On almost similarity and other related equivalence relations of operators in Hilbert spaces

Abstract

In this thesis, we study unitary equivalence, similarity, quasisimilarity, almost similarity and metric equivalence of operators acting on separable Hilbert spaces. We also study the Murray-von Neumann relation of projections and other equivalence relation of operators in Hilbert spaces. We study the relation between equivalence classes of bounded linear op- erators with respect to di erent properties such as being self-adjoint, projections, normal, unitary and having speci c rank. We will investigate the spectral picture, norms, spectral radii, numerical range, lattice of their invariant subspaces, hyperinvariant subspaces and reducing subspaces of almost similar operators and metrically equivalent operators. Simi- larly, we characterize near equivalence, Murray-von Neumann equivalence, stable unitary equivalence and stable similarity of operators