The Cesàro operator

Abstract

This paper surveys the various aspacts of the Cesàro operator with a special emphasis on the Hilbert space setting of ℓ 2 \ell ^2 . We include a discussion of summability methods of Fourier series, the Cesàro matrix and integral operator, and the Cesàro operator on the Hilbert space ℓ 2 \ell ^2 . In this setting of ℓ 2 \ell ^2 we cover the spectral properties of the Cesàro operator as well as a treatment of its hyponormality (via posinormal operators) and subnormality (via a multiplication operator of Kriete and Trutt). Using the results of Kriete and Trutt, we describe the commutant of the Cesàro operator as well as its bounded square roots. The Cesàro operator has a rich lattice of invariant subspaces whose description remains unknown and we suspect might never be fully understood. We survey some results which display the complexity of these invariant subspaces and provide the reader with several paths forward for further discussion. Though the Cesàro operator was originally explored in the ℓ 2 \ell ^2 and Hardy space settings, it has been explored in various other settings such as the general Hardy spaces as well as the Bergman spaces. Finally, we explore some of the so-called generalized Cesàro operator which are classes of integral operators on the Hardy spaces that connect to many areas of classical function theory.