Decompositions and models for Hilbert-space operators have been very active research topics in operator theory over the past three decades. This book is intended as an introduction to this crucial part of operator theory, providing for the student a unified access, from an abstract point of view, to an active research field.

It focuses on decompositions and models as if they were the main characters in a plot, chosen from a myriad of equally important characters, and highlighted for their illustrative attributes. It has been written for an audience composed mainly of graduate students taking operator theory either as their major or as a support for applications in mathematics or in one of the sciences.

The approach is elementary in the sense that all proofs use only standard results of single operator theory. However, a number of questions posed throughout the text provide the flavor of a research monograph in that they lead the reader to investigate some open problems, a number of them classical. This approach will help the reader to visualize, even if only partially, the frontiers of a few directions in which operator theory has been developing.

Although the material is mainly drawn from a variety of sources, there are some original contributions in the form of new intermediate results and simplified proofs.