Theory of non-self-adjoint operators and these applications are interested in various felds of mathematics and physics. There are many research results related to pseudo-Hermitian operators. In this feld, 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 fnd constructing operators ftting to each of physical applications.
Center for Advancing Pharmaceutical Education, Daiichi University of Pharmacy, 22-1 Tamagawacho, Minami-ku, Fukuoka 815-8511, Japan.
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