Generalized Riesz Systems and Ordered Structures of Their Constructing Operators

Abstract

Theory of non-self-adjoint operators and these applications are interested in various fields of mathematics and physics. There are many research results related to pseudo-Hermitian operators. In this field, generalized Riesz systems can be used to construct some physical operators. From this fact, it seems to be important to consider under what conditions biorthogonal sequences are generalized Riesz systems. In this chapter, we shall focus the construction of generalized Riesz systems from biorthogonal sequences and the properties of constructing operators for generalized Riesz systems. In details, we shall investigate under what conditions the ordered set of all constructing operators for a generalized Riesz system has maximal elements, minimal elements, the largest element and the smallest element in order to find constructing operators fitting to each of physical applications.