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 TA T ∗ = 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.
Highlights
We consider a Hilbert space H with an additional semi inner product defined by a positive semidefinite operator A; namely ξ| η A = Aξ| η for every ξ ; η ∈ H
It must be observed from [2] and [3] that this additional structure induces an adjoint operation. This operation is defined for not every bounded linear operator on H, unless A is invertible.For those operators T which admit an adjoint with respect to | A, we choose one, denoted by T ∗ A, which has similar, but not identical, properties as the classical T ∗
Douglas [11], which is a key for some results of this paper.At the end of this section we give some characterizations of A-quasinormal operators inspired from [26] and [27]
Summary
Department of Mathematics, College of Science, Al Jouf University P.O. Box 2014 Al Jouf, Saudi Arabia Copyright c 2014 Sidi Hamidou Jah and Ould Ahmed Mahmoud Sid Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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