On n-quasi-[m,C]-isometric operators

Junli Shen

doi:10.1186/s13660-019-2268-3

Abstract

For positive integers m and n, an operator T in B ( H ) is said to be an n-quasi-[m,C]-isometric operator if there exists some conjugation C such that . In this paper, some basic structural properties of n-quasi-[m,C]-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi-[m,C]-isometric operator is again an n-quasi-[m,C]-isometric operator. Moreover, we show that the class of n-quasi-[m,C]-isometric operators is norm closed. Finally, we examine the stability of n-quasi-[m,C]-isometric operator under perturbation by nilpotent operators commuting with T.

Highlights

1 Introduction Let N and C be the sets of natural numbers and complex numbers, respectively, and let B(H) denote the algebra of all bounded linear operators on a separable complex Hilbert space H
In 1990s, Agler and Stankus [1] studied the theory of m-isometric operators which are connected to Toeplitz operators, ordinary differential equations, classical function theory, classical conjugate point theory, distributions, Fejer–Riesz factorization, stochastic processes, and other topics
Proof Suppose that T is an invertible n-quasi-[m, C]-isometric operator

Summary

Introduction

Let N and C be the sets of natural numbers and complex numbers, respectively, and let B(H) denote the algebra of all bounded linear operators on a separable complex Hilbert space H. In [7], Cho, Ko, and Lee introduced (m, C)-isometric operators with conjugation C as follows: For an operator T ∈ B(H) and an integer m ≥ 1, T is said to be an (m, C)-isometric For positive integers m and n, an operator T ∈ B(H) is said to be an n-quasi-[m, C]isometric operator if there exists some conjugation C such that m

Results

Conclusion