A characterization of spectral operators on Hilbert spaces

Abstract

In [8] Wadhwa shows that if a bounded linear operator T T on a complex Hilbert space H H is a decomposable operator and has the condition (I), then T T is a spectral operator with a normal scalar part. In this paper, by using this result, we show that a weak decomposable operator T T is a spectral operator with a normal scalar part if and only if T T satisfies the assertion that (1) T T has the conditions ( C C ) and ( I I ) or that (2) every spectral maximal space of T T reduces T T . This result improves [1, 6 and 7]. From this result, we can get a characterization of spectral operators, but this result does not hold in complex Banach space (see Remark 2).