- Open Access
The k-quasi-∗-class contractions have property PF
Journal of Inequalities and Applications volume 2014, Article number: 433 (2014)
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.
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 . Further, T is said to be ∗-paranormal, if
for any unit vector x in H . 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.  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  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. , introduced ∗-class operator. An operator T is said to be a ∗-class operator, if
A ∗-class is a generalization of a hyponormal operator [, Theorem 1.2], and ∗-class is a subclass of the class of ∗-paranormal operators [, Theorem 1.3]. We denote the set of ∗-class by . Shen et al. in  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  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 . 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 [, 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 [, Holder-McCarthy inequality]
Let T be a positive operator. Then the following inequalities hold for all :
Lemma 1.4 [, 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.  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.  extended these results to contractions in . Jeon and Kim  extended these results to contractions . Gao and Li  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:
T is a proper contraction;
the nonnegative operator
is a strongly stable contraction.
Proof We may assume that T is a nonzero operator.
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 [, Theorem 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 
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:
For any bounded sequence such that the sequence is constant;
T has a Wold-type decomposition;
T has the PF property.
Duggal and Cubrusly in  have proved: Each k-paranormal contraction operator has a Wold-type decomposition. Pagacz in  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  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
and we have
Since , we have
then the sequence is increasing. From
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 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 [, 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 [, 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 [, Theorem 6] is hypercyclic, so . Thus . This is a contradiction, therefore . □
Furuta T: On the class of paranormal operators. Proc. Jpn. Acad. 1967, 43: 594–598. 10.3792/pja/1195521514
Arora SC, Thukral JK: On a class of operators. Glas. Mat. 1986,21(41)(2):381–386.
Furuta T, Ito M, Yamazaki T: A subclass of paranormal operators including class of log-hyponormal and several classes. Sci. Math. 1998,1(3):389–403.
Jeon IH, Kim IH:On operators satisfying . Linear Algebra Appl. 2006, 418: 854–862. 10.1016/j.laa.2006.02.040
Duggal BP, Jeon IH, Kim IH: On ∗-paranormal contractions and properties for ∗-class operators. Linear Algebra Appl. 2012,436(5):954–962. 10.1016/j.laa.2011.06.002
Shen JL, Zuo F, Yang CS:On operators satisfying . Acta Math. Sin. Engl. Ser. 2010,26(11):2109–2116. 10.1007/s10114-010-9093-4
Mecheri S: Isolated points of spectrum of k -quasi-∗-class operators. Stud. Math. 2012, 208: 87–96. 10.4064/sm208-1-6
Kubrusly CS, Levan N: Proper contractions and invariant subspace. Int. J. Math. Sci. 2001, 28: 223–230. 10.1155/S0161171201006287
Sz-Nagy B, Foiaş C: Harmonic Analysis of Operators on Hilbert Space. North-Holland, Amsterdam; 1970.
McCarthy CA:. Isr. J. Math. 1967, 5: 249–271. 10.1007/BF02771613
Duggal BP, Jeon IH, Kubrusly CS:Contractions satisfying the absolute value property . Integral Equ. Oper. Theory 2004, 49: 141–148. 10.1007/s00020-002-1202-z
Duggal BP, Jeon IH, Kim IH: On quasi-class contractions. Linear Algebra Appl. 2012, 436: 3562–3567. 10.1016/j.laa.2011.12.026
Jeon IH, Kim IH: On k -quasi-class contractions. Korean J. Math. 2014,22(1):85–89. 10.11568/kjm.2014.22.1.85
Gao F, Li X: On ∗-class contractions. J. Inequal. Appl. 2013., 2013: Article ID 239 10.1186/1029-242X-2013-239
Duggal BP:On characterising contractions with pure part. Integral Equ. Oper. Theory 1997, 27: 314–323. 10.1007/BF01324731
Duggal BP, Cubrusly CS: Paranormal contractions have property PF. Far East J. Math. Sci. 2004, 14: 237–249.
Pagacz P: On Wold-type decomposition. Linear Algebra Appl. 2012, 436: 3065–3071. 10.1016/j.laa.2011.09.001
Durszt E: Contractions as restricted shifts. Acta Sci. Math. 1985, 48: 129–134.
Conway JB Mathematical Surveys and Monographs 36. In The Theory of Subnormal Operators. Am. Math. Soc., Providence; 1991.
Miller VG: Remarks on finitely hypercyclic and finitely supercyclic operators. Integral Equ. Oper. Theory 1997, 29: 110–115. 10.1007/BF01191482
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
About this article
Cite this article
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
- proper contractions
- Wold-type decomposition
- PF property
- supercyclic operator
- hypercyclic operator