Characterizations of normal, hyponormal and EP operators

Abstract

In this paper normal and hyponormal operators with closed ranges, as well as EP operators, are characterized in arbitrary Hilbert spaces. All characterizations involve generalized inverses. Thus, recent results of S. Cheng and Y. Tian [S. Cheng, Y. Tian, Two sets of new characterizations for normal and EP matrices, Linear Algebra Appl. 375 (2003) 181-195] are extended to infinite-dimensional settings.

On EP elements, normal elements and partial isometries in rings with involution

“…The following result is well known for complex matrices and for linear operators on Hilbert spaces (see [1,5,8,9]). But it is unknown for rings with involution.…”

“…Introduction. EP matrices and normal matrices, as well as EP linear operators and normal linear operators on Banach or Hilbert spaces have been investigated by many authors (see, for example, [1,2,3,4,5,7,8,9,10,11,13,15,17,18,20]). D. Mosić et al in [21,24] use the setting of rings with involution to investigate EP elements and normal elements, giving new characterizations to them and providing simpler and more transparent proofs to already existing ones.…”

Resolvent criteria for similarity to a normal operator with spectrum on a curve

“…In [5], Campbell and Gellar studied operators T for which T * T and T + T * commute, showing, for instance, that if σ(T ) is a subset of a vertical line or R, then T is normal. In [10] Djordjević gave several conditions for an operator to be normal using the Moore-Penrose inverse. Gheondea considered operators which are the product of two normal operators in [22].…”

Weighted partial isometries and weighted-EP elements in C*-algebras

“…Some of these results were proved for complex square matrices in [3], using the rank of a matrix, or in [1], using an elegant representation of square matrices as the main technique. Moreover, the operator analogs of these results were proved in [5,6] for linear bounded operators on Hilbert spaces, using the operator matrices as the main tool.…”