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

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: ∥AX−XB−I∥≥1,for allX∈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.

Keywords

• finite operator
• numerical range
• orthogonality
• class ℛ¯1
• 47b47
• 47a30
• 47a12