Ordered Structures of Constructing Operators for Generalized Riesz Systems

Abstract

A sequence {φn} in a Hilbert space H with inner product <·,·> is called a generalized Riesz system if there exist an ONB e={en} in H and a densely defined closed operator T in H with densely defined inverse such that {en}⊂D(T)∩D((T-1)⁎) and Ten=φn, n=0,1,⋯, and (e,T) is called a constructing pair for {φn} and T is called a constructing operator for {φn}. The main purpose of this paper is to investigate under what conditions the ordered set Cφ of all constructing operators for a generalized Riesz system {φn} has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications.

Gibbs States, Algebraic Dynamics and Generalized Riesz Systems

“…In recent papers many other extensions of Riesz bases, mostly involving unbounded operators, have also been considered. In particular we mention generalized Riesz systems introduced by one of us (H.I) and analyzed in a series of papers [7][8][9][10][11][12][13][14]). For other studies on extensions of Riesz bases or on generalizations to different environments (Krein spaces, Rigged Hilbert spaces) we refer to [15][16][17].…”

Gibbs states, algebraic dynamics and generalized Riesz systems

“…In recent papers many other extensions of Riesz bases, mostly involving unbounded operators, have also been considered. In particular we mention generalized Riesz systems introduced by one of us (H.I) and analyzed in a series of papers [13,14,15,19,20,21,22,23]). For other studies on extensions of Riesz bases or on generalizations to different environments (Krein spaces, Rigged Hilbert spaces) we refer to [11,12,17].…”