. Bibliography: 8 titles. w 1. Introduction Let L(9 0 be the algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space 5s For A,B E L(9~) we introduce an operator 5A~ on L(9s by the formula ~Ad~ (X) = AX -XB. In the case A = B, the operator 5A is called the inner derivation induced by A E L (9(.). As is known, if the dimensional is finite, then the set U{R(SA)Fq{A}' :A E ,~(9-C)} is exactly the set of nilpotent operators, where {A}' is the commutant of A and R(SA) is the range of ~A-This assertion is not true in the infinite-dimensional case. As was shown in [5], in some special cases, any operator in R(~A) n {A}' is a nilpotent operator. As was shown in [1, p. 136-137], R(~A) Q {A} t = {0} ifA E L(9s is a normal operator or an isometric operator, where R(SA) is the closure of R(SA) in the norm topology. As is known, R(SA) fq {A*}' = {0}

A Note on the Range of a Derivation

Revista Colombiana de Matemáticas

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A, B ∈ L(H), define the generalized derivation δA, B ∈ L(L(H)) by δA, B(X) = AX - XB. An operator A ∈ L(H) is P-symmetric if AT = TA implies AT* = T* A for all T ∈ C1(H) (trace class operators). In this paper, we give a generalization of P-symmetric operators. We initiate the study of the pairs (A, B) of operators A, B ∈ L(H) such that R(δA, B) W* = R(δA, B) W*, where R(δA, B) W* denotes the ultraweak closure of the range of δA, B. Such pairs of operators are called generalized P-symmetric. We establish a characterization of those pairs of operators. Related properties of P-symmetric operators are also given.

Range Kernel Orthogonality of Generalized Derivations

International Journal of Open Problems in Computer Science and Mathematics, 2012

We say that the operators A, B on Hilbert space satisfy the Fuglede-Putnam theorem if AX = XB for some X implies A * X = XB *. We show that if A is k−quasihyponormal and B * is an injective p−hyponormal operator, then A, B satisfy the Fuglede-Putnam theorem. As a consequence of this result, we obtain the range of the generalized derivation induced by the above classes of operators is orthogonal to its kernel.

Range-Kernel orthogonality and elementary operators on certain Banach spaces

2021

The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , and finally, we give a counterexample to Mecheri’s result given in this context.

of Inequalities in Pure and Applied Mathematics GÂTEAUX DERIVATIVE AND ORTHOGONALITY IN C 1-CLASSES

2005

The general problem in this paper is minimizing the Cp− norm of suitable affine mappings fromB(H) to Cp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.

Classes of Operator-Smooth Functions - II. Operator-Differentiable Functions

Integral Equations and Operator Theory, 2004

This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable and their link with the space of Operator Lipschitz functions. Apart from the standard operator norm on B(H), we consider a rich variety of spaces of Operator Differentiable and Operator Lipschitz functions with respect to symmetric operator norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of Operator Differentiable functions. We apply the obtained results to the study of the functions acting on the domains of closed *-derivations of C*-algebras and prove that Operator Differentiable functions act on all such domains. We also obtain the following modification of this result: any continuously differentiable, Operator Lipschitz function acts on the domains of all weakly closed *-derivations of C*-algebras.