Remarks on &amp;lt;i&amp;gt;A&amp;lt;/i&amp;gt;-skew-adjoint, &amp;lt;i&amp;gt;A&amp;lt;/i&amp;gt;-almost Similarity Equivalence and Other Operators in Hilbert Space

Isaiah Nalianya Sitati

doi:10.11648/j.pamj.20170603.12

Abstract

In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.

Highlights

In this paper, Hilbert space(s) or subspace(s) will be denoted by capital letters, and respectively and, etc denote bounded linear operators
Hilbert space operators have been discussed by many others like [5], [16], [18] and [20] among other scholars
It is known that every normal operator is quasinormal and every quasinormal operator is binormal

Summary

Introduction

Hilbert space(s) or subspace(s) will be denoted by capital letters, and respectively and , , etc denote bounded linear operators. Two linear operators ∈ ( ) and ∈ ( ) are said to be − .- .H1 equivalent (denoted ≅ ), if there exists an − .- 1 operator 2 ∈ < ( , ) such that 2 = 2 .(For more details on this equivalence see [9] and [12]). The following classes of bounded linear operators shall be defined in this paper: An operator ∈ ( ) is said to be: self- adjoint or Hermitian if ∗ = ( equivalently, if ' , (∀ ∈ ) A projection if I= and ∗ = unitary if ∗ = ∗ = isometric if ∗ = self-adjoint unitary or a symmetry if = ∗ = 78. It has to be noted that an -isometry whose range is dense in is an − .- 1

Basic Results

A-Almost Similarity of Operators

A-Skew Adjoint and A-Normal

Some Results on A-self adjoint and A-skew-adjoint Operators

Conclusion