Products of Laurent operators and fields of values

Abstract

One of the most fundamental properties of the field of values of an operator is the inclusion of the spectrum within its closure. Obtaining information on the spectrum of products of operators in terms of this spectral inclusion region is a demanding issue. Stating general results seems difficult; however, conclusions can be derived in some special instances. In this paper, we show that the field of values of products of Laurent operators is easily related to the product of their fields of values, and the same occurs for certain classes of Laurent operators with matrix symbols. The results also apply to the class of infinite upper (lower) triangular Toeplitz matrices.