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The k-quasi--class contractions have property PF

Abstract

First, we will see that if T is a contraction of the k-quasi--class operator, then the nonnegative operator D= T k (| T 2 | | T | 2 ) T k is a contraction whose power sequence { D n } n = 1 converges strongly to a projection P and T T k P=0. Second, it will be proved that if T is a contraction of the k-quasi--class operator, then either T has a non-trivial invariant subspace or T is a proper contraction. Finally it will be proved that if T belongs to the k-quasi--class and is a contraction, then T has a Wold-type decomposition and T has the PF property.

MSC:47A10, 47B37, 15A18.

1 Introduction

Throughout this paper, let H and K be infinite dimensional separable complex Hilbert spaces with inner product ,. We denote by L(H,K) the set of all bounded operators from H into K. To simplify, we put L(H):=L(H,H). For TL(H), we denote by kerT the null space and by T(H) the range of T. The closure of a set M will be denoted by M ¯ . We shall denote the set of all complex numbers by and the set of all nonnegative integers by .

For an operator TL(H), as usual, by T we mean the adjoint of T and |T|= ( T T ) 1 2 . An operator T is said to be hyponormal, if | T | 2 | T | 2 . An operator T is said to be paranormal, if

T 2 x T x 2

for any unit vector x in H [1]. Further, T is said to be -paranormal, if

T 2 x T x 2

for any unit vector x in H [2]. T is said to be a k-paranormal operator if T x k + 1 T k + 1 x x k for all xH, and T is said to be a k--paranormal operator if T x k + 1 T k + 1 x x k , for all xH.

Furuta et al. [3] introduced a very interesting class of bounded linear Hilbert space operators: class defined by

| T 2 | | T | 2 ,

and they showed that the class is a subclass of paranormal operators and contains hyponormal operators. Jeon and Kim [4] introduced the quasi-class . An operator T is said to be a quasi-class , if

T | T 2 |T T | T | 2 T.

We denote the set of quasi-class by QA. An operator T is said to be a k-quasi-class , if

T k | T 2 | T k T k | T | 2 T k .

We denote the set of quasi-class by QA(k).

Duggal et al. [5], introduced -class operator. An operator T is said to be a -class operator, if

| T 2 || T | 2 .

A -class is a generalization of a hyponormal operator [[5], Theorem 1.2], and -class is a subclass of the class of -paranormal operators [[5], Theorem 1.3]. We denote the set of -class by A . Shen et al. in [6] introduced the quasi--class operator: an operator T is said to be a quasi--class operator, if

T | T 2 |T T | T | 2 T.

We denote the set of quasi--class by Q A . Mecheri [7] introduced the k-quasi--class operator.

Definition 1.1 An operator TL(H) is said to be a k-quasi--class operator, if

T k ( | T 2 | | T | 2 ) T k O

for a nonnegative integer k.

We denote the set of the k-quasi--class by Q A (k).

Example 1.2 Let T be an operator defined by

T=( 1 0 0 0 0 0 0 1 0 ).

Then | T 2 | | T | 2 O and so T is not a class A . However, T k (| T 2 | | T | 2 ) T k =O for every positive number k, which implies that T is a k-quasi-class A operator.

A contraction is an operator T such that Txx for all xH. A proper contraction is an operator T such that Tx<x for every nonzero xH [8]. A strict contraction is an operator such that T<1 (i.e., sup x 0 T x x <1). Obviously, every strict contraction is a proper contraction and every proper contraction is a contraction. An operator T is said to be completely non-unitary (c.n.u.) if T restricted to every reducing subspace of H is non-unitary.

An operator T on H is uniformly stable, if the power sequence { T n } n = 1 converges uniformly to the null operator (i.e., T n O). An operator T on H is strongly stable, if the power sequence { T n } n = 1 converges strongly to the null operator (i.e., T n x0, for every xH).

A contraction T is of class C 0 if T is strongly stable (i.e., T n x0 and Txx for every xH). If T is a strongly stable contraction, then T is of class C 0 . T is said to be of class C 1 if lim n T n x>0 (equivalently, if T n x0 for every nonzero x in H). T is said to be of class C 1 if lim n T n x>0 (equivalently, if T n x0 for every nonzero x in H). We define the class C α β for α,β=0,1 by C α β = C α C β . These are the Nagy-Foiaş classes of contractions [[9], p.72]. All combinations are possible leading to classes C 00 , C 01 , C 10 , and C 11 . In particular, T and T are both strongly stable contractions if and only if T is a C 00 contraction. Uniformly stable contractions are of class C 00 .

Lemma 1.3 [[10], Holder-McCarthy inequality]

Let T be a positive operator. Then the following inequalities hold for all xH:

  1. (1)

    T r x,x T x , x r x 2 ( 1 r ) for 0<r<1;

  2. (2)

    T r x,x T x , x r x 2 ( 1 r ) for r1.

Lemma 1.4 [[7], Lemma 2.1]

Let T be a k-quasi--class operator, where T k does not have a dense range, and let T have the following representation:

T=( A B O C )on H= T k ( H ) ¯ ker T k .

Then A is class A on T k ( H ) ¯ , C k =O, and σ(T)=σ(A){0}.

2 Main results

Theorem 2.1 If T is a contraction of the k-quasi--class operator, then the nonnegative operator

D= T k ( | T 2 | | T | 2 ) T k

is a contraction whose power sequence { D n } n = 1 converges strongly to a projection P and T T k P=O.

Proof Suppose that T is a contraction of the k-quasi--class operator. Then

D= T k ( | T 2 | | T | 2 ) T k O.

Let R= D 1 2 be the unique nonnegative square root of D, then for every x in H and any nonnegative integer n, we have

D n + 1 x , x = R n + 1 x 2 = D R n x , R n x = T k | T 2 | T k R n x , R n x T k | T | 2 T k R n x , R n x | T 2 | 1 2 T k R n x 2 T T k R n x 2 R n x 2 T T k R n x 2 R n x 2 = D n x , x .

Thus R (and so D) is a contraction (set n=0), and { D n } n = 1 is a decreasing sequence of nonnegative contractions. Then { D n } n = 1 converges strongly to a projection P. Moreover,

n = 0 m T T k R n x 2 n = 0 m ( R n x 2 R n + 1 x 2 ) = x 2 R m + 1 x 2 x 2

for all nonnegative integers m and for every xH. Therefore T T k R n x0 as n. Then we have

T T k Px= T T k lim n D n x= lim n T T k R 2 n x=0

for every xH. So that T T k P=O. □

A subspace M of space H is said to be non-trivial invariant (alternatively, T-invariant) under T if {0}MH and T(M)M. A closed subspace MH is said to be a non-trivial hyperinvariant subspace for T if {0}MH and is invariant under every operator SL(H), which fulfills TS=ST.

Recently Duggal et al. [11] showed that if T is a class contraction, then either T has a non-trivial invariant subspace or T is a proper contraction and the nonnegative operator D=| T 2 | | T | 2 is strongly stable. Duggal et al. [12] extended these results to contractions in QA. Jeon and Kim [13] extended these results to contractions QA(k). Gao and Li [14] have proved that if a contraction T A has a no non-trivial invariant subspace, then (a) T is a proper contraction and (b) the nonnegative operator D=| T 2 | | T | 2 is a strongly stable contraction. In this paper we extend these results to contractions in the k-quasi--class for k>0.

Theorem 2.2 Let T be a contraction of the k-quasi--class for k>0. If T has a no non-trivial invariant subspace, then:

  1. (1)

    T is a proper contraction;

  2. (2)

    the nonnegative operator

    D= T k ( | T 2 | | T | 2 ) T k

is a strongly stable contraction.

Proof We may assume that T is a nonzero operator.

  1. (1)

    If either kerT or T k ( H ) ¯ is a non-trivial subspace (i.e., kerT{0} or T k ( H ) ¯ H), then T has a non-trivial invariant subspace. Hence, if T has no non-trivial invariant subspace, then T is injective and T k ( H ) ¯ =H. Furthermore, T is a class A operator. The proof now follows from [[14], Theorem 2.2].

  2. (2)

    Let T be a contraction of the k-quasi--class . By the above theorem, we see that D is a contraction, { D n } n = 1 converges strongly to a projection P, and T T k P=O. So, P T k T=O. Suppose T has no non-trivial invariant subspaces. Since kerP is a nonzero invariant subspace for T whenever P T k T=O and TO, it follows that kerP=H. Hence P=O, and we see that { D n } n = 1 converges strongly to the null operator O, so D is a strongly stable contraction. Since D is self-adjoint, D C 00 . □

Corollary 2.3 Let T be a contraction of the k-quasi--class . If T has no non-trivial invariant subspace, then both T and the nonnegative operators

D= T k ( | T 2 | | T | 2 ) T k

are proper contractions.

Proof A self-adjoint operator T is a proper contraction if and only if T is a C 00 contraction. □

Definition 2.4 If the contraction T is a direct sum of the unitary and C 0 (c.n.u.) contractions, then we say that T has a Wold-type decomposition.

Definition 2.5 [15]

An operator TL(H) is said to have the Fuglede-Putnam commutativity property (PF property for short) if T X=XJ for any XL(K,H) and any isometry JL(K) such that TX=X J .

Lemma 2.6 [16, 17]

Let T be a contraction. The following conditions are equivalent:

  1. (1)

    For any bounded sequence { x n } n N { 0 } H such that T x n + 1 = x n the sequence { x n } n N { 0 } is constant;

  2. (2)

    T has a Wold-type decomposition;

  3. (3)

    T has the PF property.

Duggal and Cubrusly in [16] have proved: Each k-paranormal contraction operator has a Wold-type decomposition. Pagacz in [17] has proved the same and also proved that each k--paranormal operator has a Wold-type decomposition. In this paper, we extend to contractions in Q A (k).

Theorem 2.7 Let T be a contraction of the k-quasi--class . Then T has a Wold-type decomposition.

Proof Since T is a contraction operator, the decreasing sequence { T n T n } n = 1 converges strongly to a nonnegative contraction. We denote by

S= ( lim n T n T n ) 1 2 .

The operators T and S are related by T S 2 T= S 2 , OSI and S is self-adjoint operator. By [18] there exists an isometry V: S ( H ) ¯ S ( H ) ¯ such that VS=S T , and thus S V =TS, and S V m xx for every x S ( H ) ¯ . The isometry V can be extended to an isometry on H, which we still denote by V.

For an x S ( H ) ¯ , we can define x n =S V n x for nN{0}. Then for all nonnegative integers m we have

T m x n + m = T m S V m + n x=S V m V m + n x=S V n x= x n ,

and for all mn we have

T m x n = x n m .

Since T is a k-quasi--class operator and the non-trivial x S ( H ) ¯ we have

x n 4 = T k x n + k 4 = T T T k 1 x n + k , T k 1 x n + k 2 T T k x n + k 2 T k 1 x n + k 2 = T k | T | 2 T k x n + k , x n + k 2 x n + 1 2 T k | T 2 | T k x n + k , x n + k 2 x n + 1 2 | T 2 | 2 T k x n + k , T k x n + k 1 2 T k x n + k 2 ( 1 1 2 ) x n + 1 2 = T k + 2 x n + k T k x n + k x n + 1 2 = x n 2 x n x n + 1 2 .

Then

x n 3 x n 2 x n + 1 2 ;

hence

x n x n 2 1 3 x n + 1 2 3 1 3 ( x n 2 + 2 x n + 1 ) .

Thus

2 ( x n + 1 x n ) x n x n 2 = ( x n x n 1 ) + ( x n 1 x n 2 ) .

Put

b n = x n x n 1 ,

and we have

2 b n + 1 b n + b n 1 .
(1)

Since x n =T x n + 1 , we have

x n =T x n + 1 x n + 1 for every nN,

then the sequence { x n } n N { 0 } is increasing. From

S V n =S V V n + 1 =TS V n + 1

we have

x n = S V n x = T S V n + 1 x S V n + 1 x x

for every x S ( H ) ¯ and nN{0}. Then { x n } n N { 0 } is bounded. From this we have b n 0 and b n 0 as n.

It remains to check that all b n equal zero. Suppose that there exists an integer i1 such that b i >0. Using the inequality (1) we get b i + 1 >0 and b i + 2 >0, so there exists ϵ>0 such that b i + 1 >ϵ and b i + 2 >ϵ. From that, and using again the inequality (1), we can show by induction that b n >ϵ for all n>i, thus arriving at a contradiction. So b n =0 for all nN and thus x n 1 = x n for all n1. Thus the sequence { x n } n N { 0 } is constant.

From Lemma 2.6, T has a Wold-type decomposition. □

For TL(H) and xH, { T n x } n = 0 is called the orbit of x under T, and is denoted by O(x,T). When the linear span of the orbit O(x,T) is norm dense in H, x is called a cyclic vector for T and T is said to be a cyclic operator. If O(x,T) is norm dense in H, then x is called a hypercyclic vector for T. An operator TL(H) is called hypercyclic if there is at least one hypercyclic vector for T. We say that an operator TL(H) is supercyclic if there exists a vector xH such that CO(x,T)={λ T n x:λC,n=0,1,2,} is norm dense in H.

Theorem 2.8 Let TL(H) be a quasi--class such that σ(T){λC:|λ|=1}. If the inverse of T is a quasi--class , then T is not a supercyclic operator.

Proof Let TL(H) be a quasi--class . Since σ(T){λC:|λ|=1}, T is an invertible operator. From [7]T is normaloid, thus T=r(T)=1. Since T 1 Q( A ), T 1 =1. Consequently, T is unitary. Since no unitary operator on an infinite dimensional Hilbert space can be supercyclic, we see that T is not a supercyclic operator. □

Remark 2.9 The condition that the inverse of the operator T belongs to quasi--class cannot be removed from Theorem 2.8, because there are invertible operators from the quasi--class , such that their inverse does not belong to the quasi--class . This is shown in the following example.

Given a bounded sequence of complex numbers { α n :nZ} (called weights), let T be the bilateral weighted shift on an infinite dimensional Hilbert space operator H= l 2 , with the canonical orthonormal basis { e n :nZ}, defined by T e n = α n e n + 1 for all nZ.

Lemma 2.10 Let T be a bilateral weighted shift operator with weights { α n :nZ}. Then T is a quasi--class operator if and only if

| α n | 2 | α n + 1 || α n + 2 |

for all nZ.

Lemma 2.11 Let T be a non-singular bilateral weighted shift operator with weights { α n :nZ}. Then T 1 is a quasi--class operator if and only if

| α n 1 | 2 | α n 2 || α n 3 |

for all nZ.

Example 2.12 Let us denote by T the bilateral weighted shift operator, with weighted sequence { α n :nZ}, given by the relation

α n = { 1 if  n 1 , 2 if  n = 2 , 1 if  n = 3 , 4 if  n = 4 , 1 if  n = 5 , 16 if  n 6 .

From Lemma 2.10 it follows that T is a quasi--class operator. Since { α n :nZ} is a bounded sequence of positive numbers with inf{ α n :nZ}>0, T is an invertible operator [[19], Proposition II.6.8]. But T 1 is not a quasi--class operator, which follows from Lemma 2.11, for n=4.

Theorem 2.13 Let TL(H) be a quasi--class operator and D={z:|z|<1}. If T is a hypercyclic operator and for every hyperinvariant MH of T, the inverse of T | M , whenever it exists, is a normaloid operator, then σ(T | M )D and σ(T | M )(C D ¯ ).

Proof Assume that T is a hypercyclic operator. Then there exists a vector xH such that { ( T ) n x } n = 0 ¯ =H. Let S=T | M for some closed T-invariant subspace and let P be the orthogonal projection of H onto M. Since ( S ) n Px=P ( T ) n x for each nN{0} we have

{ ( S ) n ( P x ) } n = 0 ¯ =P { ( T ) n x } n = 0 ¯ =P(H)=M,

thus S is hypercyclic.

From [[20], Corollary 3] we have S >1. Since S is a quasi--class , S is normaloid, thus r(T | M )=S= S >1. Therefore σ(T | M )(C D ¯ ).

Suppose that σ(T | M )(C D ¯ ). Then σ( S 1 ) D ¯ , and since S 1 is normaloid, S 1 =r( S 1 )1. Since S is hypercyclic, from [[20], Theorem 6] ( S ) 1 is hypercyclic, so ( S ) 1 >1. Thus S 1 = ( S ) 1 >1. This is a contradiction, therefore σ(T | M )D. □

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Correspondence to Ilmi Hoxha.

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Hoxha, I., Braha, N.L. The k-quasi--class contractions have property PF. J Inequal Appl 2014, 433 (2014). https://doi.org/10.1186/1029-242X-2014-433

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Keywords

  • k-quasi--class
  • contractions
  • proper contractions
  • Wold-type decomposition
  • PF property
  • supercyclic operator
  • hypercyclic operator