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On ∗class A contractions
Journal of Inequalities and Applications volume 2013, Article number: 239 (2013)
Abstract
A Hilbert space operator T belongs to ∗class A if {T}^{2}{{T}^{\ast}}^{2}\ge 0. The famous FugledePutnam 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 HilbertSchmidt 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 phyponormal 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 phyponormal and loghyponormal 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 phyponormal and loghyponormal 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 HilbertSchmidt 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},
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,
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
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
for every x\in \mathcal{H}. By [6] Theorem 3.6, we have that
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
where the second inequality holds since T is a ∗class A operator. So, we have that
that is, {\parallel T\parallel}^{3}\parallel x\parallel \le \parallel {T}^{2}Tx\parallel. Hence we have
By (2.2), we have
that is, {\parallel T\parallel}^{2}\parallel x\parallel \le \parallel T(Tx)\parallel. Hence
Hence by (2.1) and (2.4), we have
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 selfadjoint, so that D\in {C}_{00}. □
Since a selfadjoint 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 FugledePutnam theorem for ∗class A operators
The famous FugledePutnam 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 FugledePutnam 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 FugledePutnam theorem by using Liouville’s theorem. There were various generalizations of the FugledePutnam theorem to nonnormal operators; we only cite [11–14]. For example, Radjabalipour [13] showed that the FugledePutnam theorem holds for hyponormal operators; Uchiyama and Tanahashi [14] showed that the FugledePutnam theorem holds for phyponormal and loghyponormal operators. If let X\in B(\mathcal{H}) be HilbertSchmidt class, Mecheri and Uchiyama [15] showed that normality in the FugledePutnam theorem can be replaced by A and {B}^{\ast} class A operators. In this paper, we show that if X is a HilbertSchmidt 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 HilbertSchmidt 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 nonzero 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 nonzero 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 FugledePutnam 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. □
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11071188); the Natural Science Foundation of the Department of Education, Henan Province (2011A110009), Project of Science and Technology Department of Henan province (122300410375) and Key Scientific and Technological Project of Henan Province (122102210132).
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Gao, F., Li, X. On ∗class A contractions. J Inequal Appl 2013, 239 (2013). https://doi.org/10.1186/1029242X2013239
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DOI: https://doi.org/10.1186/1029242X2013239
Keywords
 ∗class A operators
 contraction operators
 the FugledePutnam theorem