Characterizing Hermitian, normal and EP operators

Abstract

In this paper further characterizations of Hermitian, normal and EP operators on Hilbert spaces are established. Thus the recent results of O. M. Baksalary and G. Trenkler (Linear Multilin. Algebra, to appear) are extended to the infinite dimensional setting with proofs based on operator matrices.

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.…”

On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces

“…If both AG and GA satisfy any of the following statements: is bi-EP and bi-normal but it is not EP. Moreover, it is well known that -and r-quasi-normal and -and r-quasi-EP classes are different from each other as it can be seen in [5,6], even different from the normal class. The previous (strict) inclusions clarify the fact that conditions used in Theorem 2.5, which are weaker than the one of being self-adjoint, can be now adopted to define the Moore-Penrose inverse of A.…”

Reverse order law for the Moore–Penrose inverse

“…Recall that one special case of this result is proved in [7]. We prove only the result of (a), since the proof of (b) is analogous.…”