2. Properties of absolute-∗-k-paranormal operator and ∗-A(k) operator

DOI: 10.24193/subbmath.2019.1.11

Properties of absolute-∗-k-paranormal operators and contractions for ∗-A(k) operators

Ilmi Hoxha, Naim L. Braha and Agron Tato

Abstract. First, we see if T is absolute-∗-k-paranormal for k ≥ 1, then T is a normaloid operator. We also see some properties of absolute-∗-k-paranormal operator and∗-A(k) operator. Then, we will prove the spectrum continuity of the class∗-A(k) operator fork >0. Moreover, it is proved that ifT is a contraction of the class ∗-A(k) fork >0, then eitherT has a nontrivial invariant subspace orT is a proper contraction, and the nonnegative operator

D=

T^∗|T|^2kT_k+1¹

− |T^∗|²

is a strongly stable contraction. Finally ifT ∈ ∗-A(k) is a contraction fork >0, thenT is the direct sum of a unitary andC·0(c.n.u) contraction.

Mathematics Subject Classification (2010):47A10, 47B37, 15A18.

Keywords:Class∗-A(k) operators, absolute-∗-k-paranormal operators, normaloid operators, continuity spectrum, contractions.

1. Introduction

Throughout this paper, letH andKbe infinite dimensional separable complex Hilbert spaces with inner producth·,·i.We denote byL(H, K) the set of all bounded operators from H into K. To simplify, we put L(H) := L(H, H). For T ∈ L(H), we denote by ker(T) the null space and by T(H) the range of T. The null operator and the identity onH will be denoted byO andI, respectively. IfT is an operator, then T^∗ denotes its adjoint. We shall denote the set of all complex numbers by C, the set of all non-negative integers by N and the complex conjugate of a complex numberλbyλ.The closure of a setM will be denoted byM and we shall henceforth shorten T −λI to T −λ. An operator T ∈ L(H), is a positive operator, T ≥ O, if hT x, xi ≥ 0 for all x ∈ H. We write byσ(T), σ_p(T), and σ_a(T) spectrum, point spectrum and approximate point spectrum respectively. Sets of isolated points and accumulation points of σ(T) are denoted by isoσ(T) and accσ(T), respectively. We

writer(T) for the spectral radius. It is well known thatr(T)≤ kTk.The operatorT is called normaloid ifr(T) =kTk.

A contraction is an operator T such thatkT xk ≤ kxk for all x∈H. A proper contraction is an operatorT such thatkT xk<kxk for every nonzerox∈H. A strict contraction is an operator such thatkTk<1 (i.e., sup_x6=0^{kT xk}_kxk <1). 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 ofH is non-unitary.

An operator T on H is uniformly stable, if the power sequence{T^m}^∞_m=1 con- verges uniformly to the null operator (i.e., kT^mk → O). An operator T on H is strongly stable, if the power sequence{T^m}^∞_m=1 converges strongly to the null oper- ator (i.e.,kT^mxk →0, for everyx∈H).

A contraction T is of class C_0· if T is strongly stable (i.e., kT^mxk → 0 and kT xk ≤ kxkfor everyx∈H). IfT^∗ is a strongly stable contraction, thenT is of class C_·0.T is said to be of classC_1·if lim_m→∞kT^mxk>0 (equivalently, if T^mx6→0 for every nonzeroxinH).Tis said to be of classC_·1if lim_m→∞kT^∗mxk>0 (equivalently, if T^∗mx6→0 for every nonzero xin H). We define the class Cαβ for α, β = 0,1 by Cαβ = C_α·∩C_·β. These are the Nagy-Foia¸s classes of contractions [21, p.72]. All combinations are possible leading to classesC00,C01, C10 andC11. In particular,T and T^∗ are both strongly stable contractions if and only if T is a C00 contraction.

Uniformly stable contractions are of classC00.

For an operatorT ∈L(H),as usual,|T|= (T^∗T)¹².An operatorT is said to be a normal operator ifT^∗T =T T^∗ andT is said to be hyponormal, if|T|²≥ |T^∗|². An operator T ∈L(H), is said to be paranormal [11], if kT²xk ≥ kT xk² for every unit vectorxinH.Further,T is said to be∗-paranormal [1], ifkT²xk ≥ kT^∗xk² for every unit vectorxinH.

In [13] authors Furuta, Ito and Yamazaki introduced the class Aoperator, re- spectively the classA(k) operator defined as follows: For eachk >0,an operator T is from classA(k) operator if

T^∗|T|^2kT_k+1¹

≥ |T|²,

(fork= 1 it defines classAoperator), and they showed that the classAis a subclass of paranormal operators.

In the same paper, authors introduced the absolute-k-paranormal operators as follows: For eachk >0,an operator T is absolute-k-paranormal if

|T|^kT x

≥ ||T x||^k+1,

for every unit vectorx∈H.In case wherek= 1 it defines the paranormal operator.

The class A(k) operator is included in the absolute-k-paranormal operator for any k >0, [13, Theorem 2]).

B. P. Duggal, I. H. Jeon, and I. H. Kim [5], introduced∗-classAoperator. An operatorT ∈L(H) is said to be a∗-classAoperator, if|T²| ≥ |T^∗|².A∗-classAis a generalization of a hyponormal operator, [5, Theorem 1.2], and∗-classAis a subclass of the class of∗-paranormal operators, [5, Theorem 1.3]. We denote the set of∗-class

AbyA^∗.An operatorT ∈L(H) is said to be ak-quasi-∗-classAoperator [20], if T^∗k |T²| − |T^∗|²

T^k ≥O, for a nonnegative integerk.

In [24] authors, S. Panayappan and A. Radharamani introduced the class∗-A(k) operator and absolute-∗-k-paranormal operator.

Definition 1.1. For eachk >0,an operatorT is absolute-∗-k-paranormal if

|||T|^kT x|| ≥ ||T^∗x||^k+1 for every unit vectorx∈H.

In case wherek= 1 it defines the∗-paranormal operator.

Definition 1.2. For eachk >0,an operatorT is class∗-A(k), if T^∗|T|^2kT_k+1¹

≥ |T^∗|². In case wherek= 1 it defines theA^∗ class operators.

In this paper, we shall show behavior of the class∗-A(k) operator and absolute-

∗-k-paranormal operator.

2. Properties of absolute-∗-k-paranormal operator and ∗-A(k) operator

Theorem 2.1. If T is an absolute-∗-k-paranormal operator for k > 0, then T is a normaloid operator.

Proof. Let T be an absolute-∗-k-paranormal operator. In case where k = 1, T is a

∗-paranormal operator, then by [1, Theorem 1.1] it follows that T is a normaloid operator. Following, it will be proved that for k > 1 the operatorT is a normaloid operator, because for 0< k < 1, it was proved in [3](Theorem 2.9). Without losing the generality, assumekTk= 1. Since T is an absolute-∗-k-paranormal, then

kT^∗xk^k+1≤ k|T|^kT xkkxk^k ≤ k|T|^k−1kk|T|T xkkxk^k ≤ kT²xkkxk^k for allx∈H.Therefore,

kT^∗xk^k+1

kxk^k ≤ kT²xk ≤ kxk (2.1)

for allx∈H.

By definition ofkT^∗k, there exists a sequence{xi}of unit vectors such that

kT^∗xik → kT^∗k=kTk= 1. (2.2) Putx=xi in (2.1), then we have.

kT^∗xik^k+1

kxik^k ≤ kT²xik ≤ kxik= 1 (2.3) so,kT²x_ik →1, by (2.2) and (2.3), that is

kT²k= 1 =kTk².

Let us now suppose that

kTⁿ⁻¹x_ik →1 ,kTⁿ⁻²x_ik →1 andkTⁿ⁻³x_ik →1 forn≥3. (2.4) Putx=Tⁿ⁻²x_i in (2.1), then we have

kT^∗Tⁿ⁻²x_ik^k+1

kTⁿ⁻²xik^k ≤ kTⁿx_ik ≤ kTⁿ⁻²x_ik. (2.5) From Cauchy-Schwarz inequality we have

kTⁿ⁻²xk²

kTⁿ⁻³xk ≤ kT^∗Tⁿ⁻²xk. (2.6)

From relations (2.5) and (2.6) we have kTⁿ⁻²x_ik^k+2

kTⁿ⁻³xik^k+1 ≤kT^∗Tⁿ⁻²x_ik^k+1

kTⁿ⁻²xik^k ≤ kTⁿx_ik ≤ kTⁿ⁻²x_ik.

respectively:

kTⁿ⁻¹xik^k+2

kTⁿ⁻³xik^k+1 ≤ kTⁿ⁻²xik^k+2

kTⁿ⁻³xik^k+1 ≤kT^∗Tⁿ⁻²xik^k+1

kTⁿ⁻²xik^k ≤ kTⁿxik ≤ kTⁿ⁻²xik. (2.7) Hence,kTⁿxik →1,by (2.4) and (2.7) that iskTⁿk= 1 =kTkⁿ.Consequently

kTⁿk= 1 =kTkⁿ,

for all positive integersnby induction.

Example 2.2. An example of non-absolute-∗-k-paranormal operator which is a nor- maloid operator. Let us denote by

T =





1 0 0 0 0 0 0 1 0





Then||Tⁿ||=||T||ⁿ for all positive integersn.However, the relation |T|^kT x

≥ ||T^∗x||^k+1

does not hold for the unit vector e₃ = (0,0,1). With which was proved that T is a non-absolute-∗-k-paranormal operator, but it is a normaloid operator.

It is known that there exists a linear operatorT, so that Tⁿ is compact oper- ator for some n ∈ N, but T itself is not compact. For instance, take any nilpotent noncompact operator (If (e_n)_n is an orthonormal basis ofH then the shift defined by T(e_2n) =e_2n+1 andT(e_2n+1) = 0 is not a compact operator for whichT²=O).

In this context, we will show that in cases where an operatorT is an absolute-

∗-k-paranormal operator and if its exponentTⁿ is compact, for some n∈N, then T is compact too.

Theorem 2.3. If T is an absolute-∗-k-paranormal operator for k > 0 and if Tⁿ is compact for some n∈N,then it follows that T is compact too.

Proof. Compactness of Tⁿ implies countable spectrum (consisting of mutually or- thogonal eigenvalues ([26], Theorem 6)), this then impliesTⁿnormal compact, hence

T is normal compact.

Corollary 2.4. If T, Rare absolute-∗-k-paranormal operators for k >0and ifTⁿ and R^mare compact for somen, m∈N, then it follows thatT⊕R is compact too.

Corollary 2.5. If T, Rare absolute-∗-k-paranormal operators for k >0and ifTⁿ is a compact operator for some n∈NorR^m is a compact operator for somem∈N,then it follows thatT⊗R is compact too.

Lemma 2.6. [16, Hansen Inequality]If A, B∈L(H),satisfying A≥O and kBk ≤1, then

(B^∗AB)^δ ≥B^∗A^δB for all δ∈(0,1].

Lemma 2.7. [12, L¨owner-Heinz Inequality] If A, B ∈L(H), satisfying A ≥B ≥O, thenA^δ ≥B^δ for allδ∈[0,1].

A subspace M of space H is said to be nontrivial invariant(alternatively, T- invariant) underT if{0} 6=M 6=H and T(M)⊆M.

Theorem 2.8. If T is a class ∗-A(k) operator for 0 < k≤1 and M is its invariant subspace, then the restriction T |M ofT toM is also a class∗-A(k)operator.

Proof. SinceM is an invariant subspace ofT,T has the matrix representation T =

A B O C

on H =M⊕M^⊥. LetP be the projection ofH ontoM, whereA=T |M and

A O O O

=T P =P T P.

SinceT is a class∗-A(k) operator, we have P

T^∗|T|²Tk+1¹ − |T^∗|² P ≥O.

By Hansen inequality, we have |A^∗|² O

O O

≤

|A^∗|²+|B^∗|² O

O O

≤ P T^∗P|T|^2kP T Pk+1¹ . Since

P|T|^2kP≤(P|T|²P)^k, then

P T^∗P|T|^2kP T P ≤P T^∗(P|T|²P)^kT P.

By L¨owner-Heinz inequality we have

P T^∗P|T|^2kP T Pk+1¹ ≤ P T^∗(P|T|²P)^kT Pk+1¹ . So, we have

|A^∗|² O

O O

≤ A^∗|A^∗|^2kA_k+1¹ O

O O

! .

Hence,Ais a class∗-A(k) operator onM.

Theorem 2.9. If T is a class ∗-A(k)operator, has the representation T =λ⊕A on ker(T −λ)⊕(ker(T −λ))^⊥, where λ 6= 0 is an eigenvalue of T, then A is a class

∗-A(k)operator withker(A−λ) ={0}.

Proof. SinceT =λ⊕A, thenT =

λ O O A

and we have:

T^∗|T|^2kTk+1¹

− |T^∗|² =

|λ|^2(k+1) O O A^∗|A|^2kA

_k+1¹

−

|λ|² O O |A^∗|²

= |λ|² O

O A^∗|A|^2kA_k+1¹

!

−

|λ|² O O |A^∗|²

= O O

O A^∗|A|^2kA_k+1¹

− |A^∗|²

!

SinceT is a class∗-A(k) operator, thenAis a class∗-A(k) operator.

Letx2∈ker(A−λ).Then (T−λ)

0 x2

=

O O O A−λ

0 x2

= 0

0

.

Hence x₂ ∈ ker(T −λ). Since ker(A−λ) ⊆ (ker(T −λ))^⊥, this implies x₂ = 0.

Representation ofT impliesA−λis injective and by Theorem 2.8Ais∗-A(k).

3. Spectrum continuity on the set of class ∗-A(k) operator

Let {En}_n∈N be a sequence of compact subsets of C. Let’s define the infe- rior and superior limits of {En}_n∈N, denoted respectively by lim inf_n→∞{En} and lim sup_n→∞{En} as it follows:

1) lim inf_n→∞{En}={λ∈C: for every >0,there existsN ∈Nsuch that B(λ, )∩En6=∅for alln > N},

2) lim sup_n→∞{En}={λ∈C: for every >0,there existsJ ⊆Ninfinite such thatB(λ, )∩En6=∅for alln∈J}.

If

lim inf

n→∞{En}= lim sup

n→∞

{En},

then lim_n→∞{En} is said to exists and is equal to this common limit.

A mappingp, defined onL(H), whose values are compact subsets onCis said to be upper semi-continuous atT, ifTn→T then lim sup_n→∞p(Tn)⊂p(T), and lower semi-continuous at T, ifTn →T then p(T) ⊂lim inf_n→∞p(Tn). If pis both upper and lower semi-continuous atT, then it is said to be continuous atT and in this case limn→∞p(Tn) =p(T).

The spectrumσ:T →σ(T) is upper semi-continuous by [15, Problem 102], but it is not continuous in general, [25, Example 4.6]

We write α(T) = dimker(T), β(T) = dimker(T^∗). An operator T ∈ L(H) is called an upper semi-Fredholm, if it has closed range andα(T)<∞, whileT is called a lower semi-Fredholm ifβ(T)<∞. However,T is called a semi-Fredholm operator if T is either an upper or a lower semi-Fredholm, andT is said to be a Fredholm operator if it is both an upper and a lower semi-Fredholm. IfT ∈L(H) is semi-Fredholm, then the index is defined by

ind(T) =α(T)−β(T).

An operatorT ∈L(H) is said to be upper semi-Weyl operator if it is upper semi- Fredholm and ind(T)≤0, whileT ∈L(H) is said to be lower semi-Weyl operator if it is lower semi-Fredholm and ind(T)≥0. An operator is said to be Weyl operator if it is Fredholm of index zero.

Lemma 3.1. [22]If{Tn} ⊂L(H)andT ∈L(H)are such thatTn converges, according to the operator norm topology, toT then

isoσ(T)⊆lim inf

n→∞ σ(T_n).

Lemma 3.2. [2] Let H be a complex Hilbert space. Then there exists a Hilbert space Y such that H ⊂Y and a mapϕ:L(H)→L(Y)with the following properties:

1. ϕis a faithful∗-representation of the algebra L(H)on Y,so:

ϕ(IH) =IY ,ϕ(T^∗) = (ϕ(T))^∗ ,ϕ(T S) =ϕ(T)ϕ(S) ϕ(αT +βS) =αϕ(T) +βϕ(S)for any T, S∈L(H)andα, β∈C, 2. ϕ(T)≥0 for any T≥0 inL(H),

3. σa(T) =σa(ϕ(T)) =σp(ϕ(T))for any T ∈L(H),

4. If T is a positive operator, thenϕ(T^α) =|ϕ(T)|^α, for α >0,

Lemma 3.3. If T is a class∗-A(k) operator, thenϕ(T)is a class∗-A(k)operator.

Proof. Letϕ:L(H)→L(K) be Berberian’s faithful∗-representation and letT be a class∗-A(k) operator. Then, we have

(ϕ(T))^∗|ϕ(T)|^2kϕ(T)k+1¹ − |(ϕ(T))^∗|² = ϕ(T^∗)ϕ(|T|^2k)ϕ(T)k+1¹ − |ϕ(T^∗)|²

=

ϕ(T^∗|T|^2kT)^k+11 −ϕ(|T^∗|²)

= ϕ

(T^∗|T|^2kT)^k+11 − |T^∗|²

≥0

thusϕ(T) is a class ∗-A(k) operator.

Theorem 3.4. The spectrumσ is continuous on the set of class∗-A(k) operator for k >0.

Proof of the theorem is based in idea’s given in the paper [6].

Proof. Since the function σis upper semi-continuous, if {T_n} ⊂L(H) is a sequence which converges, toT ∈L(H),by operator norm topology. Then lim sup_n→∞σ(T_n)⊂ σ(T).Thus, to prove the theorem it would suffice to prove that if{T_n} is a sequence of operators so that it belongs to class∗-A(k) operator and lim_n→∞kTn−Tk= 0 for some class∗-A(k) operatorT, thenσ(T)⊂lim inf_n→∞σ(Tn). From [25, Proposition

4.9] it would suffice to proveσa(T)⊂lim infnσ(Tn).Sinceσ(T) =σ(ϕ(T)),σ(Tn) = σ(ϕ(T)n) andσa(T) =σa(ϕ(T)) we have

σ_a(T)⊂lim inf

n→∞ σ(T_n)⇐⇒σ_a(ϕ(T))⊂lim inf

n→∞ σ(ϕ(T)_n).

Letλ∈σa(ϕ(T)). Thenλ∈σp(ϕ(T)).

By Theorem 2.9,ϕ(T) has a representation

ϕ(T) =λ⊕AonH = ker(ϕ(T)−λ)⊕(ker(ϕ(T)−λ))^⊥ and ker(A−λ) ={0}.

ThereforeA−λis upper semi-Fredholm operator and α(A−λ) = 0.There exists a >0 such thatA−(λ−µ0) is upper semi-Fredholm operator with ind(A−(λ−µ0)) = ind(A−λ) andα(A−(λ−µ0)) = 0 for every µ0 such that 0 <|µ0| < . Let’s set µ = λ−µ0, and we have ϕ(T)−µ = (λ−µ)⊕(A−µ) is upper semi-Fredholm operator, ind(ϕ(T)−µ) = ind(A−µ) andα(ϕ(T)−µ) = 0.

Suppose the contrary, λ 6∈ lim inf_n→∞σ(ϕ(T)_n). Then, there exists a δ > 0, a neighborhood Dδ(λ) of λ and a subsequence {ϕ(T)_nl} of {ϕ(T)_n} such that σ(ϕ(T)_nl)∩ D_δ(λ) =∅ for everyl ≥1. This implies that ϕ(T)_nl−µis a Fredholm operator and ind(ϕ(T)_nl−µ) = 0 for everyµ∈ D_δ(λ) and

n→∞lim k(ϕ(T)_nl−µ)−(ϕ(T)−µ)k= 0.

It follows from the continuity of the index that ind(ϕ(T) −µ) = 0 and ϕ(T)−µ is a Fredholm operator. Since α(ϕ(T)−µ) = 0, µ 6∈ σ(ϕ(T)) for every µ in a -neighborhood of λ. This contradicts Lemma 3.1, therefore we must have

λ∈lim infn→∞σ(ϕ(T)n).

It is well known Index Product Theorem: ”If S and T are Fredholm operators then ST is a Fredholm operator and ind(ST) = ind(S) + ind(T)”. The converse of this theorem is not true in general. To see this, we have operators onl₂:

T(x1, x2, x3, ...) = (0, x1,0, x2,0, x3, ...) andS(x1, x2, x3, ...) = (x2, x3, x4, ...).

We see ST =I, soST is a Fredholm operator, but S andT are not Fredholm operators. However, if S and T are commuting operators and if ST is a Fredholm operator thenS andT are Fredholm operators. This fact is not true in general if S andT are Weyl operators, see [19, Remark 1.5.3].

Theorem 3.5. IfS andT are commuting class ∗-A(k) operators for0< k≤1, then S, T are Weyl operators ⇐⇒ST is Weyl operator.

Proof. IfS andT are Weyl operators, by Index Product Theorem, we have thatST is a Weyl operator.

The converse, sinceST =T S then

kerS∪kerT ⊆ker(ST) and kerS^∗∪kerT^∗⊆ker(ST)^∗, thenS andT are Fredholm operators.

Since S and T are class ∗-A(k) operators, from [26, theorem 2] S and T are class absolute-k^∗-paranormal operators and by [26, theorem 6] we have ind(S) ≤0 and ind(T)≤0. From

ind(S) + ind(T) = ind(ST) = 0,

we have ind(S) = 0 and ind(T) = 0, soS andT are Weyl operators.

4. Contractions of the class ∗-A(k) operator

Definition 4.1. If the contraction T is a direct sum of the unitary and C_·0 (c.n.u) contractions, then we say thatT has aWold-type decomposition.

Definition 4.2. [9] An operatorT ∈L(H) is said to have the Fuglede-Putnam com- mutativity property (PF propertyfor short) ifT^∗X =XJ for anyX ∈L(K, H) and any isometryJ ∈L(K) such thatT X =XJ^∗.

Lemma 4.3. [8, 23]Let T be a contraction. The following conditions are equivalent:

1. For any bounded sequence{x_n}_n∈N∪{0}⊂H such thatT x_n+1=x_n the sequence {kxnk}_n∈N∪{0} is constant,

2. T has a Wold-type decomposition, 3. T has the PF property.

Fugen Gao and Xiaochun Li [14] have proved that if a contraction T ∈ A^∗ has no nontrivial invariant subspace, then (a) T is a proper contraction and (b) The nonnegative operator D = |T²| − |T^∗|² is a strongly stable contraction. In [17] the authors proved: if T belongs tok-quasi-∗-classAand is a contraction, thenT has a Wold-type decomposition andT has the PF property. In this section we extend these results to contractions in class∗-A(k).

Lemma 4.4. [4, H¨older-McCarthy inequality] LetT be a positive operator. Then, the following inequalities hold for all x∈H:

1. hT^rx, xi ≤ hT x, xi^rkxk^2(1−r) for0< r <1, 2. hT^rx, xi ≥ hT x, xi^rkxk^2(1−r) forr≥1.

Proof of the theorems below is based in idea’s given in the paper [7].

Theorem 4.5. If T is a contraction of class ∗-A(k) operator, then the nonnegative operator

D= T^∗|T|^2kTk+1¹ − |T^∗|²

is a contraction whose power sequence{Dⁿ}^∞_n=1 converges strongly to a projectionP andT^∗P =O.

Proof. Suppose thatT is a contraction of class∗-A(k) operator. Then D= T^∗|T|^2kTk+1¹ − |T^∗|²≥O.

LetR=D¹² be the unique nonnegative square root ofD. Then for everyxinH and any nonnegative integern, we have

hDⁿ⁺¹x, xi = kRⁿ⁺¹xk²=hDRⁿx, Rⁿxi

= D

T^∗|T|^2kTk+1¹ Rⁿx, RⁿxE

−

|T^∗|²Rⁿx, Rⁿx

≤

T^∗|T|^2kT Rⁿx, Rⁿxk+1¹

kRⁿxk^{2(1−k+11 )}− kT^∗Rⁿxk²

= k|T|^kT Rⁿxk^k+12 kRⁿxk^{2(1−k+11 )}− kT^∗Rⁿxk²

≤ k|T|^kTk^k+12 kRⁿxk²− kT^∗Rⁿxk²

≤ kRⁿxk²− kT^∗Rⁿxk²

≤ kRⁿxk²=hDⁿx, xi

ThusR(and soD) is a contraction (setn= 0), and{Dⁿ}^∞_n=1is a decreasing sequence of nonnegative contractions. Then, {Dⁿ}^∞_n=1 converges strongly to a projection, say P. Moreover

m

X

n=0

kT^∗Rⁿxk²≤

m

X

n=0

kRⁿxk²− kRⁿ⁺¹xk²

=kxk²− kR^m+1xk²≤ kxk², for all nonnegative integers m and for every x ∈ H. Therefore kT^∗Rⁿxk → 0 as n→ ∞. Then, we have

T^∗P x=T^∗ lim

n→∞Dⁿx= lim

n→∞T^∗R²ⁿx= 0,

for everyx∈H. So thatT^∗P =O.

Theorem 4.6. Let T be a contraction of class∗-A(k)operator. IfT has no nontrivial invariant subspace, then

1)T is a proper contraction;

2) The nonnegative operator

D= T^∗|T|^2kTk+1¹ − |T^∗|² is a strongly stable contraction.

Proof. Suppose thatT is a class∗-A(k) operator.

1) From [18, Theorem 3.6] we have

T^∗T x=kTk²xif and only ifkT xk=kTkkxk for everyx∈H.

Put M = {x ∈ H : kT xk = kTkkxk} = ker(|T|²− kTk²), which is a closed subspace ofH. In the following, we shall show thatM is aT−invariant subspace. For allx∈M, we have

kT(T x)k² ≤ kTk²kT xk²=kTk⁴kxk²=kkTk²xk²=kT^∗T xk²

≤ k T^∗|T|^2kT_k+1¹

T xkkT xk ≤ k T^∗|T|^2kT_k+1¹

T xkkTkkxk.

So,

kTk⁴kxk²≤ k T^∗|T|^2kTk+1¹ T xkkTkkxk,

thus,

kTk³kxk ≤ k T^∗|T|^2kTk+1¹ T xk and

T^∗|T|^2kTk+1¹ T x

= D

T^∗|T|^2kTk+1² T x, T xE¹2

≤ D

T^∗|T|^2kT²

T x, T xE_2(k+1)¹

kT xk^{(1−k+11 )}

=

T^∗|T|^2kT T x

1

k+1kT xk^{(1−k+11 )}

≤ kTk^2k+3k+1kxk^k+11 kTk^k+1k kxk^k+1k

= kTk³kxk.

Hence,

kTk³kxk=k T^∗|T|^2kTk+1¹ T xk. (4.1) From relation (4.1) we have

kTk³kxk =

T^∗|T|^2kT_k+1¹ T x

= D

T^∗|T|^2kTk+1² T x, T xE¹2

≤ D

T^∗|T|^2kT²

T x, T xE_2(k+1)¹

kT xk^{(1−k+11 )}

= kT^∗|T|^2kT T xk^k+11 kT xk^{(1−k+11 )}

≤ kT^∗|T|^2kk^k+11 kT(T x)k^k+11 kT xk^{(1−k+11 )} Then,

kTk²kxk ≤ kT(T x)k=⇒ kTk²kxk=kT(T x)k, respectively,

kT(T x)k=kTk²kxk=kTkkT xk.

Thus,M is a T-invariant subspace.

Now, let T be a contraction, i.e., kT xk ≤ x, for every x ∈ H. If kTk < 1, thus T is a strict contraction, then it is trivially a proper contraction. If kTk = 1, thus T is nonstrict contraction, then M = {x ∈ H : kT xk = kxk}. Since T has no nontrivial invariant subspace, then the invariant subspace M is trivial: either M ={0} orM =H. IfM =H thenT is an isometry, and isometries have invariant subspaces. Thus M = {0} so that kT xk < kxk for every nonzero x ∈ H. So T is proper contraction.

2) Let T be a contraction of class ∗-A(k) operator. By the above theorem, we haveDis a contraction,{Dⁿ}^∞_n=1converges strongly to a projectionP, andT^∗P =O.

So,P T =O. SupposeThas no nontrivial invariant subspaces. Since kerP is a nonzero invariant subspace for T whenever P T =O andT 6=O, it follows that kerP =H. Hence P = O, and so {Dⁿ}^∞_n=1 converges strongly to null operator O, so D is a strongly stable contraction. SinceD is self-adjoint, thenD∈C00.

Corollary 4.7. Let T be a contraction of the class ∗-A(k)operator. IfT has no non- trivial invariant subspace, then bothT and the nonnegative operator

D= T^∗|T|^2kTk+1¹ − |T^∗|² are proper contractions.

Proof. Since a self−adjoint operator T is a proper contraction if and only if T is a

C00contraction.

Theorem 4.8. IfT is a contraction and class∗-A(k)operator for k >0 , thenT has a Wold-type decomposition.

Proof. SinceT is a contraction operator, the decreasing sequence {TⁿT^∗n}^∞_n=1 con- verges strongly to a nonnegative contraction. We denote by

S=

n→∞lim TⁿT^∗n1₂ .

The operators T and S are related by T^∗S²T = S², O ≤ S ≤ I and S is self-adjoint operator. By [10] there exists an isometry V :S(H)→ S(H) such that V S = ST^∗, and thus SV^∗ = T S, and kSV^mxk → kxk for every x ∈ S(H). The isometryV can be extended to an isometry onH, which we still denote byV.

For an x ∈ S(H), we can define xn = SVⁿx for n ∈ N∪ {0}. Then for all nonnegative integersmwe have

T^mxn+m=T^mSV^m+nx=SV^∗mV^m+nx=SVⁿx=xn, and for allm≤nwe have

T^mxn =x_n−m.

SinceT is class∗-A(k) operator fork >0 and nontrivialx∈A(H) we have kxnk⁴ = kT xn+1k⁴≤ kT^∗T xn+1k²kxn+1k²

≤ k T^∗|T|^2kT_k+1¹

T xn+1kkT xn+1kkxn+1k²

≤ D

T^∗|T|^2kT²

T x_n+1, T x_n+1E_2(k+1)¹

kT xn+1k^{(1−k+11 )}kT xn+1kkxn+1k²

= kT^∗|T|^2kT T x_n+1k^k+11 kx_nk^{(1−k+11 )}kx_nkkx_n+1k²

≤ kT T x_n+1k^k+11 kx_nk^2k+1k+1kx_n+1k²

= kx_n−1k^k+11 kx_nk^2k+1k+1kx_n+1k² hence

kxnk ≤ kx_n−1k^2k+31 kxn+1k^2k+22k+3 ≤ 1

2k+ 3(kx_n−1k+ (2k+ 2)kxn+1k). Thus

(2k+ 2) (kxn+1k − kxnk)≥ kxnk − kxn−1k Put,b_n=kx_nk − kx_n−1k, and we have

(2k+ 2)bn+1≥bn. (4.2)

Sincexn=T(xn+1), then

kxnk=kT xn+1k ≤ kxn+1k for everyn∈N, then sequence{kxnk}_n∈N∪{0} is increasing. From

SVⁿ =SV^∗Vⁿ⁺¹=T SVⁿ⁺¹ we have

kxnk=kSVⁿxk=kT SVⁿ⁺¹xk ≤ kSVⁿ⁺¹xk ≤ kxk,

for every x∈S(H) andn∈N∪ {0}.Then{kxnk}_n∈N∪{0} is bounded. From this we havebn≥0 andbn→0 asn→ ∞.

It remains to check that all bn equal zero. Suppose that there exists an integer i≥1 such thatbi >0. Using inequality (4.2) we getbi+1 ≥ _2k+2^bi >0, so bi+1 >0.

From that and using again inequality (4.2), we can show by induction thatbn>0 for alln > i.This is contradictory with thatbn →0 asn→ ∞.Sobn = 0 for alln∈N and thuskxn−1k=kxnk for alln≥1.Thus the sequence{kxnk}_n∈N∪{0} is constant.

From Lemma 4.3,T has aWold-type decomposition.

Acknowledgment.The authors want to thank referees for valuable comment and sug- gestion given in the paper.

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Ilmi Hoxha

Faculty of Education, University of Gjakova ”Fehmi Agani”

Avenue ”Ismail Qemali” nn, Gjakov¨e, 50000, Kosova e-mail:ilmihoxha011@gmail.com

Naim L. Braha

(Corresponding author)

Department of Mathematics and Computer Sciences, University of Prishtina

Avenue ”George Bush” nn, Prishtin¨e, 10000, Kosova e-mail:nbraha@yahoo.com

Agron Tato

Department of Mathematics

Polytechnic University of Tirana, Albania e-mail:agtato@gmail.com