2. On k-quasi class Q operators

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 6 Issue 3 (2014), Pages 31-37.

ON k−QUASI CLASS Q OPERATORS

(COMMUNICATED BY T. YAMAZAKI)

VALDETE REXH ¨EBEQAJ HAMITI

Abstract. LetTbe a bounded linear operator on a complex Hilbert spaceH. In this paper we introduce a new class of operators:k−quasi classQoperators. An operatorT is said to bek−quasi classQif it satisfies

kTk+1xk2_≤1

2(kT

k+2_xk2_+kTk_xk2_),

for allx∈ H, wherekis a natural number. We prove the basic properties of this class of operators.

1. _Introduction

Throughout this paper, letHbe a complex Hilbert space with inner producth·,·i.

LetL(H) denote the C∗ algebra of all bounded operators onH.ForT ∈ L(H),we denote by kerT the null space, byT(H) the range ofT and byσ(T) the spectrum of

T. The null operator and the identity onHwill be denoted byOandI, respectively. IfT is an operator, thenT∗ is its adjoint, andkTk=kT∗_k.

We shall denote the set of all complex numbers byC, the set of all non-negative

integers byNand the complex conjugate of a complex numberλbyλ.The closure

of a set M will be denoted byM . An operator T ∈ L(H) is a positive operator,

T ≥O,ifhT x, xi ≥0 for all x∈ H.The operatorT is an isometry ifkT xk=kxk,

for allx∈ H. The operatorT is called unitary operator ifT∗T =T T∗=I.

Duggal, Kubrusly, Levan [3] introduced a new class of operators, the class Q. An operatorT ∈ L(H) belongs to classQif

T∗2T2−2T∗T+I≥O.

It is proved that an operatorT ∈ L(H) is of classQif

kT xk2_≤ 1 2(kT

2_xk2_+kxk2_).

Devika, Sureshi [2] introduced a new class of operators, the quasi class Q. An operatorT ∈ L(H) is said to belong to the quasi class Qif

T∗3T3−2T∗2T2+T∗T ≥O.

2010Mathematics Subject Classification. 47B47, 47B20.

Key words and phrases. k−quasi class Q; quasi class Q; k−quasi−paranormal operators; quasi−paranormal operators.

c

2014 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted May 19, 2014. Published Jul 31, 2014.

It is proved that an operatorT ∈ L(H) is of the quasi classQif

kT2_xk2_≤ 1 2(kT

3_xk2_{+kT xk}2_).

Now we introduce the class of k−quasi class Q operators, which is a common generalization of classQand quasi classQoperators, defined as follows:

Definition 1.1. An operator T is said to be of thek−quasi classQif

kTk+1_xk2_≤ 1 2 kT

k+2_xk2_+kTk_xk2

,

for allx∈ H, wherek is a natural number.

A 1−quasi classQoperator is a quasi classQoperator. 2. Main results

In this section we prove some basic properties of k−quasi class Q operators. Similarly as Devika, Sureshi in [2, Theorem 1.1], we can prove the following propo-sition.

Proposition 2.1. An operatorT ∈ L(H)is of thek−quasi classQ, if and only if

T∗k(T∗2T2−2T∗T+I)Tk ≥O,

wherek is a natural number.

Proof. SinceT is of thek−quasi classQ,then

2kTk+1_xk2_{≤ kT}k+2_xk2_+kTk_xk2_, for allx∈ H, where kis a natural number.

hT∗(k+2)Tk+2x, xi −2hT∗(k+1)Tk+1x, xi +hT∗kTkx, xi ≥0 for allx∈ H, where kis a natural number.

Then

hT∗k(T∗2T2−2T∗T +I)Tkx, xi ≥0 for allx∈ H, where kis a natural number.

The last relation is equivalent to

T∗k(T∗2T2−2T∗T+I)Tk ≥O.

From the definition of the classQoperator

T∗2T2−2T∗T+I≥O,

and the proposition 2.1 we see that every operator of the classQis also an operator of thek−quasi classQ.Thus, we have the following implication:

classQ⊆quasi classQ⊆k-quasi classQ.

An operatorT ∈ L(H) is said to be paranormal, ifkT xk2_{≤ kT}2_{xkfor any unit} vectorxin H. Further, T is said to be∗−paranormal, if kT∗xk2_{≤ kT}2_{xk for any} unit vector x in H [1]. An operator T ∈ L(H) is said to be quasi−paranormal operator if

kT2_xk2_{≤ kT}3_{xkkT xk,}

for allx∈ H. An operatorT is called quasi− ∗ −paranormal if

for allx∈ H[10, 11, 12].

Mecheri, [9] introduced a new class of operators calledk−quasi paranormal op-erators. An operatorT is called k−quasi−paranormal if

kTk+1_xk2_{≤ kT}k+2_xkkTk_xk,

for allx∈ H, wherekis a natural number. It is proved that an operatorT ∈ L(H) is ak−quasi−paranormal if and only if

T∗k(T∗2T2−2λT∗T+λ2)Tk≥0, for allλ >0.

An operatorT is calledk−quasi− ∗ −paranormal if

kT∗Tkxk2_{≤ kT}k+2_xkkTk_xk, for allx∈ H, where kis a natural number [7].

Then we have that everyk−quasi−paranormal operator is operator of thek−quasi classQ. Also, every quasi−paranormal operator is operator of the quasi classQ.

In the following we will prove that ifλ−12T is an operator of the k−quasi class

Q,thenT is ak−quasi−paranormal operator for allλ >0.

Proposition 2.2. Let T ∈ L(H). If λ−12T is an operator of thek−quasi class Q,

thenT is ak−quasi −paranormal operator for all λ >0.

Proof. Letλ−1

2T be an operator ofk−quasi classQ,for allλ >0,then

(λ−12T)∗k

(λ−12T)∗2(λ− 1

2T)2−2(λ− 1 2T)∗(λ−

1 2T) +I

(λ−12T)k≥0, λ >0⇒

λ−k2_T∗k_(λ−2_T∗2_T2−_2λ−1_T∗_T+I)λ−k2_Tk≥_{0, λ >0}⇒

1

λk+2T

∗k_(T∗2_T2_−2λT∗_T+λ2_)Tk _{≥0, λ >0⇒}

T∗k(T∗2T2−2λT∗T+λ2)Tk≥0 for allλ >0.

By this it is proved that the operatorT isk−quasi−paranormal operator.

If λ−12T is an operator of the quasi class Q, then T is a quasi −paranormal

operator for allλ >0.

Kim, Duggal and Jeon [8] introduced a new class of operator quasi−classA: An operatorT ∈ L(H) is said to be a quasi−classAoperator, if

T∗|T2_{|T ≥T}∗_|T|2_T.

Gao and Fang [4] introducedk−quasi−classAoperator: An operatorT ∈ L(H) is said to be ak−quasi−classAoperator, if

T∗k|T2_|Tk_≥T∗k_|T|2_Tk_.

Gao and Li in [5] give the relation betweenk−quasi−paranormal operator and

k−quasi−classAoperator. Motivated by this in the following we give the relations betweenk−quasi classQandk−quasi−classAoperators.

Proof. SinceT belongs tok−quasi−classA, we have

T∗k|T2_|Tk_≥T∗k_|T|2_Tk_. Letx∈ H. Then

2kTk+1_xk2₌

2hT∗(k+1)_Tk+1_{x, xi =} 2hT∗k|T|2Tkx, xi ≤

2hT∗k|T2_|Tk_{x, xi ≤} 2k|T2_|Tk_{xk · kT}k_xk= 2kTk+2_{xk · kT}k_{k ≤}

kTk+2_xk2_+kTk_xk2 Therefore

2kTk+1_xk2_{≤ kT}k+2_xk2_+kTk_xk2_.

Hence,T is an operator of thek−quasi classQ.

If T ∈ L(H) belongs to the quasi−class A, then T is an operator of the quasi classQ.

In following we give an example which T is operator of the k−quasi class Q, but notk−quasi−classA.

Example 2.4. Let T =

1 0 1 0

∈ L(l2⊕l2). Then T is operator of thek−quasi classQ, but notk−quasi−classA.

By simple calculation we have that

T∗k|T2|Tk=

√

2 0

0 0

and

T∗k|T|2_Tk₌

2 0 0 0

.

Hence T is notk−quasi−classA. However,

T∗2T2−2T∗T+I=

−1 0

0 1

,

we have

T∗k(T∗2T2−2T∗T+I)Tk =

0 0 0 0

.

Therefore T is operator of thek−quasi class Q.

In [7, 12] author proved that if quasi−∗−paranormal operator double commutes with an isometric operator then their product also is a quasi − ∗ −paranormal operator. We shall give a similar result for a quasi classQoperator.

Proof. LetA=T S,T S=ST,S∗T =T S∗ andS∗S=I.

A∗3A3−2A∗2A2+A∗A

= (T S)∗3(T S)3−2(T S)∗2(T S)2+ (T S)∗(T S) = T∗3T3−2T∗2T2+T∗T ≥O,

soT S is an operator of the quasi classQ.

Proposition 2.6. If T is an operator of the quasi class Q and if T is unitarily equivalent to operatorS, thenS is an operator of thek−quasi class Q.

Proof. Since T is unitarily equivalent to operator S, there is an unitary operator

U such thatS=U∗T U. SinceT is an operator of the quasi classQ, then

T∗k(T∗2T2−2T∗T+I)Tk ≥O.

Hence,

S∗k(S∗2S2−2S∗S+I)Sk =

(U∗T U)∗k((U∗T U)∗2(U∗T U)2−2(U∗T U)∗(U∗T U) +I)(U∗T U)k=

U∗kT∗k(T∗2T2−2T∗T+I)TkUk≥O,

soS is an operator of thek−quasi classQ.

Proposition 2.7. Let T ∈ L(H). IfkTk ≤ _√1

2,then T is operator of thek−quasi classQ.

Proof. FromkTk ≤ _√1

2,we have kT xk 2_≤1

2.Then,

O≤I−2T∗T ≤T∗2T2−2T∗T+I, T∗k(T∗2T2−2T∗T+I)Tk≥0 soT is of thek−quasi classQ.

Proposition 2.8. IfT is a k−quasi classQoperator and T2_{is an isometry, then}

T isk−quasi −paranormal operator.

Proof. LetT be ak−quasi classQoperator. Then

2kTk+1_xk2_≤(kTk+2_{xk − kT}k_xk)2_{+ 2kT}k+2_xkkTk_{xk. (2.1)} Suppose thatT2_{is isometry, so kT}2_{xk=kxk,for allx∈ H. Then,}

kTk+2_xk=kTk_xk and from relation (2.1) we have

kTk+1_xk2_{≤ kT}k+2_xkkTk_xk,

soT isk−quasi−paranormal operator.

Proof. Let beu∈M. Then,

k(T |M)k+1uk2

= kTk+1_uk2_≤ 1 2 kT

k+2_uk2_+kTk_uk2

= 1

2 k(T |M)

k+2_uk2_{+k(T |} M)kuk2

This implies thatT |M is an operator of thek−quasi classQ. In the following we prove that ifT is an operator of thek−quasi classQand if the range ofTk is dense, thenT is an operator of the classQ.

Proposition 2.10. LetT ∈ L(H)be an operator of thek−quasi classQ.IfTk _has dense range, then T is an operator of the classQ.

Proof. Since Tk _{has dense range, T}k_{(H) = H. Let y ∈ H. Then there exists a} sequence{xn}∞n=1 in Hsuch that Tk(xn)→y,n→ ∞. SinceT is an operator of thek−quasi classQ, then

(T∗k(T∗2T2−2T∗T+I)Tk)xn, xn

≥0, h(T∗2T2−2T∗T+I)Tkxn, Tkxni ≥0, for alln∈N.

By the continuity of the inner product, we have

h(T∗2T2−2T∗T+I)y, yi ≥0

ThereforeT is an operator of the classQ.

In [9], S. Mecheri studied the matrix representation of k−quasi−paranormal operator with respect to the direct sum ofTk_{(H) and its orthogonal complement.} In the following we give an equivalent condition for operator ofk−quasi classQ. Proposition 2.11. LetT ∈ L(H) be ak−quasi classQoperator, the range of Tk not to be dense, and

T =

A B

O C

on H=Tk_(H)⊕kerT∗k_.

Then,Ais an operator of the classQonTk_(H),Ck_{=Oandσ(T) =σ(A)∪{0}.} Proof. Suppose thatT ∈ L(H) is an operator ofk−quasi class Q. Since that Tk

does not have dense range, we can representT as the upper triangular matrix:

T =

A B

0 C

on H=Tk_(H)⊕kerT∗k_.

SinceT is an operator ofk−quasi classQ, we have

T∗k(T∗2T2−2T∗T+I)Tk≥0.

Therefore

h(T∗2T2−2T∗T+I)x, xi=h(A∗2A2−2A∗A+I)x, xi ≥0,

for allx∈Tk_(H). Hence

A∗2A2−2A∗A+I≥0.

LetP be the orthogonal projection ofH ontoTk_{(H). For any}

x=

x1

x2

∈ H=Tk_(H)⊕kerT∗k_.

Then

hCkx2, x2i=hTk(I−P)x,(I−P)xi=h(I−P)x, T∗k(I−P)xi= 0. ThusT∗k= 0.

Sinceσ(A)∪σ(C) =σ(T)∪ϑ, whereϑis the union of the holes inσ(T), which happen to be a subset of σ(A)∩σ(C) by [6, Corollary 7]. Since σ(A)∩σ(C) has no interior points, thenσ(T) =σ(A)∪σ(C) =σ(A)∪ {0} andCk_{= 0.}

Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.

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Valdete Rexh¨ebeqaj Hamiti

Faculty of Electrical and Computer Engineering, University of Prishtina, Prishtin¨e, 10000, Kosova.