Range Kernel Orthogonality and Finite Operators

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 1,  2015, pp.63-71
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.1.63
Title & Authors
Range Kernel Orthogonality and Finite Operators
Mecheri, Salah; Abdelatif, Toualbia;

Abstract
Let H be a separable infinite dimensional complex Hilbert space, and let $\small{\mathcal{L}(H)}$ denote the algebra of all bounded linear operators on H into itself. Let $\small{A,B{\in}\mathcal{L}(H)}$ we define the generalized derivation $\small{{\delta}_{A,B}:\mathcal{L}(H){\mapsto}\mathcal{L}(H)}$ by $\small{{\delta}_{A,B}(X)=AX-XB}$, we note $\small{{\delta}_{A,A}={\delta}_A}$. If the inequality $\small{{\parallel}T-(AX-XA){\parallel}{\geq}{\parallel}T{\parallel}}$ holds for all $\small{X{\in}\mathcal{L}(H)}$ and for all $\small{T{\in}ker{\delta}_A}$, then we say that the range of $\small{{\delta}_A}$ is orthogonal to the kernel of $\small{{\delta}_A}$ in the sense of Birkhoff. The operator $\small{A{\in}\mathcal{L}(H)}$ is said to be finite [22] if $\small{{\parallel}I-(AX-XA){\parallel}{\geq}1(*)}$ for all $\small{X{\in}\mathcal{L}(H)}$, where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.
Keywords
Finite operator;dominant operator;p-hyponormal operator;Class Y;
Language
English
Cited by
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