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On k-Quasi Class Q* Operators

Valdete Rexhëbeqaj Hamiti*, Shqipe Lohaj, Qefsere Gjonbalaj

Faculty of Electrical and Computer Engineering, University of Prishtina, Prishtinë, Kosova

Abstract: Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class of operators: k-quasi class Q* operators. An operator T is said to be k-quasi class Q* if it satisfies for all x∈H, where k is a natural number. We prove the basic properties of this class of operators.

Keywords: k-quasi class Q*, quasi class Q*, k-quasi -*- paranormal operators, quasi -*- paranormal operators

Citation: Valdete Rexhëbeqaj Hamiti, Shqipe Lohaj, Qefsere Gjonbalaj, On k-Quasi Class Q* Operators. Turkish Journal of Analysis and Number Theory, 4 (4). 87-91. January 2016. doi:10.12691/tjant-4-4-1

1. Introduction

Throughout this paper, let be a complex Hilbert space with inner product . Let denote the algebra of all bounded operators on . For , we denote by the null space, by the range of and by the spectrum of . The null operator and the identity on will be denoted by and , respectively. If is an operator, then is its adjoint, and . For an operator , as usual .

We shall denote the set of all complex numbers by , the set of all non-negative integers by and the complex conjugate of a complex number by . The closure of a set will be denoted by . An operator is a positive operator, , if for all . We write for the spectral radius. It is well known that. The operator is called normaloid if. The operator is an isometry if , for all . The operator is called unitary operator if .

An operator , is said to be paranormal [4], if for any unit vector in . Further, is said to be paranormal [1,9], if for any unit vector in . An operator , is said to be quasi paranormal operator if , for all .

Mecheri [7] introduced a new class of operators called quasi paranormal operators. An operator is called quasi paranormal if , for all , where is a natural number. An operator is called quasi paranormal [8,11], if , for all .

An operator is called quasi paranormal if for all , where is a natural number, [6].

Shen, Zuo and Yang [13] introduced a new class of operator quasi class . An operator is said to be a quasiclass , if .

Mecheri [12] introduced quasi class operator. An operator is said to be a quasi class , if .

Duggal, Kubrusly, Levan [3] introduced a new class of operators, the class . An operator belongs to class if , or equivalent , for all .

Senthilkumar, Prasad [10] introduced a new class of operators, the class . An operator belongs to class if , or equivalent for all .

Senthilkumar, Naik and Kiruthika [2] introduced a new class of operators, the quasi class . An operator is said to belong to the quasi class if , or equivalent for all .

Now we introduce the class of quasi class operators defined as follows:

Definition 1.1. An operator is said to be of the quasi class if

for all , where is a natural number.

Remark 1.2. For , a quasi class operators is a quasi class operators.

2. Main Results

Proposition 2.1. An operator is of the quasi class , if and only if

where is a natural number.

Proof: Since is operator of the quasi class , then

for all , where is a natural number.

for all , where is a natural number.

The last relation is equivalent to

From the definition of the class operators, quasi class operators and the proposition 2.1 we see that every operator of the class and every operator of the quasi class is also an operator of the quasi class . Thus, we have the following implication:

Corollary 2.2. A weighted shift operator with decreasing weighted sequence is an operator of the quasi class if and only if

for all .

Proof: Since is a weighted shift operator, its adjoint is also a wighted shift operator, then:

Since, is an operator of the quasi class then after some calculations we have:

Now we will give an example of quasi class operator which is not quasi class operator.

Example 2.3. Consider the operator in defined by , where

Then is an operator of the quasi class but this operator is not quasi class .

Given:

Now from the proposition 2.1 and corollary 2.2, for quasi class operator we have:

But for quasi class operator we have:

In the following we prove that if is an operator of the quasi class and if the range of is dense, then is an operator of the class .

Proposition 2.4. Let be an operator of the quasi class . If has dense range, then is an operator of the class .

Proof: Since has dense range, then Let be . Then there exist a sequence in such that Since is an operator of the quasi class , then:

By the continuity of the inner product, we have

So,

Therefore is an operator of the class .

In the following we give the relations between quasi class and quasi paranormal operators.

Hoxha and Braha [[6], Proposition 2.1] prove that an operator is of the quasi paranormal if and only if for all

From this we have that every quasi paranormal is operator of the quasi class . Also, every quasi paranormal is operator of the quasi class .

Proposition 2.5. Let . If is an operator of the quasi class , then is a quasi paranormal operator for all .

Proof: Let be an operator of quasi class , for all , then:

By this it is proved that the is quasi paranormal operator.

Remark 2.6. If is an operator of the quasi class , then is a quasi paranormal operator for all .

Proposition 2.7. If is an operator of the quasi class and is an isometry, then is quasi paranormal operator.

Proof: Let be an operator of the quasi class , then:

Since operator is an isometry, then for all .

Then,

so we have,

So, is quasi paranormal operator.

In the following we give the relation between quasi class and quasi class operators.

Proposition 2.8. If belongs to the quasi class , for a natural number, then is an operator of quasi class .

Proof: Since belongs to quasi class operators, we have:

where is a natural number.

Let . Then,

Therefore,

Hence, is an operator of the quasi class .

Remark 2.9. If belongs to the quasi class , then is an operator of quasi class .

In following we give an example of operator which is operator of the quasi class , but not quasi class .

Example 2.10. Let , where . Given positive operators , , define the operator on as follows:

The operator is quasi class if and only if .

Let and be operator as

Then,

Hence is not quasi class .

Then a computation shows that the operator is quasi class if and only if

So,

Therefore is operator of the quasi class .

Proposition 2.11. Let If then is operator of the quasi class .

Proof: From we have

Then,

so is operator of the quasi class .

Proposition 2.12. If is an operator of the quasi class and if commutes with an isometric operator , then is an operator of the quasi class .

Proof: Let . Then

Hence is an operator of the quasi class .

Proposition 2.13. Let be an operator of the quasi class and if is unitarily equivalent to operator , then is an operator of the quasi class .

Proof: Since is unitarily equivalent to operator , there is an unitary operator such that .

Since is an operator of the quasi class , then

Hence,

so, is an operator of the quasi class .

Proposition 2.14. Let be a closed invariant subset of . Then, the restriction of a quasi class operator to is a quasi class operator.

Proof: Let be . Then

This implies that is an operator of quasi class .

Proposition 2.15. Let be a quasi class operator, the range of not to be dense, and

Then, is an operator of the class on and .

Proof: Suppose that is an operator of quasi class . Since does not have dense range, we can represent as the upper triangular matrix:

Since is an operator of quasi class , we have

Therefore

for all .

Hence

This shows that is an operator of the class on .

Let be the orthogonal projection of onto .

For any

We have

Thus .

Since,

where is the union of the holes in , which happen to be a subset of by [[5], Corollary 7].

Since, have no interior points, then and .

3. Conclusion

In this paper we introduce a new class of operators: quasi class operators. It is proved that the following impication is true

With example it is shown that, exist a quasi class operator which is not quasi class (Example 2.3). Further, it is proved that if is an operator of the quasi class and if the range of is dense, then is an operator of the class (Proposition 2.4).

It is shown the relation between quasi class and quasi paranormal operators (Proposition 2.5, Remark 2.6 and Proposition 2.7). Also it is shown the relation between quasi class and quasi class operators (Proposition 2.8, Remark 2.9 and Example 2.10).

Finally is proved that every operator which satisfy the condition is operator of the quasi class (Proposition 2.11).

References

[1]   | S. C. Arora and J. K. Thukral, “On a class of operators,” Glasnik Matematicki, 21(41), (1986), 381-386.
[2]   | D. Senthilkumar, P. Maheswari Naik and D. Kiruthika,“Quasi class Q* composition operators,” International Journal of Math. Sci. and Engg. Appls. (IJMSEA), ISSN 0973-9424, Vol. 5, No IV, July, 2011, pp. 1-9.
[3]   | B. P. Duggal, C. S. Kubrusly, and N. Levan, “Contractions of class Q and invariant subspaces,” Bull. Korean Math. Soc. 42(2005), No. 1, pp. 169-177.
[4]   | T. Furuta,“On the class of paranormal operators,” Proc. Jap. Acad. 43(1967), 594-598.
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[6]   | Ilmi Hoxha and Naim L. Braha, “A note on k- quasi -*- paranormal operators,” Journal of Inequalities and Applications 2013, 2013:350.
[7]   | Salah Mecheri, “Bishops property β and Riesz idempotent for k-quasi-paranormal operators,” Banach J. Math. Anal., 6(2012), No. 1, 147-154.
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[9]   | S. M. Patel, “Contributions to the study of spectraloid operators,” Ph. D. Thesis, Delhi University, 1974.
[10]   | D. Senthilkumar and T. Prasad, “Composition operators of class Q*,” Int. Journal of Math. Analysis, Vol. 4, 2010, no. 21, 1035-1040.
[11]   | T. Veluchamy and A. Devika, “Some properties of quasi -*- paranormal operators,” Journal of Modern Mathematics and Statistics, 1 (1-4), 35-38, 2007.
[12]   | S. Mecheri, “Isolated points of spectrum of k- quasi -*- class A operators,” Studia Mathematica, 208(2012), 87-96.
[13]   | J. L. Shen, F. Zuo and C. S. Yang, “On operators satisfying ,” Acta Mathematica Sinica, English Series, Nov., Vol. 26(2010), no.11, pp. 2109-2116.