It should be emphasised at the start that this book does not claim to be an exhaustive treatise on either linear operators or linear systems, but it presents an introduction to the common ground between the two subjects, one pure mathematical and one applied, by regarding a linear system as a (causal) shift-invariant operator on a Hilbert space such as ℓ2(ℤ+) or L2(0, ∞). It therefore includes material on Hardy spaces, shift-invariant operators, the commutant lifting theorem, and almost-periodic functions, which might traditionally be regarded as “pure” mathematics, and is suitable for those working in analysis who wish to learn more advanced material on linear operators.

At the same time, it is hoped that students and researchers in systems and control will find the approach taken attractive, including as it does much recent material on the mathematical side of systems theory, which cannot easily be found elsewhere: these include recent developments in robust control, power signal spaces, and the input–output approach to time-delay systems. Parts of this book have been expounded in graduate courses and other lectures at that level and could be used for a similar purpose elsewhere.

Chapter 1 begins with a review of basic operator theory without proofs. All this material can be found in any introductory course and many textbooks, and so is included mostly for reference. The other main topic of this chapter, which is treated in considerably more detail, is that of Hardy spaces, which are Banach spaces of analytic functions on the disc or half-plane.