By a Hilbert-space operator we mean a bounded linear transformation be-
tween separable complex Hilbert spaces. Decompositions and models for
Hilbert-space operators have been very active research topics in
operator theory over the past three decades. The main motivation behind
them is the in- variant subspace problem: does every Hilbert-space
operator have a nontrivial invariant subspace? This is perhaps the most
celebrated open question in op- erator theory. Its relevance is easy to
explain: normal operators have invariant subspaces (witness: the
Spectral Theorem), as well as operators on finite- dimensional Hilbert
spaces (witness: canonical Jordan form). If one agrees that each of
these (i. e. the Spectral Theorem and canonical Jordan form) is
important enough an achievement to dismiss any further justification,
then the search for nontrivial invariant subspaces is a natural one; and
a recalcitrant one at that. Subnormal operators have nontrivial
invariant subspaces (extending the normal branch), as well as compact
operators (extending the finite-dimensional branch), but the question
remains unanswered even for equally simple (i. e. simple to define)
particular classes of Hilbert-space operators (examples: hyponormal and
quasinilpotent operators). Yet the invariant subspace quest has
certainly not been a failure at all, even though far from being settled.
The search for nontrivial invariant subspaces has undoubtly yielded a
lot of nice results in operator theory, among them, those concerning
decompositions and models for Hilbert-space operators. This book
contains nine chapters.
