Asymmetric Putnam-Fuglede Theorem for (n,k)-Quasi-∗-Paranormal Operators

Ahmed Bachir,Abdelkader Segres

doi:10.3390/sym11010064

Abstract

T ∈ B ( H ) is said to be ( n , k ) -quasi-∗-paranormal operator if, for non-negative integers k and n, ∥ T ∗ ( T k x ) ∥ ( 1 + n ) ≤ ∥ T ( 1 + n ) ( T k x ) ∥ ∥ T k x ∥ n ; for all x ∈ H . In this paper, the asymmetric Putnam-Fuglede theorem for the pair ( A , B ) of power-bounded operators is proved when (i) A and B ∗ are n-∗-paranormal operators (ii) A is a ( n , k ) -quasi-∗-paranormal operator with reduced kernel and B ∗ is n-∗-paranormal operator. The class of ( n , k ) -quasi-∗-paranormal operators properly contains the classes of n-∗-paranormal operators, ( 1 , k ) -quasi-∗-paranormal operators and k-quasi-∗-class A operators. As a consequence, it is showed that if T is a completely non-normal ( n , k ) -quasi-∗-paranormal operator for k = 0 , 1 such that the defect operator D T is Hilbert-Schmidt class, then T ∈ C 10 .

Highlights

Throughout this paper, H denotes an infinite dimensional complex Hilbert space with inner product h·, ·i and B(H) denotes the algebra of all bounded linear operators acting on H
For any operator T ∈ B(H), let | T | = ( T ∗ T )1/2, and consider the following standard definitions: normal if T ∗ T = TT ∗ and T is hyponormal if | T ∗ |2 ≤ | T |2
If T is a contraction with the above matrix form (25) such that the defect operator DT = ( I − T ∗ T ) 2 is of Hilbert-Schmidt class

Summary

Introduction

Throughout this paper, H denotes an infinite dimensional complex Hilbert space with inner product h·, ·i and B(H) denotes the algebra of all bounded linear operators acting on H. If T is a power-bounded n-∗-paranormal operator and there is an invariant subspace M for which the restriction T |M = N of T on M is a normal operator, M reduces T and N = U ⊕ 0 where U is unitary. If T is a power-bounded (n, k)-quasi-∗-paranormal operator and there is an invariant subspace M for which the restriction T |M = N of T on M is an injective normal operator M reduces T and N is a unitary operator. Let T be a power-bounded n-∗-paranormal operator and let us consider an invariant subspace. Let T be a power-bounded (n, k )-quasi-∗-paranormal operator. T, N is a normal operator and A is a power-bounded (n, k )-quasi-∗-paranormal operator with σr ( A∗ ) = ∅.

Main Theorems

Application

Discussions and Further Studies