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# A note on *k*-quasi-∗-paranormal operators

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 350 (2013)

## Abstract

Let *T* be a bounded linear operator on a complex Hilbert space ℋ. In this paper we introduce a new class of operators satisfying

for all x\in \mathcal{H}, where *k* is a natural number. This class includes the classes of ∗-paranormal and *k*-quasi-∗-class . We prove some of the properties of these operators.

**MSC:**47A10, 47B37, 15A18.

## 1 Introduction

Throughout this paper, let ℋ be a complex separable Hilbert space with inner product \u3008\cdot ,\cdot \u3009. Let \mathcal{L}(\mathcal{H}) denote the {C}^{\ast} algebra of all bounded operators on ℋ. For T\in \mathcal{L}(\mathcal{H}), we denote by ker*T* the null space and by T(\mathcal{H}) the range of *T*. We shall denote the set of all complex numbers and the complex conjugate of a complex number *μ* by ℂ and \overline{\mu}, respectively. The closure of a set *M* will be denoted by \overline{M}, and we shall henceforth shorten T-\mu I to T-\mu. We write \alpha (T)=dimkerT, \beta (T)=dim[\mathcal{H}/T(\mathcal{H})]=dimker{T}^{\ast}, and let \sigma (T), {\sigma}_{p}(T) and {\sigma}_{a}(T) denote the spectrum, point spectrum and approximate point spectrum. Sets of isolated points and accumulation points of \sigma (T) are denoted by iso\sigma (T) and acc\sigma (T), respectively. We write r(T) for the spectral radius. It is well known that r(T)\le \parallel T\parallel. The operator *T* is called *normaloid* if r(T)=\parallel T\parallel.

For an operator T\in \mathcal{L}(\mathcal{H}), as usual, |T|={({T}^{\ast}T)}^{\frac{1}{2}} and [{T}^{\ast},T]={T}^{\ast}T-T{T}^{\ast} (the self-commutator of *T*). An operator T\in \mathcal{L}(\mathcal{H}) is said to be normal if [{T}^{\ast},T] is zero, and *T* is said to be hyponormal if [{T}^{\ast},T] is nonnegative (equivalently if |T|\ge |{T}^{\ast}|). Furuta *et al.* [1] introduced a very interesting class of bounded linear Hilbert space operators: class defined by |{T}^{2}|\ge {|T|}^{2}, which is called the absolute value of *T*, and they showed that the class *A* is a subclass of paranormal operators. Jeon and Kim [2] introduced quasi-class (*i.e.*, {T}^{\ast}|{T}^{2}|T\ge {T}^{\ast}{|T|}^{2}T) operators as an extension of the notion of class operators. Dugall *et al.* [3] introduced ∗-class operator. An operator T\in \mathcal{L}(\mathcal{H}) is said to be a ∗-class operator if

A ∗-class operator is a generalization of a hyponormal operator [[3], Theorem 1.2], and ∗-class is a subclass of the class of ∗-paranormal operators [[3], Theorem 1.3]. We denote the set of ∗-class by {\mathcal{A}}^{\ast}. Shen *et al.* [4] introduced quasi-∗-class operator: An operator T\in \mathcal{L}(\mathcal{H}) is said to be a quasi-∗-class operator if

We denote the set of quasi-∗-class by \mathcal{Q}({\mathcal{A}}^{\ast}). Mecheri [5] introduced *k*-quasi-∗-class operator: An operator T\in \mathcal{L}(\mathcal{H}) is said to be a *k*-quasi-∗-class operator if

We denote the set of *k*-quasi-∗-class operator by \mathcal{Q}({\mathcal{A}}_{k}^{\ast}).

An operator T\in \mathcal{L}(\mathcal{H}) is said to be paranormal if {\parallel Tx\parallel}^{2}\le \parallel {T}^{2}x\parallel for any unit vector *x* in ℋ. Further, *T* is said to be ∗-paranormal if {\parallel {T}^{\ast}x\parallel}^{2}\le \parallel {T}^{2}x\parallel for any unit vector *x* in ℋ. An operator T\in \mathcal{L}(\mathcal{H}) is said to be a quasi-paranormal operator if

for all x\in \mathcal{H}. Mecheri [6] introduced a new class of operators called *k*-quasi-paranormal operators. An operator *T* is called *k*-quasi-paranormal if

for all x\in \mathcal{H}, where *k* is a natural number. Also, Mecheri [7] introduced a new class of operators called quasi-∗-paranormal operators. An operator *T* is called quasi-∗-paranormal if

for all x\in \mathcal{H}. In order to extend the class of paranormal and ∗-paranormal operators, we introduce the class of *k*-quasi-∗-paranormal operators defined as follows.

**Definition 1.1** An operator *T* is called *k*-quasi-∗-paranormal if

for all x\in \mathcal{H}, where *k* is a natural number.

A 1-quasi-∗-paranormal operator is quasi-∗-paranormal.

## 2 Main results

It is well known that *T* is ∗-paranormal if and only if {T}^{\ast 2}{T}^{2}-2\lambda T{T}^{\ast}+{\lambda}^{2}\ge 0 for all \lambda \in \mathbb{R} [8]. Similarly, we can prove the following proposition.

**Proposition 2.1** *An operator* T\in \mathcal{L}(\mathcal{H}) *is* *k*-*quasi*-∗-*paranormal if and only if*

*Proof* Let us suppose that *T* is *k*-quasi-∗-paranormal. Then it follows that the following relation holds:

for all x\in \mathcal{H}, where *k* is a natural number.

The last relation is equivalent to

for every \lambda \in \mathbb{R}. □

**Proposition 2.2** *Let* *M* *be a closed* *T*-*invariant subspace of* ℋ. *Then the restriction* {T}_{|M} *of a* *k*-*quasi*-∗-*paranormal operator* *T* *to* *M* *is a* *k*-*quasi*-∗-*paranormal operator*.

*Proof* Let

Since *T* is *k*-quasi-∗-paranormal, we have

for all \lambda \in \mathbb{R}.

Therefore

for some operators *E*, *F* and *G*.

Hence,

for all \lambda >0. This implies that A={T}_{|M} is a *k*-quasi-∗-paranormal operator. □

**Proposition 2.3** *Let* T\in \mathcal{L}(\mathcal{H}), *k*-*quasi*-∗-*paranormal operator*. *If* {T}^{k} *has dense range*, *then* *T* *is a* ∗-*paranormal operator*.

*Proof* Since {T}^{k} has dense range, \overline{{T}^{k}(\mathcal{H})}=\mathcal{H}. Let y\in \mathcal{H}. Then there exists a sequence {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} in ℋ such that {T}^{k}({x}_{n})\to y as n\to \mathrm{\infty}. Since *T* is a *k*-quasi-∗-paranormal operator, then

By the continuity of the inner product, we have

Therefore *T* is a ∗-paranormal operator. □

**Proposition 2.4** *Let* T\in \mathcal{L}(\mathcal{H}) *be a* *k*-*quasi*-∗-*paranormal operator*, *let the range of* {T}^{k} *not be dense*, *and*

*Then* *A* *is* ∗-*paranormal on* \overline{{T}^{k}(\mathcal{H})}, {C}^{k}=0 *and* \sigma (T)=\sigma (A)\cup \{0\}.

*Proof* Since *T* is a *k*-quasi-∗-paranormal operator and {T}^{k} does not have dense range, we can represent *T* as follows:

Since *T* is a *k*-quasi-∗-paranormal operator, from Proposition 2.1 we have

Therefore

for all \lambda \in \mathbb{R} and for all x\in \overline{{T}^{k}(\mathcal{H})}.

Hence, {A}^{\ast 2}{A}^{2}-2\lambda A{A}^{\ast}+{\lambda}^{2}\ge 0 for all \lambda \in \mathbb{R}. This shows that *A* is ∗-paranormal on \overline{{T}^{k}(\mathcal{H})}.

Let x=\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)\in \mathcal{H}=\overline{{T}^{k}(\mathcal{H})}\oplus ker{T}^{\ast k}. Then

Thus {T}^{\ast k}=0.

Since \sigma (A)\cup \sigma (C)=\sigma (T)\cup \vartheta, where *ϑ* is the union of the holes in \sigma (T), which happen to be a subset of \sigma (A)\cap \sigma (C) by [[9], Corollary 7]. Since \sigma (A)\cap \sigma (C) has no interior points, then \sigma (T)=\sigma (A)\cup \sigma (C)=\sigma (A)\cup \{0\} and {C}^{k}=0. □

The converse of the above proposition is valid when k=1.

**Proposition 2.5** *If* *T* *is a quasi*-∗-*paranormal operator*, *which commutes with an isometric operator* *S*, *then* *TS* *is a quasi*-∗-*paranormal operator*.

*Proof* Let A=TS, TS=ST, {S}^{\ast}{T}^{\ast}={T}^{\ast}{S}^{\ast} and {S}^{\ast}S=I.

so that *T* is a quasi-∗-paranormal operator. □

It is known that there exists a linear operator *T*, so that {T}^{n} is a compact operator for some n\in \mathbb{N}, but *T* itself is not compact. In this context, we will show that in cases where an operator *T* is *k*-quasi-∗-paranormal and if its exponent {T}^{n} is compact, for some n\in \mathbb{N}, then *T* is compact too.

**Proposition 2.6** *Let* *T* *be a* *k*-*quasi*-∗-*paranormal operator such that* {T}^{n} *is compact for some* n\ge k+2. *Then* {T}^{k} *is compact if* k\ge 2 *and* *T* *is compact if* k=0,1.

*Proof* To prove this proposition, it is enough to prove that {T}^{n-1} is compact. Let us consider the unit vector \frac{{T}^{n-k-2}x}{\parallel {T}^{n-k-2}x\parallel}\in \mathcal{H} for n\ge k+2. Since *T* is *k*-quasi-∗-paranormal, then

hence

Let ({x}_{m}) be any sequence in ℋ, satisfying \parallel {x}_{m}\parallel =1 and {x}_{m}\to 0 weakly as m\to \mathrm{\infty}. Now, by the compactness of {T}^{n} and from relation (2.1), we have

If n=2, relation (2.2) implies the compactness of {T}^{\ast}, hence *T* is compact. If n\ge 3, relation (2.2) implies the compactness of {T}^{\ast}{T}^{n-2}, hence {T}^{\ast (n-1)}{T}^{n-1}={T}^{\ast (n-2)}{T}^{\ast}{T}^{n-2}T is compact. Then {T}^{n-1} is a compact operator. □

## 3 SVEP property

Let Hol(\sigma (T)) be the space of all analytic functions in an open neighborhood of \sigma (T). We say that T\in \mathcal{L}(\mathcal{H}) has the single-valued extension property (SVEP) at \mu \in \mathbb{C} if for every open neighborhood *U* of *μ*, the only analytic function f:U\to \mathbb{C} which satisfies the equation (T-\mu )f(\mu )=0 is the constant function f\equiv 0. The operator *T* is said to have SVEP if *T* has SVEP at every \mu \in \mathbb{C}. An operator T\in \mathcal{L}(\mathcal{H}) has SVEP at every point of the resolvent \rho (T)=\mathbb{C}\setminus \sigma (T). Every operator *T* has SVEP at an isolated point of the spectrum.

**Proposition 3.1** *Let* *T* *be a* *k*-*quasi*-∗-*paranormal operator*. *If* \mu \ne 0 *and* (T-\mu )x=0, *then* {(T-\mu )}^{\ast}x=0.

*Proof* We may assume x\ne 0 and (T-\mu )x=0. Since *T* is a *k*-quasi-∗-paranormal operator, then {\parallel {T}^{\ast}{T}^{k}x\parallel}^{2}\le \parallel {T}^{k+2}x\parallel \parallel {T}^{k}x\parallel for all x\in \mathcal{H}. Hence, {\parallel {T}^{\ast}x\parallel}^{2}\le {|\mu |}^{2}{\parallel x\parallel}^{2}. So,

Therefore, {(T-\mu )}^{\ast}x=0. □

For T\in \mathcal{L}(\mathcal{H}), the smallest nonnegative integer *p* such that ker{T}^{p}=ker{T}^{p+1} is called the ascent of *T* and is denoted by p(T). If no such integer exists, we set p(T)=\mathrm{\infty}. We say that T\in \mathcal{L}(\mathcal{H}) is of finite ascent (finitely ascensive) if p(T-\mu )<\mathrm{\infty} for all \mu \in \mathbb{C}.

**Proposition 3.2** *Let* T\in \mathcal{L}(\mathcal{H}) *be a* *k*-*quasi*-∗-*paranormal operator*. *Then* T-\mu *has finite ascent for all* \mu \in \mathbb{C}.

*Proof* We have to tell that ker{(T-\mu )}^{k}=ker{(T-\mu )}^{k+1}. To do that, it is sufficient enough to show that ker{(T-\mu )}^{k+1}\subseteq ker{(T-\mu )}^{k} since ker{(T-\mu )}^{k}\subseteq ker{(T-\mu )}^{k+1} is clear.

Let x\in ker{(T-\mu )}^{k+1}, then {(T-\mu )}^{k+1}x=0. We consider two cases:

If \mu \ne 0, then from Proposition 3.1 we have {(T-\mu )}^{\ast}{(T-\mu )}^{k}x=0. Hence,

so we have {(T-\mu )}^{k}x=0, which implies ker{(T-\mu )}^{k+1}\subseteq ker{(T-\mu )}^{k}.

If \mu =0, then {T}^{k+1}x=0, hence {T}^{k+2}x=0.

Since *T* is a *k*-quasi-∗-paranormal operator, then

so

Since {T}^{k+2}x=0, then {T}^{k}x=0, which implies ker{T}^{k+1}\subseteq ker{T}^{k}. □

**Corollary 3.3** *Let* T\in \mathcal{L}(\mathcal{H}) *be a* *k*-*quasi*-∗-*paranormal operator*. *Then* *T* *has the SVEP property*.

*Proof* The proof of the corollary follows directly from Proposition 3.2. □

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Hoxha, I., Braha, N.L. A note on *k*-quasi-∗-paranormal operators. *J Inequal Appl* **2013**, 350 (2013). https://doi.org/10.1186/1029-242X-2013-350

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DOI: https://doi.org/10.1186/1029-242X-2013-350