A Study on EP and Hypo - EP Operators

Abstract

A complex square matrix A is said to be EP (EP stands for Equal Projections) if ranges of A and its adjoint are equal. The class of EP matrices was introduced by Schwerdtfeger (Schwerdtfeger, 1950) which contains the class of normal matrices. Later the notion of EP matrix was extended to bounded linear operators on Hilbert spaces with the additional assumption that the operators have closed ranges and then this class of operators was generalized to hypo-EP operators. In this thesis, we characterize the hypo-EP operators with the aid of factorization of bounded linear operators. Precisely, two kinds of factorizations are involved in these characterizations. One is factorization involving direct sum of operators whereas another is similar to full rank factorization in matrix theory. Also, we prove that for a given subspace M of Cn, there exists an EP matrix whose range space is M. The product of two hypo-EP operators is not necessarily hypo-EP and hence we derive necessary and sufficient conditions for product of two hypo-EP to be hypo-EP. Also we come up with some conditions which are necessary or sufficient for sum and restriction of hypo-EP operators to be again hypo-EP. One of the classical results concerning normal operators is Fuglede theorem which states that if a bounded linear operator commutes with a normal operator then the bounded operator commutes with adjoint of the normal operator. We show that this celebrated result is not true for EP operators and we find some conditions so that the Fuglede-Putnam theorem is true for EP operators. Also, we evince that if we replace adjoint operation by Moore-Penrose inverse, we arrive at Fuglede-Putnam type theorems for EP operators. We generalize quite a number of characterizations of EP operators on Hilbert spaces into Krein space settings. We extend some of the results of EP and hypoEP bounded operators into unbounded densely defined closed operators on Hilbert spaces.