Positive-normal operators in semi-Hilbertian spaces

Abstract

Given a bounded positive linear operator A on a Hilbert space H we consider the semi-Hilbertian space (H , | A ), where ξ | η A := Aξ | η . In this paper we introduce a class of operators on a semi Hilbertian space H with inner product | A . We call the elements of this class A-positive-normal or A-posinormal. An operator T ∈ B(H) is said to be A-posinormal if there exists a A-positive operator P ∈ B(H) (i.e., AP ≥ 0) such that T AT * = T * AP T. We study some basic properties of these operators. Also we study the relationship between a special case of this class with the other kinds of classes of operators in semi-Hilbertian spaces.Mathematics Subject Classification: Primary 47A05, 47A30 Secondary 47A62

(n,m) -Power Quasi Normal Operators in Semi- Hilbertian Spaces

“…The set of all ‫-ܣ‬ bounded operators which admit an ‫-ܣ‬adjoint is denoted by ‫ܤ‬ ‫. )ܪ(‬ By Douglas theorem [2,8] we have that ‫ܤ‬ ‫)ܪ(‬ = {ܶ ∈ ‫ܶ(ܴ/)ܪ(ܤ‬ * ‫})ܣ(ܴ‪)ϲ‬ܣ‬…”

“…2. ‫ܣ(‬ ௧ ) ♯ = ‫ܣ‬ ௧ for every ‫ݐ‬ > 0, if ‫ܣ‬ ∈ ‫)ܪ(ܤ‬ ା and if ‫ܶܣ‬ = ‫ܣܶ‬ then ܶ ♯ = ܲܶ * , here ܲ = ܲ ோ() തതതതതതത [8]. The classes of normal, quasi normal, n-quasi normal, isometries, partial isometries, etc., on Hilbert Spaces have been generalized to semi-Hilbertain spaces by many authors in many papers.…”