Metric Properties of Projections in Semi-Hilbertian Spaces

Abstract

Several results on norms of projections on a Hilbert space H are extended for the operator seminorm defined by a positive semidefinite operator A ∈ L(H) + . Mathematics Subject Classification (2000). Primary 46C05; Secondary 47A05, 47A30.

A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications

“…where ω( T ) is the numerical radius of T and ω A (T ) is the A-numerical radius of T , defined as below. [2,3,14]. It is useful to recall that an operator T is called A-self-adjoint if AT is selfadjoint (i.e.…”

Spectral radius of semi-Hilbertian space operators and its applications

“…For the sake of completeness, we will give examples that show that the above inclusions are in general strict. For an account of results, we refer to [1,2,2] and the references therein.…”

Positive-normal operators in semi-Hilbertian spaces

“…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. However, 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 * .…”

“…These classes of operators have been generalized to semi -Hilbertian spaces by many authors. Such operators appear in different contexts in [2], [3], [4], [14], [28], [29], [32] and other papers. The aim of this work is to continue this process of generalization to posinormal operators.…”

“…The aim of this work is to continue this process of generalization to posinormal operators. The inspiration for our investigation comes from [2], [3], [4], [14], [28], [29], [32]. In this paper section 2 contains basic results on A-operators.…”