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.)