I am beginning the book Operator Theory by Example (Garcia, Mashreghi, and Ross 2023). The book’s special feature is that it presents Operator Theory through concrete examples, rather than as a string of abstract theory.
All the “known” operators are covered: the Bishop, Cesaro, Hankel, Toeplitz and Volterra Operators, the Bergman and Dirichlet Shifts, the Fourier and Hilbert Transforms, along with the combinations by multiplication, operator matrices, composition operators, etc. These are applied to various Hilbert spaces: the sequence space (), Lebesgue Measure spaces (, , , ), and the Hardy Spaces (, ). Various properties are computed for the operators, including the spectrum and invariant spaces.
My first impressions are positive - the book appears quite accessible to someone who has studied Functional Analysis. The first chapter reviews various aspects of Hilbert spaces, and the second begins discussion of the diagonal operator - including a look at its spectrum and the more general class of compact self-adjoint operators. The next few chapters appear to contain a good bit of material and exercises which can be understood without deep knowledge of measure theory or specific function analysis. Mixed in is a recap of much of the required background material, e.g. Chapter 2 gives a summary of the main Banach Space theorems, and Chapter 3 proves the Riesz Representation Theorem.
The authors write in the preface that they aim for proofs and examples to be “instructive”, not holding back on the details. Certainly, the earlier proofs are remarkably complete. Consider Proposition 1.1.1 (Cauchy-Schwartz): we have the following lines:
Notice how the proof does not skimp on the algebraic manipulations: each step is shown. But the biggest surprise is in the conclusion, which is written with an explanation: “Since the left side is non-negative, it follows that:” - it’s a small point, but that “Since …” is so valuable when you are self-studying. There’s enough complexity already with the new material, that a brief comment to keep the reader orientated is appreciated.
Resources
Introduction to the book by W. Ross at OTTER.
Video course about six of these operators, by W. Ross: part 1, part 2, and part 3
W. Ross has also published an extended treatment of Chapter 6, on Cesàro Operators (Ross 2024).