This is a set of lecture notes to accompany a series of talks given as part of the Fields Institute session on Operators on Function Spaces from July–December 2021). These notes are also part of a book project by myself, Stephan R. Garcia, and Javad Mashreghi titled Operator Theory by Example. Many of the technical details as well as other examples of operators on function spaces can be found there.

Operator theory on function spaces is important since it can be used to represent abstract operators on Hilbert spaces in a concrete setting as well as change the point of view to obtain more information about an operator. For example, a version of the spectral theorem says that any bounded normal operator is unitarily equivalent to a multiplication operator Mϕf = ϕf on an L 2 (X, µ) space. From here one can glean information about normal operators by looking at this very specific tangible class of multiplication operators. As another example, if one considers the shift operator Sen = en+1 on the sequence space ` 2 = ` 2 (N0) (where en is the standard basis vector for ` 2 ), one seems at a loss to come up with examples of invariant subspaces for S, besides the obvious ones defined by the closed linear span of {ek : k > N}. However, when one views S as the multiplication operator (Sf)(z) = zf(z) on a certain Hilbert space of analytic functions on the open unit disk D (the Hardy space H2 ), more invariant subspaces appear and one is off to characterize them all – as Beurling did in 1949.

This set of notes explores a selection of well-studied operators on Hilbert spaces to give the novice a taste of the subject and encourage them to explore these ideas further. As this is a sample of operators on function spaces, I do not intend to be thorough on covering both the breadth as well as the depth of the subject. As such, these are a collection of operators that I find interesting and have encountered at the many conferences I have attended over the years.

To get the most out of these notes, the reader should be familiar with the basics of real, complex, and functional analysis. Perhaps have the books [1, 2] at their side while reading these notes. In addition, in order to make things more manageable, I restrict myself to the operators on two basic Hilbert spaces, the Lebesgue space L 2 (T) and the Hardy space H2 . I also only sample results concerning the Volterra, Cesaro, Toeplitz, Hankel, Fourier transform, and Hilbert ` transform operators on these spaces. These spaces and operators are easy to describe which allows me to quickly get to some meaningful results.

As a matter of apology, I’m leaving out a multi-volume treatise one can write on the subject. Indeed, there are many other Hilbert function spaces one can cover as well as many more interesting operators on these function spaces. Of course, there is the whole arena of Banach spaces of functions and their corresponding operators which would take up another several volumes. Indeed, even the operators I do cover, are classical and there is a large literature on each of them. I will do my best to point out good books the reader can consider in order to learn more.

References:

[1] W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill Book Co., New York, 1987. MR 88k:00002

[2] Walter Rudin, Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815