This is a problem book on Hilbert space operators (Le., on bounded
linear transformations of a Hilbert space into itself) where theory and
problems are investigated together. We tre!l: t only a part of the
so-called single operator theory. Selected prob- lems, ranging from
standard textbook material to points on the boundary of the subject, are
organized into twelve chapters. The book begins with elementary aspects
of Invariant Subspaces for operators on Banach spaces 1. Basic
properties of Hilbert Space Operators are introduced in in Chapter
Chapter 2, Convergence and Stability are considered in Chapter 3, and
Re- ducing Subspaces is the theme of Chapter 4. Primary results about
Shifts on Hilbert space comprise Chapter 5. These are introductory
chapters where the majority of the problems consist of auxiliary results
that prepare the ground for the next chapters. Chapter 6 deals with
Decompositions for Hilbert space contractions, Chapter 7 focuses on
Hyponormal Operators, and Chapter 8 is concerned with Spectral
Properties of operators on Banach and Hilbert spaces. The next three
chapters (as well as Chapter 6) carry their subjects from an
introductory level to a more advanced one, including some recent
results. Chapter 9 is about Paranormal Operators, Chapter 10 covers
Proper Contractions, and Chapter 11 searches through Quasi- reducible
Operators. The final Chapter 12 commemorates three decades of The
Lomonosov Theorem on nontrivial hyperinvariant subspaces for compact
operators.
