Generalized Riesz Systems and Ordered Structures of Their Constructing Operators

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 nd constructing operators Stting to each of physical applications.