Abstract
First, we will see that if T is a contraction of the k-quasi-∗-class operator, then the nonnegative operator is a contraction whose power sequence converges strongly to a projection P and . Second, it will be proved that if T is a contraction of the k-quasi-∗-class operator, then either T has a non-trivial invariant subspace or T is a proper contraction. Finally it will be proved that if T belongs to the k-quasi-∗-class and is a contraction, then T has a Wold-type decomposition and T has the PF property.
MSC:47A10, 47B37, 15A18.
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1 Introduction
Throughout this paper, let H and K be infinite dimensional separable complex Hilbert spaces with inner product . We denote by the set of all bounded operators from H into K. To simplify, we put . For , we denote by kerT the null space and by the range of T. The closure of a set M will be denoted by . We shall denote the set of all complex numbers by ℂ and the set of all nonnegative integers by ℕ.
For an operator , as usual, by we mean the adjoint of T and . An operator T is said to be hyponormal, if . An operator T is said to be paranormal, if
for any unit vector x in H [1]. Further, T is said to be ∗-paranormal, if
for any unit vector x in H [2]. T is said to be a k-paranormal operator if for all , and T is said to be a k-∗-paranormal operator if , for all .
Furuta et al. [3] introduced a very interesting class of bounded linear Hilbert space operators: class defined by
and they showed that the class is a subclass of paranormal operators and contains hyponormal operators. Jeon and Kim [4] introduced the quasi-class . An operator T is said to be a quasi-class , if
We denote the set of quasi-class by . An operator T is said to be a k-quasi-class , if
We denote the set of quasi-class by .
Duggal et al. [5], introduced ∗-class operator. An operator T is said to be a ∗-class operator, if
A ∗-class is a generalization of a hyponormal operator [[5], Theorem 1.2], and ∗-class is a subclass of the class of ∗-paranormal operators [[5], Theorem 1.3]. We denote the set of ∗-class by . Shen et al. in [6] introduced the quasi-∗-class operator: an operator T is said to be a quasi-∗-class operator, if
We denote the set of quasi-∗-class by . Mecheri [7] introduced the k-quasi-∗-class operator.
Definition 1.1 An operator is said to be a k-quasi-∗-class operator, if
for a nonnegative integer k.
We denote the set of the k-quasi-∗-class by .
Example 1.2 Let T be an operator defined by
Then and so T is not a class . However, for every positive number k, which implies that T is a k-quasi-class operator.
A contraction is an operator T such that for all . A proper contraction is an operator T such that for every nonzero [8]. A strict contraction is an operator such that (i.e., ). Obviously, every strict contraction is a proper contraction and every proper contraction is a contraction. An operator T is said to be completely non-unitary (c.n.u.) if T restricted to every reducing subspace of H is non-unitary.
An operator T on H is uniformly stable, if the power sequence converges uniformly to the null operator (i.e., ). An operator T on H is strongly stable, if the power sequence converges strongly to the null operator (i.e., , for every ).
A contraction T is of class if T is strongly stable (i.e., and for every ). If is a strongly stable contraction, then T is of class . T is said to be of class if (equivalently, if for every nonzero x in H). T is said to be of class if (equivalently, if for every nonzero x in H). We define the class for by . These are the Nagy-Foiaş classes of contractions [[9], p.72]. All combinations are possible leading to classes , , , and . In particular, T and are both strongly stable contractions if and only if T is a contraction. Uniformly stable contractions are of class .
Lemma 1.3 [[10], Holder-McCarthy inequality]
Let T be a positive operator. Then the following inequalities hold for all :
-
(1)
for ;
-
(2)
for .
Lemma 1.4 [[7], Lemma 2.1]
Let T be a k-quasi-∗-class operator, where does not have a dense range, and let T have the following representation:
Then A is class on , , and .
2 Main results
Theorem 2.1 If T is a contraction of the k-quasi-∗-class operator, 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 the k-quasi-∗-class operator. Then
Let be the unique nonnegative square root of D, then for every x in H and any nonnegative integer n, we have
Thus R (and so D) is a contraction (set ), and is a decreasing sequence of nonnegative contractions. Then converges strongly to a projection P. Moreover,
for all nonnegative integers m and for every . Therefore as . Then we have
for every . So that . □
A subspace M of space H is said to be non-trivial invariant (alternatively, T-invariant) under T if and . A closed subspace is said to be a non-trivial hyperinvariant subspace for T if and is invariant under every operator , which fulfills .
Recently Duggal et al. [11] showed that if T is a class contraction, then either T has a non-trivial invariant subspace or T is a proper contraction and the nonnegative operator is strongly stable. Duggal et al. [12] extended these results to contractions in . Jeon and Kim [13] extended these results to contractions . Gao and Li [14] have proved that if a contraction has a no non-trivial invariant subspace, then (a) T is a proper contraction and (b) the nonnegative operator is a strongly stable contraction. In this paper we extend these results to contractions in the k-quasi-∗-class for .
Theorem 2.2 Let T be a contraction of the k-quasi-∗-class for . If T has a no non-trivial invariant subspace, then:
-
(1)
T is a proper contraction;
-
(2)
the nonnegative operator
is a strongly stable contraction.
Proof We may assume that T is a nonzero operator.
-
(1)
If either kerT or is a non-trivial subspace (i.e., or ), then T has a non-trivial invariant subspace. Hence, if T has no non-trivial invariant subspace, then T is injective and . Furthermore, T is a class operator. The proof now follows from [[14], Theorem 2.2].
-
(2)
Let T be a contraction of the k-quasi-∗-class . By the above theorem, we see that D is a contraction, converges strongly to a projection P, and . So, . Suppose T has no non-trivial invariant subspaces. Since kerP is a nonzero invariant subspace for T whenever and , it follows that . Hence , and we see that converges strongly to the null operator O, so D is a strongly stable contraction. Since D is self-adjoint, . □
Corollary 2.3 Let T be a contraction of the k-quasi-∗-class . If T has no non-trivial invariant subspace, then both T and the nonnegative operators
are proper contractions.
Proof A self-adjoint operator T is a proper contraction if and only if T is a contraction. □
Definition 2.4 If the contraction T is a direct sum of the unitary and (c.n.u.) contractions, then we say that T has a Wold-type decomposition.
Definition 2.5 [15]
An operator is said to have the Fuglede-Putnam commutativity property (PF property for short) if for any and any isometry such that .
Let T be a contraction. The following conditions are equivalent:
-
(1)
For any bounded sequence such that the sequence is constant;
-
(2)
T has a Wold-type decomposition;
-
(3)
T has the PF property.
Duggal and Cubrusly in [16] have proved: Each k-paranormal contraction operator has a Wold-type decomposition. Pagacz in [17] has proved the same and also proved that each k-∗-paranormal operator has a Wold-type decomposition. In this paper, we extend to contractions in .
Theorem 2.7 Let T be a contraction of the k-quasi-∗-class . Then T has a Wold-type decomposition.
Proof Since T is a contraction operator, the decreasing sequence converges strongly to a nonnegative contraction. We denote by
The operators T and S are related by , and S is self-adjoint operator. By [18] there exists an isometry such that , and thus , and for every . The isometry V can be extended to an isometry on H, which we still denote by V.
For an , we can define for . Then for all nonnegative integers m we have
and for all we have
Since T is a k-quasi-∗-class operator and the non-trivial we have
Then
hence
Thus
Put
and we have
Since , we have
then the sequence is increasing. From
we have
for every and . Then is bounded. From this we have and as .
It remains to check that all equal zero. Suppose that there exists an integer such that . Using the inequality (1) we get and , so there exists such that and . From that, and using again the inequality (1), we can show by induction that for all , thus arriving at a contradiction. So for all and thus for all . Thus the sequence is constant.
From Lemma 2.6, T has a Wold-type decomposition. □
For and , is called the orbit of x under T, and is denoted by . When the linear span of the orbit is norm dense in H, x is called a cyclic vector for T and T is said to be a cyclic operator. If is norm dense in H, then x is called a hypercyclic vector for T. An operator is called hypercyclic if there is at least one hypercyclic vector for T. We say that an operator is supercyclic if there exists a vector such that is norm dense in H.
Theorem 2.8 Let be a quasi-∗-class such that . If the inverse of T is a quasi-∗-class , then T is not a supercyclic operator.
Proof Let be a quasi-∗-class . Since , T is an invertible operator. From [7]T is normaloid, thus . Since , . Consequently, T is unitary. Since no unitary operator on an infinite dimensional Hilbert space can be supercyclic, we see that T is not a supercyclic operator. □
Remark 2.9 The condition that the inverse of the operator T belongs to quasi-∗-class cannot be removed from Theorem 2.8, because there are invertible operators from the quasi-∗-class , such that their inverse does not belong to the quasi-∗-class . This is shown in the following example.
Given a bounded sequence of complex numbers (called weights), let T be the bilateral weighted shift on an infinite dimensional Hilbert space operator , with the canonical orthonormal basis , defined by for all .
Lemma 2.10 Let T be a bilateral weighted shift operator with weights . Then T is a quasi-∗-class operator if and only if
for all .
Lemma 2.11 Let T be a non-singular bilateral weighted shift operator with weights . Then is a quasi-∗-class operator if and only if
for all .
Example 2.12 Let us denote by T the bilateral weighted shift operator, with weighted sequence , given by the relation
From Lemma 2.10 it follows that T is a quasi-∗-class operator. Since is a bounded sequence of positive numbers with , T is an invertible operator [[19], Proposition II.6.8]. But is not a quasi-∗-class operator, which follows from Lemma 2.11, for .
Theorem 2.13 Let be a quasi-∗-class operator and . If is a hypercyclic operator and for every hyperinvariant of T, the inverse of , whenever it exists, is a normaloid operator, then and .
Proof Assume that is a hypercyclic operator. Then there exists a vector such that . Let for some closed T-invariant subspace and let P be the orthogonal projection of H onto M. Since for each we have
thus is hypercyclic.
From [[20], Corollary 3] we have . Since S is a quasi-∗-class , S is normaloid, thus . Therefore .
Suppose that . Then , and since is normaloid, . Since is hypercyclic, from [[20], Theorem 6] is hypercyclic, so . Thus . This is a contradiction, therefore . □
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Hoxha, I., Braha, N.L. The k-quasi-∗-class contractions have property PF. J Inequal Appl 2014, 433 (2014). https://doi.org/10.1186/1029-242X-2014-433
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DOI: https://doi.org/10.1186/1029-242X-2014-433