A note on k-quasi-∗-paranormal operators

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

$${\parallel T^{\ast }{T}^{k}x\parallel }^2\le \parallel T^{k+2}x\parallel \parallel T^{k}x\parallel$$

for all $x\in \mathcal{H}$, 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 $〈\cdot ,\cdot 〉$. 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 kerT 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

$$\left|T^2\right|\ge {\left|T^{\ast }\right|}^2.$$

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

$$T^{\ast }\left|T^2\right|T\ge T^{\ast }{\left|T^{\ast }\right|}^{2}T.$$

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

$$T^{\ast k}\left|T^2\right|T^k\ge T^{\ast k}{\left|T^{\ast }\right|}^2{T}^k.$$

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

$${\parallel T^{2}x\parallel }^2\le \parallel T^{3}x\parallel \parallel Tx\parallel$$

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

$${\parallel T^{k+1}x\parallel }^2\le \parallel T^{k+2}x\parallel \parallel T^{k}x\parallel$$

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

$${\parallel T^{\ast }Tx\parallel }^2\le \parallel T^{3}x\parallel \parallel Tx\parallel$$

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

$${\parallel T^{\ast }{T}^{k}x\parallel }^2\le \parallel T^{k+2}x\parallel \parallel T^{k}x\parallel$$

for all $x\in \mathcal{H}$, 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

$$T^{\ast k}(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)T^k\ge 0\mathit{\text{for all}}\lambda \in \mathbb{R}.$$

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

$${\parallel T^{\ast }{T}^{k}x\parallel }^2\le \parallel T^{k+2}x\parallel \parallel T^{k}x\parallel$$

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

$$\begin{array}{c}{\parallel T^{\ast }{T}^{k}x\parallel }^2\le \parallel T^{k+2}x\parallel \parallel T^{k}x\parallel \iff 4{\parallel T^{\ast }{T}^{k}x\parallel }^2-4\parallel T^{k+2}x\parallel \parallel T^{k}x\parallel \le 0\hfill \\ \iff {\parallel T^{k+2}x\parallel }^2-2\lambda {\parallel T^{\ast }{T}^{k}x\parallel }^2+{\lambda }^2{\parallel T^{k}x\parallel }^2\ge 0.\hfill \end{array}$$

The last relation is equivalent to

$$T^{\ast k}(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)T^k\ge 0$$

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

$$T=\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)\text{on }\mathcal{H}=M\oplus M^{\perp }.$$

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

$${\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)}^{\ast k}\{{\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)}^{\ast 2}{\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)}^2-2\lambda \left(\begin{array}{cc}A& B\\ 0& C\end{array}\right){\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)}^{\ast }+{\lambda }^2\}{\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)}^k\ge 0$$

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

Therefore

$$\left(\begin{array}{cc}{A}^{\ast k}(A^{\ast 2}{A}^2-2\lambda (A{A}^{\ast }+B{B}^{\ast })+{\lambda }^2)A^k& E\\ F& G\end{array}\right)\ge 0$$

for some operators E, F and G.

Hence,

$$〈\left(A^{\ast k}(A^{\ast 2}{A}^2-2\lambda A{A}^{\ast }+{\lambda }^2)A^k\right)x,x〉\ge 〈\left(A^{\ast k}2\lambda B{B}^{\ast }{A}^k\right)x,x〉=2|\lambda |{\parallel B^{\ast }{A}^{k}x\parallel }^2\ge 0$$

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

$$\begin{array}{c}〈\left(T^{\ast k}(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)T^k\right)x_n,x_n〉\ge 0\text{for all }\lambda \in \mathbb{R},\hfill \\ 〈(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)T^k{x}_n,T^k{x}_n〉\ge 0\text{for all }\lambda \in \mathbb{R}\text{ and for all }n\in \mathbb{N}.\hfill \end{array}$$

By the continuity of the inner product, we have

$$〈(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)y,y〉\ge 0\text{for all }\lambda \in \mathbb{R}.$$

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

$$T=\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)\mathit{\text{on}}\mathcal{H}=\overline{T^k(\mathcal{H})}\oplus ker{T}^{\ast k}.$$

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:

$$T=\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right)\text{on }\mathcal{H}=\overline{T^k(\mathcal{H})}\oplus ker{T}^{\ast k}.$$

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

$$T^{\ast k}(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)T^k\ge 0\text{for all }\lambda \in \mathbb{R}.$$

Therefore

$$〈(T^{\ast 2}{T}^2-2\lambda T{T}^{\ast }+{\lambda }^2)x,x〉=〈(A^{\ast 2}{A}^2-2\lambda A{A}^{\ast }+{\lambda }^2)x,x〉\ge 0$$

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

$$〈C^k{x}_2,x_2〉=〈T^k(I-P)x,(I-P)y〉=〈(I-P)x,T^{\ast k}(I-P)y〉=0.$$

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 $S^{\ast }S=I$.

$$\begin{array}{rcl}{A}^{\ast 3}{A}^3-2\lambda {\left(A^{\ast }A\right)}^2+{\lambda }^2{A}^{\ast }A& =& {(TS)}^{\ast 3}{(TS)}^3-2\lambda {({(TS)}^{\ast }(TS))}^2+{\lambda }^2{(TS)}^{\ast }(TS)\\ =& T^{\ast 3}{T}^3-2\lambda T^{\ast }T{T}^{\ast }T+{\lambda }^2{T}^{\ast }T\ge 0,\end{array}$$

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

$${\parallel T^{\ast }{T}^k\frac{T^{n-k-2}x}{\parallel T^{n-k-2}x\parallel }\parallel }^2\le \parallel T^{k+2}\frac{T^{n-k-2}x}{\parallel T^{n-k-2}x\parallel }\parallel \parallel T^k\frac{T^{n-k-2}x}{\parallel T^{n-k-2}x\parallel }\parallel ,$$

hence

$${\parallel T^{\ast }{T}^{n-2}x\parallel }^2\le \parallel T^{n}x\parallel \parallel T^{n-2}x\parallel \text{for all }x\in \mathcal{H}.$$

(2.1)

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

$$\parallel T^{\ast }{T}^{n-2}{x}_m\parallel \to 0\text{as }m\to \mathrm{\infty }.$$

(2.2)

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,

$$\begin{array}{rcl}{\parallel {(T-\mu )}^{\ast }x\parallel }^2& =& 〈\overline{\mu }x,\overline{\mu }x〉-〈\overline{\mu }x,T^{\ast }x〉-〈T^{\ast }x,\overline{\mu }x〉+〈T^{\ast }x,T^{\ast }x〉\\ \le & {|\mu |}^2{\parallel x\parallel }^2-{|\mu |}^2{\parallel x\parallel }^2-{|\mu |}^2{\parallel x\parallel }^2+{|\mu |}^2{\parallel x\parallel }^2=0.\end{array}$$

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,

$${\parallel {(T-\mu )}^k\parallel }^2=〈{(T-\mu )}^{\ast }{(T-\mu )}^{k}x,{(T-\mu )}^{k-1}x〉=0,$$

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

$$\begin{array}{rcl}{\parallel T^{k}x\parallel }^4& =& {〈T^{k}x,T^{k}x〉}^2={〈T^{\ast }{T}^{k}x,T^{k-1}x〉}^2\\ \le & {\parallel T^{\ast }{T}^{k}x\parallel }^2{\parallel T^{k-1}x\parallel }^2\le \parallel T^{k+2}x\parallel \parallel T^{k}x\parallel {\parallel T^{k-1}x\parallel }^2\end{array}$$

so

$${\parallel T^{k}x\parallel }^3\le \parallel T^{k+2}x\parallel {\parallel T^{k-1}x\parallel }^2.$$

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. □