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."
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