Spectral Theory of Generalized Toeplitz Operators

R. G. Douglas, Carl Pearcy
1965 Transactions of the American Mathematical Society  
1. Introduction. In this note we consider a class of (bounded) operators on Hilbert space that properly contains the much-studied class of Toeplitz operators, and we show that virtually all of the results of [2] concerning spectral properties of Toeplitz operators are valid for operators in this larger class. We devote §2 to preliminary definitions and examples. The concept of a "Riesz system" is first introduced and then is used to define the class of generalized Toeplitz operators. Several
more » ... erators. Several examples of Riesz systems are set forth to illustrate the generality of the concept. In §3 the spectral inclusion theorem for generalized Toeplitz operators is proved, i.e., it is shown that the spectrum of any given generalized Laurent operator is a subset of the spectrum of the associated generalized Toeplitz operator. This result is then used to recover several of the theorems of [ 2] for generalized Toeplitz operators. §4 is devoted to questions concerning isomorphisms between Riesz systems and ways in which these systems can differ from the "classical" Riesz system that gives rise to Toeplitz operators. In §5 we make some concluding remarks.
doi:10.2307/1994279 fatcat:u6pgsf3j6zdkxpcx7bbpbkkgmi