A Characterization of Spectral Operators on Hilbert Spaces

Abstract

In [8] Wadhwa shows that if a bounded linear operator $T$ on a complex Hilbert space $H$ is a decomposable operator and has the condition (I), then $T$ is a spectral operator with a normal scalar part. In this paper, by using this result, we show that a weak decomposable operator $T$ is a spectral operator with a normal scalar part if and only if $T$ satisfies the assertion that (1) $T$ has the conditions ($C$) and ($I$) or that (2) every spectral maximal space of $T$ reduces $T$. 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).