On ∗-class A contractions

Abstract

A Hilbert space operator T belongs to ∗-class A if $|T^2|-{|T^{\ast }|}^2\ge 0$. The famous Fuglede-Putnam theorem is as follows: the operator equation $AX=XB$ implies $A^{\ast }X=X{B}^{\ast }$ when A and B are normal operators. In this paper, firstly we prove that if T is a contraction of ∗-class A operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator $D=|T^2|-{|T^{\ast }|}^2$ is a strongly stable contraction; secondly, we show that if X is a Hilbert-Schmidt operator, A and ${(B^{\ast })}^{-1}$ are ∗-class A operators such that $AX=XB$, then $A^{\ast }X=X{B}^{\ast }$.

MSC:47B20, 47A63.

1 Introduction

Let ℋ be a complex Hilbert space and let ℂ be the set of complex numbers. Let $B(\mathcal{H})$ denote the $C^{\ast }$-algebra of all bounded linear operators acting on ℋ. For operators $T\in B(\mathcal{H})$, we shall write kerT and ranT for the null space and the range of T, respectively. Also, let $\sigma (T)$ denote the spectrum of T.

Recall that $T\in B(\mathcal{H})$ is called p-hyponormal for $p>0$ if ${(T^{\ast }T)}^p-{(T{T}^{\ast })}^p\ge 0$ [1]; when $p=1$, T is called hyponormal. And T is called paranormal if ${\parallel Tx\parallel }^2\le \parallel T^{2}x\parallel \parallel x\parallel$ for all $x\in \mathcal{H}$ [2, 3]. And T is called normaloid if $\parallel T^n\parallel ={\parallel T\parallel }^n$ for all $n\in \mathbb{N}$ (equivalently, $\parallel T\parallel =r(T)$, the spectral radius of T). In order to discuss the relations between paranormal and p-hyponormal and log-hyponormal operators (T is invertible and $log{T}^{\ast }T\ge logT{T}^{\ast }$), Furuta et al. [4] introduced a very interesting class of operators: class A defined by $|T^2|-{|T|}^2\ge 0$, where $|T|={(T^{\ast }T)}^{\frac{1}{2}}$ which is called the absolute value of T; and they showed that the class A is a subclass of paranormal and contains p-hyponormal and log-hyponormal operators. Recently Duggal et al. [5] introduced ∗-class A operators (i.e., $|T^2|-{|T^{\ast }|}^2\ge 0$) and ∗-paranormal operators (i.e., ${\parallel T^{\ast }x\parallel }^2\le \parallel T^{2}x\parallel \parallel x\parallel$ for all $x\in \mathcal{H}$); and they proved that a ∗-class A operator is a generalization of hyponormal operator and ∗-class A operators form a subclass of the class of ∗-paranormal operators.

A contraction is an operator T such that $\parallel T\parallel \le 1$. A contraction T is said to be a proper contraction if $\parallel Tx\parallel <\parallel x\parallel$ for every nonzero $x\in \mathcal{H}$. A strict contraction is an operator T such that $\parallel T\parallel <1$. A strict contraction is a proper contraction, but a proper contraction is not necessary a strict contraction, although the concepts of strict and proper contractions coincide for compact operators. A contraction T is of class $C_{0\cdot }$ if $\parallel T^{n}x\parallel \to 0$ when $n\to \mathrm{\infty }$ for every $x\in \mathcal{H}$ (i.e., T is a strongly stable contraction) and it is said to be of class $C_{1\cdot }$ if ${lim}_{n\to \mathrm{\infty }}\parallel T^{n}x\parallel >0$ for every nonzero $x\in \mathcal{H}$. Classes $C_{\cdot 0}$ and $C_{\cdot 1}$ are defined by considering $T^{\ast }$ instead of T, and we define the class $C_{\alpha \beta }$ for $\alpha ,\beta =0,1$ by $C_{\alpha \beta }=C_{\alpha \cdot }\cap C_{\cdot \beta }$. An isometry is a contraction for which $\parallel Tx\parallel =\parallel x\parallel$ for every $x\in \mathcal{H}$.

In this paper, firstly we prove that if T is a contraction of ∗-class A operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator $D=|T^2|-{|T^{\ast }|}^2$ is a strongly stable contraction; secondly, we show that if X is a Hilbert-Schmidt operator, A and ${(B^{\ast })}^{-1}$ are ∗-class A operators such that $AX=XB$, then $A^{\ast }X=X{B}^{\ast }$.

2 On ∗-class A contractions

Theorem 2.1 If T is a contraction of ∗-class A operators, then the nonnegative operator $D=|T^2|-{|T^{\ast }|}^2$ is a contraction whose power sequence $\{D^n\}$ converges strongly to a projection P, and $T^{\ast }P=0$.

Proof Suppose that T is a contraction of ∗-class A operators, then $D=|T^2|-{|T^{\ast }|}^2\ge 0$. Let $R=D^{\frac{1}{2}}$. Then for every $x\in \mathcal{H}$,

$$\begin{array}{rl}〈D^{n+1}x,x〉& ={\parallel R^{n+1}x\parallel }^2=〈D{R}^{n}x,R^{n}x〉\\ =〈\left|T^2\right|R^{n}x,R^{n}x〉-〈{\left|T^{\ast }\right|}^2{R}^{n}x,R^{n}x〉\\ ={\parallel {\left|T^2\right|}^{\frac{1}{2}}{R}^{n}x\parallel }^2-{\parallel \left|T^{\ast }\right|R^{n}x\parallel }^2\\ \le {\parallel R^{n}x\parallel }^2-{\parallel T^{\ast }{R}^{n}x\parallel }^2\\ \le {\parallel R^{n}x\parallel }^2\\ =〈D^{n}x,x〉.\end{array}$$

Thus R (and so D) is a contraction and $\{D^n\}$ is a decreasing sequence of nonnegative contractions. Hence $\{D^n\}$ converges strongly to a projection P. Moreover,

$$\sum _{n=0}^m{\parallel T^{\ast }{R}^{n}x\parallel }^2\le \sum _{n=0}^m({\parallel R^{n}x\parallel }^2-{\parallel R^{n+1}x\parallel }^2)={\parallel x\parallel }^2-{\parallel R^{m+1}x\parallel }^2\le {\parallel x\parallel }^2$$

for all nonnegative integers m and every $x\in \mathcal{H}$. Therefore $\parallel T^{\ast }{R}^{n}x\parallel \to 0$ as $n\to \mathrm{\infty }$, hence we have

$$T^{\ast }Px=T^{\ast }\underset{n\to \mathrm{\infty }}{lim}{D}^{n}x=\underset{n\to \mathrm{\infty }}{lim}{T}^{\ast }{R}^{2n}x=0$$

for every $x\in \mathcal{H}$. So that $T^{\ast }P=0$. □

Theorem 2.2 Let T be a contraction of ∗-class A operators. If T has no nontrivial invariant subspace, then

• (i) T is a proper contraction;

• (ii) the nonnegative operator $D=|T^2|-{|T^{\ast }|}^2$ is a strongly stable contraction (so that $D\in C_{00}$).

Proof (i) Suppose that T is a ∗-class A operator, then ${|T^{\ast }|}^2\le |T^2|$. We have

$${\parallel T^{\ast }x\parallel }^2=〈{\left|T^{\ast }\right|}^{2}x,x〉\le 〈\left|T^2\right|x,x〉\le \parallel \left|T^2\right|x\parallel \parallel x\parallel =\parallel T^{2}x\parallel \parallel x\parallel$$

for every $x\in \mathcal{H}$. By [6] Theorem 3.6, we have that

$$T^{\ast }Tx={\parallel T\parallel }^{2}x\text{if and only if}\parallel Tx\parallel =\parallel T\parallel \parallel x\parallel .$$

Put $\mathcal{U}=\{x\in \mathcal{H}:\parallel Tx\parallel =\parallel T\parallel \parallel x\parallel \}=ker({|T|}^2-{\parallel T\parallel }^2)$, which is a subspace of ℋ. In the following, we shall show that $\mathcal{U}$ is an invariant subspace of T. For every $x\in \mathcal{U}$, we have

$$\begin{array}{rl}{\parallel T(Tx)\parallel }^2& \le {\parallel T\parallel }^2{\parallel Tx\parallel }^2={\parallel T\parallel }^4{\parallel x\parallel }^2={\parallel {\parallel T\parallel }^{2}x\parallel }^2={\parallel T^{\ast }Tx\parallel }^2\\ \le \parallel T^{2}Tx\parallel \parallel Tx\parallel =\parallel T^{2}Tx\parallel \parallel T\parallel \parallel x\parallel ,\end{array}$$

(2.1)

where the second inequality holds since T is a ∗-class A operator. So, we have that

$${\parallel T\parallel }^4{\parallel x\parallel }^2\le \parallel T^{2}Tx\parallel \parallel T\parallel \parallel x\parallel ,$$

that is, ${\parallel T\parallel }^3\parallel x\parallel \le \parallel T^{2}Tx\parallel$. Hence we have

$${\parallel T\parallel }^3\parallel x\parallel =\parallel T^{2}Tx\parallel .$$

(2.2)

By (2.2), we have

$${\parallel T\parallel }^3\parallel x\parallel =\parallel T^{2}Tx\parallel \le \parallel T\parallel \parallel T(Tx)\parallel ,$$

(2.3)

that is, ${\parallel T\parallel }^2\parallel x\parallel \le \parallel T(Tx)\parallel$. Hence

$${\parallel T\parallel }^2\parallel x\parallel =\parallel T(Tx)\parallel .$$

(2.4)

Hence by (2.1) and (2.4), we have

$$\parallel T^{2}Tx\parallel \parallel T\parallel \parallel x\parallel ={\parallel T(Tx)\parallel }^2.$$

So, we have that $\parallel T(Tx)\parallel =\parallel T\parallel \parallel Tx\parallel$. That is, $\mathcal{U}$ is an invariant subspace of T. Now suppose T is a contraction of ∗-class A operators. If T is a strict contraction, then it is trivially a proper contraction. If T is not a strict contraction (i.e., $\parallel T\parallel =1$) and T has no nontrivial invariant subspace, then $\mathcal{U}=\{x\in \mathcal{H}:\parallel Tx\parallel =\parallel x\parallel \}=\{0\}$ (actually, if $\mathcal{U}=\mathcal{H}$, then T is an isometry, and isometries have nontrivial invariant subspaces). Thus, for every nonzero $x\in \mathcal{H}$, $\parallel Tx\parallel <\parallel x\parallel$, so T is a proper contraction.

• (ii) Let T be a contraction of ∗-class A operators. By Theorem 2.1 we have D is a contraction, $\{D^n\}$ converges strongly to a projection P, and $T^{\ast }P=0$. So, $PT=0$. Suppose T has no nontrivial invariant subspace. Since kerP is a nonzero invariant subspace for T whenever $PT=0$ and $T\ne 0$, it follows that $kerP=\mathcal{H}$. Hence $P=0$ and so $D^n$ converges strongly to 0, that is, $D=|T^2|-{|T^{\ast }|}^2$ is a strongly stable contraction. D is self-adjoint, so that $D\in C_{00}$. □

Since a self-adjoint operator T is a proper contraction if and only if T is a $C_{00}$-contraction, we have the following corollary by Theorem 2.2.

Corollary 2.3 Let T be a contraction of ∗-class A operators. If T has no nontrivial invariant subspace, then both T and the nonnegative operator $D=|T^2|-{|T^{\ast }|}^2$ are proper contractions.

3 The Fuglede-Putnam theorem for ∗-class A operators

The famous Fuglede-Putnam theorem is as follows [3, 7, 8].

Theorem 3.1 Let A and B be normal operators and X be an operator such that $AX=XB$, then $A^{\ast }X=X{B}^{\ast }$.

The Fuglede-Putnam theorem was first proved in the case $A=B$ by Fuglede [7] and then a proof in the general case was given by Putnam [8]. Berberian [9] proved that the Fuglede theorem was actually equivalent to that of Putnam by a nice operator matrix derivation trick. Rosenblum [10] gave an elegant and simple proof of the Fuglede-Putnam theorem by using Liouville’s theorem. There were various generalizations of the Fuglede-Putnam theorem to nonnormal operators; we only cite [11–14]. For example, Radjabalipour [13] showed that the Fuglede-Putnam theorem holds for hyponormal operators; Uchiyama and Tanahashi [14] showed that the Fuglede-Putnam theorem holds for p-hyponormal and log-hyponormal operators. If let $X\in B(\mathcal{H})$ be Hilbert-Schmidt class, Mecheri and Uchiyama [15] showed that normality in the Fuglede-Putnam theorem can be replaced by A and $B^{\ast }$ class A operators. In this paper, we show that if X is a Hilbert-Schmidt operator, A and ${(B^{\ast })}^{-1}$ are ∗-class A operators such that $AX=XB$, then $A^{\ast }X=X{B}^{\ast }$.

Let ${\mathcal{C}}_2(\mathcal{H})$ denote the Hilbert-Schmidt class. For each pair of operators $A,B\in B(\mathcal{H})$, there is an operator ${\mathrm{\Gamma }}_{A,B}$ defined on ${\mathcal{C}}_2(\mathcal{H})$ via the formula ${\mathrm{\Gamma }}_{A,B}(X)=AXB$ in [11]. Obviously, $\parallel {\mathrm{\Gamma }}_{A,B}\parallel \le \parallel A\parallel \parallel B\parallel$. The adjoint of ${\mathrm{\Gamma }}_{A,B}$ is given by the formula ${\mathrm{\Gamma }}_{A,B}^{\ast }(X)=A^{\ast }X{B}^{\ast }$; see details [11].

Let $A\otimes B$ denote the tensor product on the product space $\mathcal{H}\otimes \mathcal{H}$ for non-zero $A,B\in B(\mathcal{H})$. In [5], Duggal et al. give a necessary and sufficient condition for $A\otimes B$ to be a ∗-class A operator.

Lemma 3.2 (see [5])

Let $A,B\in B(\mathcal{H})$ be non-zero operators. Then $A\otimes B$ belongs to ∗-class A operators if and only if A and B belong to ∗-class A operators.

Theorem 3.3 Let $A\mathit{\text{and}}B\in B(\mathcal{H})$. Then ${\mathrm{\Gamma }}_{A,B}$ is a ∗-class A operator on ${\mathcal{C}}_2(\mathcal{H})$ if and only if A and $B^{\ast }$ belong to ∗-class A operators.

Proof The unitary operator $U:{\mathcal{C}}_2(\mathcal{H})\to \mathcal{H}\otimes \mathcal{H}$ by a map ${(x\otimes y)}^{\ast }\to x\otimes y$ induces the ∗-isomorphism $\mathrm{\Psi }:B({\mathcal{C}}_2(\mathcal{H}))\to B(\mathcal{H}\otimes \mathcal{H})$ by a map $X\to UX{U}^{\ast }$. Then we can obtain $\mathrm{\Psi }({\mathrm{\Gamma }}_{A,B})=A\otimes B^{\ast }$; see details [16]. This completes the proof by Lemma 3.2. □

Lemma 3.4 (see [17])

Let $T\in B(\mathcal{H})$ be a ∗-class A operator. If $\lambda \ne 0$ and $(T-\lambda )x=0$ for some $x\in \mathcal{H}$, then ${(T-\lambda )}^{\ast }x=0$.

Now we are ready to extend the Fuglede-Putnam theorem to ∗-class A operators.

Theorem 3.5 Let A and ${(B^{\ast })}^{-1}$ be ∗-class A operators. If $AX=XB$ for $X\in {\mathcal{C}}_2(\mathcal{H})$, then $A^{\ast }X=X{B}^{\ast }$.

Proof Let Γ be defined on ${\mathcal{C}}_2(\mathcal{H})$ by $\mathrm{\Gamma }Y=AY{B}^{-1}$. Since A and ${(B^{-1})}^{\ast }={(B^{\ast })}^{-1}$ are ∗-class A operators, we have that Γ is a ∗-class A operator on ${\mathcal{C}}_2(\mathcal{H})$ by Theorem 3.3. Moreover, we have $\mathrm{\Gamma }X=AX{B}^{-1}=X$ because of $AX=XB$. Hence X is an eigenvector of Γ. By Lemma 3.4 we have ${\mathrm{\Gamma }}^{\ast }X=A^{\ast }X{(B^{-1})}^{\ast }=X$, that is, $A^{\ast }X=X{B}^{\ast }$. The proof is complete. □