RIESZ IDEMPOTENT OF (n,k)-QUASI-?-PARANORMAL OPERATORS?

Qingping ZENGCollege of Computer and Information Sciences,Fujian Agriculture and Forestry University, Fuzhou 350002,China E-mail:zqpping2003@163.com

Huaijie ZHONGSchool of Mathematics and Computer Science,Fujian Normal University,Fuzhou 350007,China E-mail:zhonghuaijie@sina.com

RIESZ IDEMPOTENT OF (n,k)-QUASI-?-PARANORMAL OPERATORS?

A bounded linear operator T on a complex Hilbert space H is called(n,k)-quasi-?-paranormal if

where n,k are nonnegative integers.This class of operators has many interesting properties and contains the classes of n-?-paranormal operators and quasi-?-paranormal operators.The aim of this note is to show that every Riesz idempotent Eλwith respect to a non-zero isolated spectral point λ of an(n,k)-quasi-?-paranormal operator T is self-adjoint and satisfies ranEλ=ker(T?λ)=ker(T?λ)?.

?-class A operator;?-paranormal operator;Riesz idempotent

2010 MR Subject Classification47A10;47B20

1　Introduction

There is a growing interest concerning nonhyponormal operators.Let L(H)be the C?lgebra of all bounded linear operators on an infinite dimensional complex Hilbert space H. Below we list some of these nonhyponormal operators.Recall that an operator T∈L(H)is aid to behere and henceforth,n,k denote nonnegative integers.

As an extension of the above operator classes,we introduced and studied in[21]the following definition.

Definition 1.1An operator T∈L(H)is said to be(n,k)-quasi-?-paranormal if

The class of(n,k)-quasi-?-paranormal operators has many interesting properties(see[21]), such as inclusion relations,SVEP(single valued extension property),matrix representation, joint point spectrum,and so on.

In the present note,we continue to investigate the properties of(n,k)-quasi-?-paranormal operators.We show that every Riesz idempotent Eλwith respect to a non-zero isolated spectral point λ of an(n,k)-quasi-?-paranormal operator T is self-adjoint and satisfies ranEλ=ker(T?λ)=ker(T?λ)?.

2　Riesz Idempotent of(n,k)-Quasi-?-Paranormal Operators

The self-adjointness of Riesz idempotent with respect to the isolated spectral point of an operator was investigated by a number of mathematicians around the world.For an isolated spectral point λ∈isoσ(T),the Riesz idempotent Eλwith respect to λ is defined by

where D is a closed disk with center λ and its radius is small enough such that D∩σ(T)={λ}. In general,the Riesz idempotent Eλis not orthogonal and Eλis orthogonal if and only if Eλis self-adjoint.Stampfli[12]showed that the Riesz idempotent Eλfor an isolated spectral point λ of a hyponormal operator T is self-adjoint.Stampfli’s result was extended to p-hyponormal operators and log-hyponormal operators by Chˉo and Tanahashi[4],to M-hyponormal operators by Chˉo and Han[3],to?-paranormal operators by Tanahashi and Uchiyama[14].In the case λ/=o,Stampfli’s result was extended to p-quasihyponormal operators by Tanahashi and Uchiyama[15],to(p,k)-quasihyponormal operators by Tanahashi,Uchiyama and Chˉo[16],to w-hyponormal operators by Han,Lee and Wang[6],to class A operators by Uchiyama and Tanahashi[18],to quasi-class A operators by Jeon and Kim[7],to quasi-class(A,k)operators by Tanahashi,Jeon,Kim and Uchiyama[13],to paranormal operators by Uchiyama[17],to k-quasi-?-class A operators by Mecheri[9].

In this section,we will extend Stampfli’s result to n-?-paranormal operators and,to(n,k)-quasi-?-paranormal operators in the case λ/=o.

Theorem 2.1(1)Let T be(n,k)-quasi-?-paranormal and o/=λ∈isoσ(T).Then the Riesz idempotent Eλis self-adjoint and

(2)Let T be n-?-paranormaland λ∈isoσ(T).Then the Riesz idempotent Eλis self-adjoint and

It is worth to note that Theorem 2.1(1)does not hold even for(o,1)-quasi-?-paranormal (that is,quasi-hyponormal)operators when λ=o(see[15,Example 6]).

To give the proof of Theorem 2.1,we prepare the following lemmas.Recall that an operator T∈L(H)is said to be n-paranormal if‖T1+nx‖1/(1+n)‖x‖n/(1+n)≥‖Tx‖for all x∈H(see [19,2o]).

Lemma 2.2Let T be n-?-paranormal and λ∈isoσ(T).Then λ is a pole of order one of the resolvent of T.

ProofLet λ∈isoσ(T).Then

By[1,Theorem 3.74],it follows that ranEλ=Ho(T?λ)and kerEλ=K(T?λ),where

Since T is n-?-paranormal,it follows from[21]that T is(n+1)-paranormal.Hence by[14, Theorem 2]we have that

Since H=ranEλ⊕kerEλ,

That is,λ is a pole of order one.

Lemma 2.3(see[21])Suppose that ranTkis not dense.Letis the closure of ranTk.If T is(n,k)-quasi-?-paranormal,then T1is n-?-paranormal,

Lemma 2.4If T is(n,k)-quasi-?-paranormal and quasi-nilpotent,then Tk+1=o.

ProofAssume that ranTkis dense,then T is n-?-paranormal,hence by[21],T is(n+1)-paranormal.Thus by[14,Corollary 1],T=o.So we may suppose that ranTkis not dense. Hence by Lemma 2.3,we can write

where T1is n-?-paranormal,σ(T)={o},we see that σ(T)={o}.Hence T=o,and so T=An easy computation yields that

Lemma 2.5Let T be(n,k)-quasi-?-paranormal and λ∈isoσ(T).Then λ is a pole of the resolvent of T,and the order of λ is no more than k+1 when λ=o,and 1 when λ/=o.

ProofLet λ∈isoσ(T).Then

Write T=T1⊕T2with respect to this direct decomposition.Since σ(T1)={λ}and σ(T2)= σ(T){λ},T1?λ is quasi-nilpotent and T2?λ is invertible.

where T11is n-?-paranormal,={λ}implies that λ=o.It follows from Lemma 2.4 thatand hence λ=o is a pole of order no more than k+1.

Lemma 2.6(see[21])Let T be(n,k)-quasi-?-paranormal and o/=λ∈?.Then

Proof of Theorem 2.1(1)By Lemma 2.5,λ is an eigenvalue of T,ranEλ=ker(T?λ) and kerEλ=ran(T?λ).Since ker(T?λ)?ker(T??λ)by Lemma 2.6,ranEλ=ker(T?λ) is a reducing subspace of T.Write T=λ⊕T1with respect to the orthogonal decomposition

where T1is(n,k)-quasi-?-paranormal with ker(T1?λ)={o}.Since λ∈isoσ(T)and σ(T)= {λ}∪σ(T1),it then follows from Lemma 2.5 that T1?λ is invertible.Consequently,ker(T?λ)=we have ker(T?λ)⊥ran(T?λ),and hence ranEλ⊥kerEλ.That is,Eλis self-adjoint.

(2)It needs only to consider the case when λ=o.Applying the above argument and using Lemma 2.2 and the fact that ker(T)?ker(T?),the assertion follows.

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?October 16,2014;revised March 12,2016.This work has been supported by National Natural cience Foundation of China(11301077,11301078,11401097,11501108)and Natural Science Foundation of Fujian Province(2015J01579,2016J05001).