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On ∗-class A contractions
Journal of Inequalities and Applications volume 2013, Article number: 239 (2013)
A Hilbert space operator T belongs to ∗-class A if . The famous Fuglede-Putnam theorem is as follows: the operator equation implies 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 is a strongly stable contraction; secondly, we show that if X is a Hilbert-Schmidt operator, A and are ∗-class A operators such that , then .
Let ℋ be a complex Hilbert space and let ℂ be the set of complex numbers. Let denote the -algebra of all bounded linear operators acting on ℋ. For operators , we shall write kerT and ranT for the null space and the range of T, respectively. Also, let denote the spectrum of T.
Recall that is called p-hyponormal for if ; when , T is called hyponormal. And T is called paranormal if for all [2, 3]. And T is called normaloid if for all (equivalently, , the spectral radius of T). In order to discuss the relations between paranormal and p-hyponormal and log-hyponormal operators (T is invertible and ), Furuta et al.  introduced a very interesting class of operators: class A defined by , where 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.  introduced ∗-class A operators (i.e., ) and ∗-paranormal operators (i.e., for all ); 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 . A contraction T is said to be a proper contraction if for every nonzero . A strict contraction is an operator T such that . 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 if when for every (i.e., T is a strongly stable contraction) and it is said to be of class if for every nonzero . Classes and are defined by considering instead of T, and we define the class for by . An isometry is a contraction for which for every .
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 is a strongly stable contraction; secondly, we show that if X is a Hilbert-Schmidt operator, A and are ∗-class A operators such that , then .
2 On ∗-class A contractions
Theorem 2.1 If T is a contraction of ∗-class A operators, then the nonnegative operator is a contraction whose power sequence converges strongly to a projection P, and .
Proof Suppose that T is a contraction of ∗-class A operators, then . Let . Then for every ,
Thus R (and so D) is a contraction and is a decreasing sequence of nonnegative contractions. Hence converges strongly to a projection P. Moreover,
for all nonnegative integers m and every . Therefore as , hence we have
for every . So that . □
Theorem 2.2 Let T be a contraction of ∗-class A operators. If T has no nontrivial invariant subspace, then
T is a proper contraction;
the nonnegative operator is a strongly stable contraction (so that ).
Proof (i) Suppose that T is a ∗-class A operator, then . We have
for every . By  Theorem 3.6, we have that
Put , which is a subspace of ℋ. In the following, we shall show that is an invariant subspace of T. For every , we have
where the second inequality holds since T is a ∗-class A operator. So, we have that
that is, . Hence we have
By (2.2), we have
that is, . Hence
Hence by (2.1) and (2.4), we have
So, we have that . That is, 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., ) and T has no nontrivial invariant subspace, then (actually, if , then T is an isometry, and isometries have nontrivial invariant subspaces). Thus, for every nonzero , , so T is a proper contraction.
Let T be a contraction of ∗-class A operators. By Theorem 2.1 we have D is a contraction, converges strongly to a projection P, and . So, . Suppose T has no nontrivial invariant subspace. Since kerP is a nonzero invariant subspace for T whenever and , it follows that . Hence and so converges strongly to 0, that is, is a strongly stable contraction. D is self-adjoint, so that . □
Since a self-adjoint operator T is a proper contraction if and only if T is a -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 are proper contractions.
3 The Fuglede-Putnam theorem for ∗-class A operators
Theorem 3.1 Let A and B be normal operators and X be an operator such that , then .
The Fuglede-Putnam theorem was first proved in the case by Fuglede  and then a proof in the general case was given by Putnam . Berberian  proved that the Fuglede theorem was actually equivalent to that of Putnam by a nice operator matrix derivation trick. Rosenblum  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  showed that the Fuglede-Putnam theorem holds for hyponormal operators; Uchiyama and Tanahashi  showed that the Fuglede-Putnam theorem holds for p-hyponormal and log-hyponormal operators. If let be Hilbert-Schmidt class, Mecheri and Uchiyama  showed that normality in the Fuglede-Putnam theorem can be replaced by A and class A operators. In this paper, we show that if X is a Hilbert-Schmidt operator, A and are ∗-class A operators such that , then .
Let denote the tensor product on the product space for non-zero . In , Duggal et al. give a necessary and sufficient condition for to be a ∗-class A operator.
Lemma 3.2 (see )
Let be non-zero operators. Then belongs to ∗-class A operators if and only if A and B belong to ∗-class A operators.
Theorem 3.3 Let . Then is a ∗-class A operator on if and only if A and belong to ∗-class A operators.
Proof The unitary operator by a map induces the ∗-isomorphism by a map . Then we can obtain ; see details . This completes the proof by Lemma 3.2. □
Lemma 3.4 (see )
Let be a ∗-class A operator. If and for some , then .
Now we are ready to extend the Fuglede-Putnam theorem to ∗-class A operators.
Theorem 3.5 Let A and be ∗-class A operators. If for , then .
Proof Let Γ be defined on by . Since A and are ∗-class A operators, we have that Γ is a ∗-class A operator on by Theorem 3.3. Moreover, we have because of . Hence X is an eigenvector of Γ. By Lemma 3.4 we have , that is, . The proof is complete. □
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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).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.
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Gao, F., Li, X. On ∗-class A contractions. J Inequal Appl 2013, 239 (2013). https://doi.org/10.1186/1029-242X-2013-239
- ∗-class A operators
- contraction operators
- the Fuglede-Putnam theorem