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The kquasi∗class contractions have property PF
Journal of Inequalities and Applications volume 2014, Article number: 433 (2014)
Abstract
First, we will see that if T is a contraction of the kquasi∗class operator, then the nonnegative operator $D={T}^{\ast k}({T}^{2}{{T}^{\ast}}^{2}){T}^{k}$ is a contraction whose power sequence ${\{{D}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to a projection P and $T{T}^{\ast k}P=0$. Second, it will be proved that if T is a contraction of the kquasi∗class operator, then either T has a nontrivial invariant subspace or T is a proper contraction. Finally it will be proved that if T belongs to the kquasi∗class and is a contraction, then T has a Woldtype decomposition and T has the PF property.
MSC:47A10, 47B37, 15A18.
1 Introduction
Throughout this paper, let H and K be infinite dimensional separable complex Hilbert spaces with inner product $\u3008\cdot ,\cdot \u3009$. We denote by $L(H,K)$ the set of all bounded operators from H into K. To simplify, we put $L(H):=L(H,H)$. For $T\in L(H)$, we denote by kerT the null space and by $T(H)$ the range of T. The closure of a set M will be denoted by $\overline{M}$. We shall denote the set of all complex numbers by ℂ and the set of all nonnegative integers by ℕ.
For an operator $T\in L(H)$, as usual, by ${T}^{\ast}$ we mean the adjoint of T and $T={({T}^{\ast}T)}^{\frac{1}{2}}$. An operator T is said to be hyponormal, if ${T}^{2}\ge {{T}^{\ast}}^{2}$. 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 kparanormal operator if ${\parallel Tx\parallel}^{k+1}\le \parallel {T}^{k+1}x\parallel {\parallel x\parallel}^{k}$ for all $x\in H$, and T is said to be a k∗paranormal operator if ${\parallel {T}^{\ast}x\parallel}^{k+1}\le \parallel {T}^{k+1}x\parallel {\parallel x\parallel}^{k}$, for all $x\in H$.
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 quasiclass . An operator T is said to be a quasiclass , if
We denote the set of quasiclass by $\mathcal{Q}\mathcal{A}$. An operator T is said to be a kquasiclass , if
We denote the set of quasiclass by $\mathcal{Q}\mathcal{A}(k)$.
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 ${\mathcal{A}}^{\ast}$. 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 $\mathcal{Q}{\mathcal{A}}^{\ast}$. Mecheri [7] introduced the kquasi∗class operator.
Definition 1.1 An operator $T\in L(H)$ is said to be a kquasi∗class operator, if
for a nonnegative integer k.
We denote the set of the kquasi∗class by $\mathcal{Q}{\mathcal{A}}^{\ast}(k)$.
Example 1.2 Let T be an operator defined by
Then ${T}^{2}{{T}^{\ast}}^{2}\ngeqq O$ and so T is not a class ${\mathcal{A}}^{\ast}$. However, ${T}^{\ast k}({T}^{2}{{T}^{\ast}}^{2}){T}^{k}=O$ for every positive number k, which implies that T is a kquasiclass ${\mathcal{A}}^{\ast}$ operator.
A contraction is an operator T such that $\parallel Tx\parallel \le \parallel x\parallel $ for all $x\in H$. A proper contraction is an operator T such that $\parallel Tx\parallel <\parallel x\parallel $ for every nonzero $x\in H$ [8]. A strict contraction is an operator such that $\parallel T\parallel <1$ (i.e., ${sup}_{x\ne 0}\frac{\parallel Tx\parallel}{\parallel x\parallel}<1$). Obviously, every strict contraction is a proper contraction and every proper contraction is a contraction. An operator T is said to be completely nonunitary (c.n.u.) if T restricted to every reducing subspace of H is nonunitary.
An operator T on H is uniformly stable, if the power sequence ${\{{T}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges uniformly to the null operator (i.e., $\parallel {T}^{n}\parallel \to O$). An operator T on H is strongly stable, if the power sequence ${\{{T}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to the null operator (i.e., $\parallel {T}^{n}x\parallel \to 0$, for every $x\in H$).
A contraction T is of class ${C}_{0\cdot}$ if T is strongly stable (i.e., $\parallel {T}^{n}x\parallel \to 0$ and $\parallel Tx\parallel \le \parallel x\parallel $ for every $x\in H$). If ${T}^{\ast}$ is a strongly stable contraction, then T is of class ${C}_{\cdot 0}$. T is said to be of class ${C}_{1\cdot}$ if ${lim}_{n\to \mathrm{\infty}}\parallel {T}^{n}x\parallel >0$ (equivalently, if ${T}^{n}x\nrightarrow 0$ for every nonzero x in H). T is said to be of class ${C}_{\cdot 1}$ if ${lim}_{n\to \mathrm{\infty}}\parallel {T}^{\ast n}x\parallel >0$ (equivalently, if ${T}^{\ast n}x\nrightarrow 0$ for every nonzero x in H). We define the class ${C}_{\alpha \beta}$ for $\alpha ,\beta =0,1$ by ${C}_{\alpha \beta}={C}_{\alpha \cdot}\cap {C}_{\cdot \beta}$. These are the NagyFoiaş classes of contractions [[9], p.72]. All combinations are possible leading to classes ${C}_{00}$, ${C}_{01}$, ${C}_{10}$, and ${C}_{11}$. In particular, T and ${T}^{\ast}$ are both strongly stable contractions if and only if T is a ${C}_{00}$ contraction. Uniformly stable contractions are of class ${C}_{00}$.
Lemma 1.3 [[10], HolderMcCarthy inequality]
Let T be a positive operator. Then the following inequalities hold for all $x\in H$:

(1)
$\u3008{T}^{r}x,x\u3009\le {\u3008Tx,x\u3009}^{r}{\parallel x\parallel}^{2(1r)}$ for $0<r<1$;

(2)
$\u3008{T}^{r}x,x\u3009\ge {\u3008Tx,x\u3009}^{r}{\parallel x\parallel}^{2(1r)}$ for $r\ge 1$.
Lemma 1.4 [[7], Lemma 2.1]
Let T be a kquasi∗class operator, where ${T}^{k}$ does not have a dense range, and let T have the following representation:
Then A is class ${\mathcal{A}}^{\ast}$ on $\overline{{T}^{k}(H)}$, ${C}^{k}=O$, and $\sigma (T)=\sigma (A)\cup \{0\}$.
2 Main results
Theorem 2.1 If T is a contraction of the kquasi∗class operator, then the nonnegative operator
is a contraction whose power sequence ${\{{D}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to a projection P and ${T}^{\ast}{T}^{k}P=O$.
Proof Suppose that T is a contraction of the kquasi∗class operator. Then
Let $R={D}^{\frac{1}{2}}$ 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 $n=0$), and ${\{{D}^{n}\}}_{n=1}^{\mathrm{\infty}}$ is a decreasing sequence of nonnegative contractions. Then ${\{{D}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to a projection P. Moreover,
for all nonnegative integers m and for every $x\in H$. Therefore $\parallel {T}^{\ast}{T}^{k}{R}^{n}x\parallel \to 0$ as $n\to \mathrm{\infty}$. Then we have
for every $x\in H$. So that ${T}^{\ast}{T}^{k}P=O$. □
A subspace M of space H is said to be nontrivial invariant (alternatively, Tinvariant) under T if $\{0\}\ne M\ne H$ and $T(M)\subseteq M$. A closed subspace $M\subseteq H$ is said to be a nontrivial hyperinvariant subspace for T if $\{0\}\ne M\ne H$ and is invariant under every operator $S\in L(H)$, which fulfills $TS=ST$.
Recently Duggal et al. [11] showed that if T is a class contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator $D={T}^{2}{T}^{2}$ is strongly stable. Duggal et al. [12] extended these results to contractions in $\mathcal{Q}\mathcal{A}$. Jeon and Kim [13] extended these results to contractions $\mathcal{Q}\mathcal{A}(k)$. Gao and Li [14] have proved that if a contraction $T\in {\mathcal{A}}^{\ast}$ has a no nontrivial invariant subspace, then (a) T is a proper contraction and (b) the nonnegative operator $D={T}^{2}{{T}^{\ast}}^{2}$ is a strongly stable contraction. In this paper we extend these results to contractions in the kquasi∗class for $k>0$.
Theorem 2.2 Let T be a contraction of the kquasi∗class for $k>0$. If T has a no nontrivial invariant subspace, then:

(1)
T is a proper contraction;

(2)
the nonnegative operator
$$D={T}^{\ast k}\left(\right{T}^{2}{T}^{\ast}{}^{2}){T}^{k}$$
is a strongly stable contraction.
Proof We may assume that T is a nonzero operator.

(1)
If either kerT or $\overline{{T}^{k}(H)}$ is a nontrivial subspace (i.e., $kerT\ne \{0\}$ or $\overline{{T}^{k}(H)}\ne H$), then T has a nontrivial invariant subspace. Hence, if T has no nontrivial invariant subspace, then T is injective and $\overline{{T}^{k}(H)}=H$. Furthermore, T is a class ${\mathcal{A}}^{\ast}$ operator. The proof now follows from [[14], Theorem 2.2].

(2)
Let T be a contraction of the kquasi∗class . By the above theorem, we see that D is a contraction, ${\{{D}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to a projection P, and ${T}^{\ast}{T}^{k}P=O$. So, $P{T}^{\ast k}T=O$. Suppose T has no nontrivial invariant subspaces. Since kerP is a nonzero invariant subspace for T whenever $P{T}^{\ast k}T=O$ and $T\ne O$, it follows that $kerP=H$. Hence $P=O$, and we see that ${\{{D}^{n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to the null operator O, so D is a strongly stable contraction. Since D is selfadjoint, $D\in {C}_{00}$. □
Corollary 2.3 Let T be a contraction of the kquasi∗class . If T has no nontrivial invariant subspace, then both T and the nonnegative operators
are proper contractions.
Proof A selfadjoint operator T is a proper contraction if and only if T is a ${C}_{00}$ contraction. □
Definition 2.4 If the contraction T is a direct sum of the unitary and ${C}_{\cdot 0}$ (c.n.u.) contractions, then we say that T has a Woldtype decomposition.
Definition 2.5 [15]
An operator $T\in L(H)$ is said to have the FugledePutnam commutativity property (PF property for short) if ${T}^{\ast}X=XJ$ for any $X\in L(K,H)$ and any isometry $J\in L(K)$ such that $TX=X{J}^{\ast}$.
Let T be a contraction. The following conditions are equivalent:

(1)
For any bounded sequence ${\{{x}_{n}\}}_{n\in \mathbb{N}\cup \{0\}}\subset H$ such that $T{x}_{n+1}={x}_{n}$ the sequence ${\{\parallel {x}_{n}\parallel \}}_{n\in \mathbb{N}\cup \{0\}}$ is constant;

(2)
T has a Woldtype decomposition;

(3)
T has the PF property.
Duggal and Cubrusly in [16] have proved: Each kparanormal contraction operator has a Woldtype decomposition. Pagacz in [17] has proved the same and also proved that each k∗paranormal operator has a Woldtype decomposition. In this paper, we extend to contractions in $\mathcal{Q}{\mathcal{A}}^{\ast}(k)$.
Theorem 2.7 Let T be a contraction of the kquasi∗class . Then T has a Woldtype decomposition.
Proof Since T is a contraction operator, the decreasing sequence ${\{{T}^{n}{T}^{\ast n}\}}_{n=1}^{\mathrm{\infty}}$ converges strongly to a nonnegative contraction. We denote by
The operators T and S are related by ${T}^{\ast}{S}^{2}T={S}^{2}$, $O\le S\le I$ and S is selfadjoint operator. By [18] there exists an isometry $V:\overline{S(H)}\to \overline{S(H)}$ such that $VS=S{T}^{\ast}$, and thus $S{V}^{\ast}=TS$, and $\parallel S{V}^{m}x\parallel \to \parallel x\parallel $ for every $x\in \overline{S(H)}$. The isometry V can be extended to an isometry on H, which we still denote by V.
For an $x\in \overline{S(H)}$, we can define ${x}_{n}=S{V}^{n}x$ for $n\in \mathbb{N}\cup \{0\}$. Then for all nonnegative integers m we have
and for all $m\le n$ we have
Since T is a kquasi∗class operator and the nontrivial $x\in \overline{S(H)}$ we have
Then
hence
Thus
Put
and we have
Since ${x}_{n}=T{x}_{n+1}$, we have
then the sequence ${\{\parallel {x}_{n}\parallel \}}_{n\in \mathbb{N}\cup \{0\}}$ is increasing. From
we have
for every $x\in \overline{S(H)}$ and $n\in \mathbb{N}\cup \{0\}$. Then ${\{\parallel {x}_{n}\parallel \}}_{n\in \mathbb{N}\cup \{0\}}$ is bounded. From this we have ${b}_{n}\ge 0$ and ${b}_{n}\to 0$ as $n\to \mathrm{\infty}$.
It remains to check that all ${b}_{n}$ equal zero. Suppose that there exists an integer $i\ge 1$ such that ${b}_{i}>0$. Using the inequality (1) we get ${b}_{i+1}>0$ and ${b}_{i+2}>0$, so there exists $\u03f5>0$ such that ${b}_{i+1}>\u03f5$ and ${b}_{i+2}>\u03f5$. From that, and using again the inequality (1), we can show by induction that ${b}_{n}>\u03f5$ for all $n>i$, thus arriving at a contradiction. So ${b}_{n}=0$ for all $n\in \mathbb{N}$ and thus $\parallel {x}_{n1}\parallel =\parallel {x}_{n}\parallel $ for all $n\ge 1$. Thus the sequence ${\{\parallel {x}_{n}\parallel \}}_{n\in \mathbb{N}\cup \{0\}}$ is constant.
From Lemma 2.6, T has a Woldtype decomposition. □
For $T\in L(H)$ and $x\in H$, ${\{{T}^{n}x\}}_{n=0}^{\mathrm{\infty}}$ is called the orbit of x under T, and is denoted by $\mathcal{O}(x,T)$. When the linear span of the orbit $\mathcal{O}(x,T)$ is norm dense in H, x is called a cyclic vector for T and T is said to be a cyclic operator. If $\mathcal{O}(x,T)$ is norm dense in H, then x is called a hypercyclic vector for T. An operator $T\in L(H)$ is called hypercyclic if there is at least one hypercyclic vector for T. We say that an operator $T\in L(H)$ is supercyclic if there exists a vector $x\in H$ such that $\mathbb{C}\mathcal{O}(x,T)=\{\lambda {T}^{n}x:\lambda \in \mathbb{C},n=0,1,2,\dots \}$ is norm dense in H.
Theorem 2.8 Let $T\in L(H)$ be a quasi∗class such that $\sigma (T)\subseteq \{\lambda \in \mathbb{C}:\lambda =1\}$. If the inverse of T is a quasi∗class , then T is not a supercyclic operator.
Proof Let $T\in L(H)$ be a quasi∗class . Since $\sigma (T)\subseteq \{\lambda \in \mathbb{C}:\lambda =1\}$, T is an invertible operator. From [7]T is normaloid, thus $\parallel T\parallel =r(T)=1$. Since ${T}^{1}\in \mathcal{Q}({\mathcal{A}}^{\ast})$, $\parallel {T}^{1}\parallel =1$. 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 $\{{\alpha}_{n}:n\in \mathbb{Z}\}$ (called weights), let T be the bilateral weighted shift on an infinite dimensional Hilbert space operator $H={l}_{2}$, with the canonical orthonormal basis $\{{e}_{n}:n\in \mathbb{Z}\}$, defined by $T{e}_{n}={\alpha}_{n}{e}_{n+1}$ for all $n\in \mathbb{Z}$.
Lemma 2.10 Let T be a bilateral weighted shift operator with weights $\{{\alpha}_{n}:n\in \mathbb{Z}\}$. Then T is a quasi∗class operator if and only if
for all $n\in \mathbb{Z}$.
Lemma 2.11 Let T be a nonsingular bilateral weighted shift operator with weights $\{{\alpha}_{n}:n\in \mathbb{Z}\}$. Then ${T}^{1}$ is a quasi∗class operator if and only if
for all $n\in \mathbb{Z}$.
Example 2.12 Let us denote by T the bilateral weighted shift operator, with weighted sequence $\{{\alpha}_{n}:n\in \mathbb{Z}\}$, given by the relation
From Lemma 2.10 it follows that T is a quasi∗class operator. Since $\{{\alpha}_{n}:n\in \mathbb{Z}\}$ is a bounded sequence of positive numbers with $inf\{{\alpha}_{n}:n\in \mathbb{Z}\}>0$, T is an invertible operator [[19], Proposition II.6.8]. But ${T}^{1}$ is not a quasi∗class operator, which follows from Lemma 2.11, for $n=4$.
Theorem 2.13 Let $T\in L(H)$ be a quasi∗class operator and $\mathbb{D}=\{z:z<1\}$. If ${T}^{\ast}$ is a hypercyclic operator and for every hyperinvariant $M\subseteq H$ of T, the inverse of $T{}_{M}$, whenever it exists, is a normaloid operator, then $\sigma (T{}_{M})\cap \mathbb{D}\ne \mathrm{\varnothing}$ and $\sigma (T{}_{M})\cap (\mathbb{C}\setminus \overline{\mathbb{D}})\ne \mathrm{\varnothing}$.
Proof Assume that ${T}^{\ast}$ is a hypercyclic operator. Then there exists a vector $x\in H$ such that $\overline{{\{{({T}^{\ast})}^{n}x\}}_{n=0}^{\mathrm{\infty}}}=H$. Let $S=T{}_{M}$ for some closed Tinvariant subspace and let P be the orthogonal projection of H onto M. Since ${({S}^{\ast})}^{n}Px=P{({T}^{\ast})}^{n}x$ for each $n\in \mathbb{N}\cup \{0\}$ we have
thus ${S}^{\ast}$ is hypercyclic.
From [[20], Corollary 3] we have $\parallel {S}^{\ast}\parallel >1$. Since S is a quasi∗class , S is normaloid, thus $r(T{}_{M})=\parallel S\parallel =\parallel {S}^{\ast}\parallel >1$. Therefore $\sigma (T{}_{M})\cap (\mathbb{C}\setminus \overline{\mathbb{D}})\ne \mathrm{\varnothing}$.
Suppose that $\sigma (T{}_{M})\subset (\mathbb{C}\setminus \overline{\mathbb{D}})$. Then $\sigma ({S}^{1})\subset \overline{\mathbb{D}}$, and since ${S}^{1}$ is normaloid, $\parallel {S}^{1}\parallel =r({S}^{1})\le 1$. Since ${S}^{\ast}$ is hypercyclic, from [[20], Theorem 6] ${({S}^{\ast})}^{1}$ is hypercyclic, so $\parallel {({S}^{\ast})}^{1}\parallel >1$. Thus $\parallel {S}^{1}\parallel =\parallel {({S}^{\ast})}^{1}\parallel >1$. This is a contradiction, therefore $\sigma (T{}_{M})\cap \mathbb{D}\ne \mathrm{\varnothing}$. □
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Hoxha, I., Braha, N.L. The kquasi∗class contractions have property PF. J Inequal Appl 2014, 433 (2014). https://doi.org/10.1186/1029242X2014433
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Keywords
 kquasi∗class
 contractions
 proper contractions
 Woldtype decomposition
 PF property
 supercyclic operator
 hypercyclic operator