ON ALMOST SIMILARITY AND SOME CLASSES OF OPERATORS IN HILBERT SPACES

Abstract

Similarity is a recent area of study on classes of operators in Hilbert spaces. The study of operators, under similarity and quasisimilarity concepts, motivated researchers to extend their research to almost similarity property which is undergoing current research. Almost similarity has been shown to be an equivalence relation and to preserve nullity and co nullity of operators. Though similarity preserves nontrivial subspaces and quasisimilarity preserves hyperinvariant subspaces of operators, there is scanty literature linking such conclusion to almost similarity. Properties of almost similarity on some classes of operators, namely, partial isometries, θ-operators, posinormal operators among others and conditions under which almost similarity gives equality of spectra remain open for more research. The main purpose of this research was to investigate almost similarity properties on partial isometries, θ-operators, posinormal operators and conditions yielding equality of spectra. Comparative and analytic approaches were used by considering known results on similarity and quasisimilarity concepts. Among other results, unitary equivalence of both θ-operators and posinormal operators under isometry and co-isometry properties were established. Results from the study are fundamental as they will bring about more understanding on properties of operators, which is the basis for those applying these operators in quantum mechanics, spectral analysis of functions and unitary group representation.