Abstract. A Hilbert space operator $T\in{\cal B}[{\cal H}]$ is said to be totally hereditarily normaloid, or $\cal THN$, if for every $T$-invariant subspace ${\cal M}\subseteq{\cal H}$ the restriction $T|_{\cal M}$ of $T$ to ${\cal M}$ is normaloid and, whenever $T|_{\cal M}\in{\cal B}[{\cal M}]$ is invertible, the inverse $(T|_{\cal M})^{-1}$ is normaloid as well. In this paper we explore the structure of $\cal THN$ contractions, and conclude some properties which follow from such a structure, specially for $\cal THN$ contractions with either compact or Hilbert--Schmidt defect operators.
AMS Subject Classification
(1991): 47A45, 47B20
Keyword(s):
Hereditarily normaloid,
contractions,
defect operator,
decompositions
Received August 15, 2003, and in final form December 28, 2004. (Registered under 5872/2009.)
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