# On Some Normality-Like Properties and Bishop's Property (β) for a Class of Operators on Hilbert Spaces

Authors: Sid Ahmed Ould Ahmed Mahmoud
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We prove some further properties of the operatorT∈[nQN]
(n-power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator
T∈[nQN] satisfying the translation invariant property is normal and that the
operator T∈[nQN] is not supercyclic provided that it is not invertible. Also, we
study some cases in which an operator T∈[2QN] is subscalar of order m; that is, it is
similar to the restriction of a scalar operator of order m to an invariant subspace.


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## Body

## 1. Introduction

Although normality of operators(T*T=TT*) makes things easier, it rarely occurs and relaxing the normality condition is essential in the theory of operators on Hilbert spaces. One of the most important subclasses of the algebra of all bounded linear operators acting on a Hilbert space, the class of hyponormal operators, has been studied by many authors (see [1]). In recent years this class has been generalized, in some sense, to larger classes of the so-called p-hyponormal, log-hyponormal, posinormal, k-quasihyponormal classes, and so forth (see [2–6]).In [7], Putinar showed that hyponormal operators are subscalar. This fact has led to far-reaching results, discovering deep properties of these operators. In this paper we extend that result to other generalized classes of operators.LetH be an infinite dimensional separable complex Hilbert space, let K be a complex Hilbert space, and let ℒ(H,K) be the algebra of all bounded linear operators from H to K. We write ℒ(H) for ℒ(H,H). If T∈ℒ(H,K), we will write N(T) and R(T) for the null space (also referred to as the reducing subspace) and the range of T, respectively. The spectrum, the point spectrum, and the approximate point spectrum of an operator T are denoted by σ(T),σp(T),σap(T), respectively. T* means the adjoint of T.An operatorT∈ℒ(H) is(1)
hyponormal if and only ifTT*≤T*T⇔∥T*x∥≤∥Tx∥,forallx∈H,(2)
posinormal if and only ifR(T)⊂R(T*), or equivalently TT*≤λ2T*T for some λ>0,(3)
p-hyponormal if and only if (TT*)p≤(T*T)p,0<p≤1,(4)
p-quasihyponormal if and only if T*[(TT*)p-(T*T)p]T≤0,0<p<1,(5)
log-hyponormal if T is invertible and log(TT*)≤log(T*T),(6)
2-isometry if and only ifT*2T2-2T*T+I=0 (see [8]), where I indicates the identity operator that is, if and only if(1.1)T*2T2=2T*T-I.
An operator T∈ℒ(H) is said to be n-power quasinormal (abbreviated as nQN), n=1,2,…, if(1.2)TnT*T=T*TTn=T*Tn+1.
If n=1, T is called quasinormal. This class of operators being denoted by [nQN], that is,(1.3)[nQN]:={T∈L(H):TnT*T-T*TTn=0}
was studied by the author [9].T is called an m-partial isometry if T satisfies(1.4)TBm(T)=T∑k=0m(mk)(-1)kT*m-kTm-k=0,
where Bm(T) is obtained formally from the binomial expansion of Bm(T)=(T*T-I)m by understanding (T*T)m-k=T*m-kTm-k. The case when m=1 is called the partial isometries class. The class of m-partial isometries was defined by Saddi and Sid Ahmed [10] who proved some properties of the class. See Proposition 5.4.This paper is divided into five sections. Section2 deals with some preliminary facts concerning function spaces. Section 3 includes our main results. There we study some properties of [nQN]. In particular we show that an operator T∈[nQN] satisfying the translation invariant property is normal, and an invertible operator T∈[nQN] and its inverse T-1 have a common nontrivial invariant closed set provided that T*∈[nQN]. Also we show that some of class [2QN] satisfy an analogue of the single-valued extension for W2m(D,H) and have scalar extension. In Section 4, we give some results about the Berberian extension. In Section 5, we shall use some properties of the approximate spectrum to obtain some results on single-valued extension (SVEP) (see Section 2) property for the m-partial isometries operators.


## 2. Spaces of Vector-Valued Functions

We will need the following function spaces.Letλ be the coordinate in ℂ and let dμ(λ) denote the planar Lebesgue measure. Let D be a bounded open subset of ℂ. We will denote by L2(D,H) the Hilbert space of square measurable (or summable) functions f:D→H such that(2.1)‖f‖2,D={∫D‖f(λ)‖2dμ(λ)}1/2<∞.LetO(D,H) denote the space of H-valued functions analytic on D, that is, ∂¯f=∂f/∂z¯=0. Equipped with the topology of uniform convergence on compact subsets of D, O(D,H) is a Frechet space. Let(2.2)A2(D,H)=L2(D,H)∩O(D,H)
denote the Bergman space for D consisting of square measurable functions f that are analytic on D.We denote byP the orthogonal projection of L2(D,H) onto A2(D,H).Let us define now a special Sobolev type space. Letm be a fixed nonnegative integer. The Sobolev space W2m(D,H) of order m of vector-valued functions with respect to ∂¯ will be the space of those functions f∈L2(D,H) whose derivatives ∂¯f,…,∂¯mf in the sense of distributions also belong to L2(D,H), that is,(2.3)W2m(D,H)={f∈L2(D,H):∂¯kf∈L2(D,H),fork=0,1,…,m}.Endowed with the norm(2.4)‖f‖W2m2=∑k=0m‖∂¯kf‖2,D2.W2m(D,H) becomes a Hilbert space contained continuously in L2(D,H); that is, there is a constant 0<C<∞ such that ∥f∥L2(D,H)≤C∥f∥W2m(D,H)forallf∈W2m(D,H).Let(2.5)Tλ=T-λ=T-λI
for λ∈ℂ once and for all, whenever the definition is meaningful. We say that T has the single-valued extension property at λ0∈ℂ (abbreviated SVEP at λ0) if, for every open neighborhood U of λ0, the only analytic solution f to the equation(2.6)(Tλf)(λ)=(T-λ)f(λ)=0
for all λ in U is the constant function f≡0. We say that T has SVEP if T has a SVEP at every λ∈ℂ.It is easily seen that an operatorT∈ℒ(H) has SVEP if and only if, for each open D⊆ℂ.The operatorTD:O(D,H)→O(D,H) defined by(2.7)TD(f)(λ)=Tλf(λ)∀f∈O(D,H),λ∈D.
is one to one.Recall that, for a bounded operatorT on H, the local resolvent set ρT(x) of T at the point x∈H is defined as the union of all open subsets D of ℂ such that there exists an analytic function f:D→H which satisfies(2.8)(Tλ)f(λ)=x∀λ∈D.
The local spectrum σT(x) of T at x∈H is the set defined by σT(x):=ℂ∖ρT(x) and obviously σT(x)⊂σ(T). It is clear from the definition that, T has SVEP if and only if zero is the unique vector x∈H such that σT(x)=∅ (see for more details [11]).Recall that a bounded operatorT∈ℒ(H) is said to have the Bishop’s property (β) if for every open subset D of the complex plane ℂ and every sequence of analytic functions fn:D→H with the property that(2.9)Tλfn(λ)⟶0asn⟶∞,
uniformly on all compact subsets of D, fn(λ)→0 as n→∞ locally uniformly on D or equivalently, for every open subset D of ℂ, the operator TD defined in (2.7) is one to one and has the closed range [11, Proposition 3.3.5]. It is a very important notion in spectral theory. It is wellknown that every normal operator has Bishop’s property (β).A bounded operatorT on H is called scalar of order m if it possesses a spectral distribution of order m, that is, if there is a continuous unital morphism,(2.10)Φ:C0m(C)⟶L(H),
such that Φ(z)=T, where z stands for the identity function on ℂ and C0m(ℂ) for the space of compactly supported functions on ℂ, continuously differentiable of order m,0≤m≤∞. An operator is subscalar if it is similar to the restriction of a scalar operator to an invariant subspace.LetMz be the operator on W2m(D;H) such that (Mzf)(z)=zf(z) for f∈W2m(D;H). This has a spectral distribution of order m, defined by the functional calculus ΦM:C0m(ℂ)→ℒ(W2m(D,H));ΦM(f)=Mf. Therefore Mz is a scalar operator of order m. Consider a bounded open disk D which contains σ(T) and the quotient space(2.11)H(D)=W2m(D,H)TzW2m(D,H)¯
endowed with the Hilbert space norm. We denote the class containing a vector f or an operator A on H(D) by f̂ or Â, respectively. Let Mz be the operator of multiplication by z on W2m(D;H). As noted above, Mz is a scalar of order m and has a spectral distribution Φ. Let S≡Mẑ. Since TzW2m(D,H)¯ is invariant under every operator Mf; f∈C0m(ℂ), we infer that S is a scalar operator of order m with spectral distribution Φ̂. Consider the natural map V:H→H(D) defined by Vh=[1⊗h]̂, for h∈H, where 1⊗h denotes the constant function identically equal to h. In [7], Putinar showed that if T∈ℒ(H) is a hyponormal operator then V is one to one and has closed range such that VT=SV, and so T is subscalar of order m.


## 3. Further Properties of the Class[nQN]

We start this section with some properties ofn-power quasinormal operators.Theorem 3.1.
The class[nQN] has the following properties.(1)
The class[nQN] is closed under unitary equivalence and scalar multiplication.(2)
IfT is of class [nQN] and M is a closed subspace of H that reduces T, then T∣M (the restriction ofT to M) is of class [nQN].Proof.
(1) LetS∈ℒ(H) be unitary equivalent to T. Then there is a unitary operator V∈ℒ(H) such that T=V*SV which implies that T*=V*S*V. Noting that Tn=V*SnV and inserting I=VV* suitably, we deduce from (1.2) that
(3.1)V*SnS*SV=TnT*T=T*Tn+1=V*S*Sn+1V,
and (1.2) follows for S. Since
(3.2)(T∣M)Δ=(TΔ∣M)
for Δ as the n-th power or the adjoint, it follows that the left-hand side of (1.2) for (T∣M) reads
(3.3)(TnT*T∣M),
which is (T*Tn+1∣M)=(T∣M)*(T∣M)n+1, which is the right-hand side of (1.2). Thus, T∣M is of class [nQN].Next we characterize a matrix on a 2-dimensional complex Hilbert space which is in[nQN]. Since every matrix on a finite dimensional complex Hilbert space is unitarily equivalent to an upper triangular matrix and an n-power quasinormal operator is unitarily invariant, it suffices to characterize an upper triangular matrix T. From the direct calculation, we get the following characterization.Proposition 3.2.
Forn≥2 one has
(3.4)T=(xy0z)∈[nQN]⟺{y¯x(xn-1+zxn-2+⋯+zn-1)=0,xy¯(zn-xn)=0,x¯y(zn-xn)-(xn-1+zxn-2+⋯+zn-1)(|y|2+|z|2-|x|2)=0.We remark here that Proposition3.2 offers the convenient criterion to find some examples of operators in [nQN]. Also we observe that [nQN] is not necessarily normal on a finite dimensional space.Next couple of results show that[nQN] does not have a translation invariant property.Theorem 3.3 (see [9]).
IfT and T-I are of class [2QN], then T is normal.Theorem 3.4 (see [9]).
IfT is of class [2QN]∩[3QN] such that T-I is of class [nQN], then T is normal.It is natural to ask the following question: what is the operators in[nQN] satisfying the translation invariant property? The answer to this question is provided by the following theorem.Theorem 3.5.
Tλ is of class [nQN] for every λ∈ℂ if and only if T is a normal operator.Proof.
Assume that(Tλ) is of class [nQN] for every λ∈ℂ. Then (1.2) reads (T-λ)n(T-λ)*(T-λ)=(T-λ)*(T-λ)(T-λ)n, which reduces on eliminating the common factor -λ¯(T-λ)n+1 to
(3.5)(T-λ)n(T*T-λT*)=(T*T-λT*)(T-λ)n.
By the binomial expansion,
(3.6)∑k=0n(-1)k(nk)λkTn-k(T*T-λT*)=(T*T-λT*)∑k=0n(-1)k(nk)λkTn-k,
whence by arranging terms suitably, the extremal terms vanishing in view of (1.2),
(3.7)∑k=1n-1(-1)k(nk)λk(Tn-kT*T-T*TTn-k)-∑k=1n-1(-1)k(nk)λk+1(Tn-kT*-T*Tn-k)=0.
Now note that from the second summand in (3.7), we may extract the extremal term (-1)nnλn(T*T-TT*) and express it in terms of the remaining terms which contain λ to the power <n. Hence dividing (3.7) by λn and letting λ→∞, we conclude that T*T-TT*→0, whence the normality of T follows.Conversely it is known that normality is a translation invariant property; that is, ifT is normal, then (Tλ) is normal for every λ∈ℂ, and hence (Tλ) is of class [nQN].The following proposition gives a characterization of ann-power quasinormal operator.Proposition 3.6.
LetT∈ℒ(H), A=Tn+T*T, and B=Tn-T*T. Then T is of class [nQN] if and only if A commutes with B.Proof.
Commutativity ofA and B is equivalent to (1.2).Proposition 3.7.
LetT,A,B be as in Proposition 3.6. Then if T is of class [nQN], then TnT*T commutes with A and B.Proof.
By (1.2),
(3.8)TnT*T(Tn±T*T)=(Tn±T*T)TnT*T.
In general, the two classes[nQN] and [(n+1)QN] are not the same (see [9]).Proposition 3.8.
IfT is both of class [nQN] and [(n+1)QN], then it is of class [(n+2)QN], that is, [nQN]∩[(n+1)QN]⊂[(n+2)QN].Proof.
By (1.2),
(3.9)Tn+1T*T=TnT*T2
so that Tn+2T*T may be transformed into T*TTn+2.
It is known that ifT belongs to [nQN] some n>0, T2 does not necessarily belong to the same class.Theorem 3.9 (see [9]).
IfT and T* are of class [nQN], then Tn is normal.Proposition 3.10.
If an operatorT of class [2QN] is a 2-isometry, then T2 is of class [nQN] for all integers n≥2.Proof.
From Proposition3.8 it suffices to prove that T2 is of class [2QN] and of class [3QN] because we may then proceed inductively.
SinceT is a 2-isometry, we have T4(T*2T2)=T4(2T*T-I). Using T2T*T=T*T3, we may shift the power of T2 to the left, arriving at T4(T*2T2)=T*2T6; that is, T2 is of class [2QN].
In the same way, we may deduce thatT6(T*2T2)=T*2T8 whence T2 is of class [3QN].Lemma 3.11 (see [9]).
IfT is of class [nQN], then N(Tn)⊂N(T*n).Proposition 3.12.
IfT is both of class [nQN] and [(n+1)QN] such that T is injective or T* is injective, then T is quasinormal.Proof.
SinceT is of class [nQN]∩[(n+1)QN], we have (3.9), which reads Tn(TT*T-T*T2)=0. If T is injective, then so is Tn and we have TT*T-T*T2=0, whence T is quasi-normal. If T* is injective, we may appeal to Lemma 3.11.In the following theorem we prove some topological properties of the class[nQN].Theorem 3.13.
The class[nQN] is an arcwise-connected, closed subset of ℒ(H) equipped with the uniform operator (norm) topology.Proof.
By Theorem3.1, (1), we see that the ray aT in ℒ(H) through T is contained in [nQN] for every complex number a, and therefore [nQN] is arcwise-connected.
To see that[nQN] is closed, we prove that any strong limit T∈ℒ(H) of a sequence (Tp) in [nQN] also belongs to [nQN]; that is, we let (Tp) be a sequence of operators in [nQN] converging to T∈ℒ(H) in norm:
(3.10)‖Tpx-Tx‖⟶0asp⟶∞,foreachx∈H.
Hence it follows that
(3.11)‖Tp*x-T*x‖=‖(Tp-T)*x‖≤‖(Tp-T)*‖‖x‖=‖Tp-T‖‖x‖⟶0,
whence (Tp*) converges strongly to T*.
Since the product of operators is sequentially continuous in the strong topology (see [12, page 62]), one concludes thatTpnTp*Tp converge strongly to TnT*T. Similarly Tp*Tpn+1 converges strongly to T*Tn+1. Hence the limiting case of (1.2) shows that T belongs to [nQN], completing the proof.Proposition 3.14.
IfT1,T2,…,Tp are of class [nQN], then both the direct sum T1⊕T2⊕⋯⊕Tp and the tensor product T1⊗T2⊗⋯⊗Tp are of class [nQN].Proof.
By the compatibility principle similar to (3.2),
(3.12)(T1⊕T2)Δ=(T1Δ⊕T2Δ),(T1⊗T2)Δ=(T1Δ⊗T2Δ),
where Δ indicates either the nth power or adjoint, the proof follows.A linear operatorT on H is hypercyclic if there is a vector with dense orbit; that is, if there exists an x∈H such that orbit Orb(T,x):={x,Tx,T2x,…} is dense in H, and in this case x is called a hypercyclic vector for T.An operatorT on H is supercyclic if there exists a vector whose scaled orbit is dense; that is, if there exists an x∈H such that {λTnx,n≥0,λ∈ℂ} is dense in H and in this case x is called a supercyclic vector for T.Kitai [13] showed that hyponormal operators are not hypercyclic. We generalize Kitai’s theorem to the class [nQN].Proposition 3.15.
IfT is of class [nQN] with σp(Tn)≠∅, then T is not hypercyclic.Proof.
IfT is hypercyclic, Tn is hypercyclic, and hence σp(T*n)=∅ by [13, corollary 2.4]. From Lemma 3.11 we have σp(Tn)⊂σp(T*n) and, hence, σp(Tn)=∅, a contradiction.Theorem 3.16 (see [14, Theorem 2]).
IfT is not hypercyclic, then T and T-1 have a common non-trivial invariant closed subset.Proposition 3.17.
IfT is of class [nQN] and 0∉σ(T), then T and T-1 have a common nontrivial closed invariant subset.Proof .
SinceT is of class [nQN] and 0∉σ(T), it follows that Tn is normal, and hence (T-1)n is normal. By [13, Corollary 4.5] Tn and (T-1)n have no hypercyclic vector. Thus by [15], neither T or T-1 has a hypercyclic vector. Therefore by [15] T and T-1 have a common nontrivial closed invariant subset. Hence Theorem 3.16 completes the proof.Proposition 3.18.
OperatorsT that are of class [nQN] such that T is not invertible are not supercyclic.Proof.
Assume thatT is of class [nQN] and supercyclic. Considering the class [nQN] being closed under multiplication by nonzero scalars, we may assume that ∥T∥=1. Since the supercyclic contraction T satisfies property (β), σ(T) is contained in the boundary ∂𝔻 of the unit disk 𝔻 [11, Proposition 3.3.18]. Thus T is invertible, and we have a contradiction.Definition 3.19.
An operatorT∈ℒ(H) is algebraic if there is non-zero-polynomial p such that p(T)=0.The following proposition shows that some quasinilpotentn-power quasi-normal operators are subscalar.Proposition 3.20.
If bothT and T* are of class [nQN] such that T is quasinilpotent, then T is nilpotent, and hence T is subscalar.Proof.
SinceT is quasinilpotent, σ(T)={0}. Hence by the spectral mapping theorem we get σ(Tn)=σ(T)n={0}. Thus Tn is quasinilpotent and normal. So Tn=0;thatis,T is nilpotent, and T is algebraic operator, and hence T is subscalar.Proposition 3.21 (see [7, Proposition 2.1]).
For every bounded diskD in ℂ, there is a constant CD such that for an arbitrary operator T∈ℒ(H) and f∈W2m(D,H) we have
(3.13)‖(I-P)∂¯jf‖2,D≤CD(‖(Tλ)*∂¯j+1f‖2,D+‖(Tλ)*∂¯j+2f‖2,D),j=0,1,…,m-2,
where P is the orthogonal projection of L2(D,H) onto A2(D,H).The next theorem is important for the proof of our main theorem, Theorem3.27.Theorem 3.22.
LetD be an arbitrary bounded disk in ℂ. If T is of class [2QN] and σ(T)∩(-σ(T))=∅, then the operator
(3.14)Tλ:W2m(D,H)⟶W2m(D,H)
is one to one.Proof.
Letg∈W2m(D,H) such that Tλg=0, that is,
(3.15)‖Tλg‖W2m=0.
Then for j=0,1,…,m we have
(3.16)‖Tλ∂¯jg‖2,D=0.
Hence for j=1,…,m we get ∥Tλ22∂¯jg∥2,D=0. Since σ(T)∩(-σ(T)) is empty then T2 is normal [9, Theorem 2.2]. Hence,
(3.17)‖(Tλ22)*∂j¯g‖2,D=0.
Now we claim that
(3.18)‖(Tλ)*∂j¯g‖2,D=0.
Indeed, since Tλ is invertible for λ∈D∖σ(T), (3.16) implies that
(3.19)‖∂j¯g‖2,D∖σ(T)=0.
Therefore
(3.20)‖(Tλ)*∂j¯g‖2,D∖σ(T)=0.
Sinceσ(T)∩(-σ(T))=∅ and σ(T*)=σ(T)*,(T-λ)* is invertible for λ∈σ(T), therefore; from (3.17) we have
(3.21)‖(Tλ)*∂¯jg‖2,σ(T)=0.
It is clear form (3.20); and (3.21) that
(3.22)‖(Tλ)*∂j¯g‖2,D=0,forj=0,1,…,m.
Thus Proposition 3.21 and (3.21) imply
(3.23)‖(I-P)∂¯jg‖2,D=0forj=0,1,…,m-2,
where P denotes the orthogonal projection of L2(D,H) onto A2(D,H).
Hence(Tλ22)Pg=Tλ22g=0. Since T2 has SVEP, T has SVEP. Also g=Pg is analytic and (T-λ)g(λ)=0 for λ∈D. Hence g=0. Thus, Tλ is one to one.Corollary 3.23.
IfT1 and T2 are of class [2QN] with σ(Ti)∩(-σ(Ti))=∅, for i=1,2 and T2T1=0. Then
(3.24)(T1+T2)λ:W2m(D,H)⟶W2m(D,H)
is one to one.Proof.
Iff∈W2m(D,H) is such that (T1+T2)λf=0. Since T2T1=0, we get (T2-λ)T2f=0. Since (T2)λ is one to one, T2f=0. Hence, (T1)λf=0. Since (T1)λ is one to one, f=0.The following corollary shows that the nilpotent perturbation of operators in[2QN] satisfying SVEP satisfies SVEP.Corollary 3.24.
If an operatorT∈ℒ(H) is a nilpotent perturbation of a 2-power quasi-normal operator S, that is, T=S+N, where S is of class [2QN], S and N commute, and Nm=0. If σ(S)∩(-σ(S))=∅, then Tλ is one-to-one.Proof.
Ifg∈W2m(D,H) is such that Tλg=0, then
(3.25)Sλg=-Ng.
Hence SλNj-1g=-Njg for j=1,2,…,m. We prove that Njg=0 for j=0,1,…,m-1 by indication. Since Nm=0, SλNm-1g=-Nmg=0. Since Sλ is one-to-one by Theorem 3.22Nm-1g=0. Assume it is true when j=k, that is, Nkg=0. From (3.25), we get
(3.26)SλNk-1g=-Nkg=0.
Since Sλ is one-to-one from Theorem 3.22, Nk-1g=0. By indication we have g=0. Hence Tλ is one-to-one.An operatorT∈ℒ(H) is said to be the following.(1)
It is left invertible if there is an operatorS∈ℒ(H) such that ST=I, where I denotes the identity operator. The operator S is called a left inverse of T.(2)
It is right invertible if there is an operatorR∈ℒ(X) such that TR=I. The operator R is called a right inverse of T (see [16]).Corollary 3.25.
IfT is of class [2QN] with the property σ(T)∩(-σ(T))=∅, and if S is a left invertible operator with the left inverse R, then the operator (STR)λ:W2m(D,H)→W2m(D,H) is one-to-one.Proof.
Ifg∈W2m(D,H) is such that (STR)λg=0, then
(3.27)TλRg=0.
Hence for j=0,1,…,m we have TλR∂¯jg=0. From Theorem 3.22, we get R∂¯jg=0 for j=0,1,…,m.
Thus,STR∂¯jg=0 for j=0,1,…,m. It follows that λ∂¯jg=0 for j=0,1,…,m. By application of [7, Proposition 2.1] with T=(0), we have
(3.28)‖(I-P)g‖2,D=0,
where P denotes the orthogonal projection of L2(D,H) onto the Bergman space A2(D,H). Hence λg=λPg=0. From [17, Corollary 10.7], there exists a constant c>0 such that
(3.29)c‖Pg‖2,D≤‖λPg‖2,D.
So g=Pg=0. Thus, (STR)λ is one-to-one.Corollary 3.26.
IfT is of class [2QN] with the property σ(T)∩(-σ(T))=∅, and if S is a right invertible operator with the right inverse R then the operator (RTS)λ:W2m(D,H)→W2m(D,H), is one-to-one.Now we are ready to prove our main theorem.Theorem 3.27.
IfT is of class [2QN] with the property that σ(T)∩(-σ(T))=∅ and let D be a bounded disk which contains σ(T), then the operator V:H→H(D), defined by
(3.30)Vg=1⊗g+TλW2m(D,H)¯=[1⊗g]̂,
is one to one and has closed range, where H(D) is as in (2.11).Proof.
First, we will prove that if{gk}1∞⊂H and {fk}1∞⊂W2m(D,H) are sequences such that
(3.31)lim⁡k→∞‖1⊗gk+Tλfk‖W2m=0,
then limk→∞gk=0.
By the definition of the norm of a Sobolev space, (3.31) implies that
(3.32)lim⁡k→∞‖Tλ∂¯jfk‖2,D=0forj=1,…,m.
From (3.32) we get
(3.33)lim⁡k→∞‖Tλ22∂¯jfk‖2,D=0forj=1,…,m.
SinceT2 is normal,
(3.34)lim⁡k→∞‖(Tλ22)*∂¯jfk‖2,D=0forj=1,…,m.
Since Tλ is invertible for λ∈D∖σ(T), (3.32) implies that
(3.35)lim⁡k→∞‖∂¯jfk‖2,D∖σ(T)=0.
Therefore
(3.36)lim⁡k→∞‖(Tλ)*∂¯jfk‖2,D∖σ(T)=0forj=1,…,m.
Sinceσ(T)∩(-σ(T))=∅ and σ(T*)=σ(T)*, it is clear that (T-λ)* is invertible for λ∈σ(T). Therefore from (3.34), we have
(3.37)lim⁡k→∞‖(Tλ)*∂¯jfk‖2,σ(T)=0.
Hence, from (3.36) and (3.37) we get
(3.38)lim⁡k→∞‖(Tλ)*∂¯jfk‖2,D=0,j=1,…,m.
Then by Proposition 3.21, we have
(3.39)lim⁡k→∞‖(I-P)∂¯jfk‖2,D=0,j=1,…,m-2.
By (3.31) and (3.39), we have
(3.40)lim⁡k→∞‖1⊗gk+TλPfk‖2,D=0.
Let Γ be a curve in D surrounding σ(T). Then
(3.41)lim⁡k→∞‖Pfk+(Tλ)-1(1⊗gk)‖=0
uniformly for λ∈Γ by (3.40). Hence by Riesz-Dunford functional calculus
(3.42)lim⁡k→∞‖12πi∫ΓPfk(z)dz+gk‖=0.
But by Cauchy’s theorem
(3.43)∫ΓPfk(z)dz=0.
Hence limk→∞gk=0. Thus the map V is one-to-one and has closed range.Corollary 3.28.
IfT is of class [2QN] with the property that σ(T)∩(-σ(T))=∅, then T is subscalar of order m≥2.Proof.
Consider an arbitrary bounded open diskD in ℂ that contains σ(T) and the quotient space given in (2.11).
LetMλ be the multiplication operator by λ on W2m(D,H). Then Mλ is a scalar operator of order m, and its spectral distribution is
(3.44)ΦM:C0m(C)⟶L(W2m(D;H)),ΦM(f)=Mf,
where Mf is the multiplication operator by f∈C0m(ℂ). Let S≡Mλ̂. Since TλW2m(D;H) is invariant under every operator Mf, we infer that S is a scalar operator of order m with spectral distribution Φ̂.
LetV be the operator
(3.45)Vg=1⊗g+TλW2m(D,H)¯
from H into H(D). Then we have the following commutative diagram
(3.46)
By the previous theorem the operator V is a topological isomorphism of H into R(V). The relation VT=SV shows that R(V) is S-invariant. Hence S is an extension of the operator V, so this operator is subscalar. Since V is invertible on R(V), then the operator T is subscalar of order m. On the other hand from [18, Theorem 4.3] we deduce that m≥2 and the theorem is proved.Corollary 3.29.
IfT is of class [2QN] with the property that σ(T)∩(-σ(T))=∅, then T has Bishop’s property (β).Proof.
It follows from Corollary3.28 and [18, Lemma 2.1].In [19] the authors study some operators with the single-valued extension property. In the following propositions we extend some of these results to operators with the Bishop’s property (β).Proposition 3.30.
LetT∈ℒ(⊕i=1kH) be the following k×k triangular operator matrix
(3.47)T=(T11T12T13……T1k0T22T23……T2k00T33……T3k000………0…………Tkk).
Assume that Tii is of class [2QN] and satisfies σ(Tii)∩(-σ(Tii))=∅ for i=1,2,…,k-1 and Tkk is nilpotent. Then T has Bishop’s property (β).Proof.
Letfp=⊕i=1kfpi:D→⊕i=1kH be a sequence of analytic functions such that Tλfp(λ)→0 uniformly on every compact subset K of an open set D of ℂ; then we have
(3.48)(T11)λfp1(λ)+T12fp2(λ)+T13fp3(λ)⋯+T1kfpk(λ)⟶0,(T22)λfp2(λ)+T23fp3(λ)+⋯+T2kfpk⟶0,(T33)λfp3(λ)+⋯+T3kfpk(λ)⟶0,⋮⋮⋮⋮(Tk-1k-1)λfpk-1(λ)+Tk-1kfpk(λ)⟶0,(Tkk)λfpk(λ)⟶0.
Since Tkkm=0,λTkkm-1fpk(λ)→0 and hence Tkkm-1fpk(λ)→0 if λ≠0. Since (Tkk)λfpk(λ)→0 from (3.48) λTkkm-2fpk(λ)→0. By the same reason, Tkkm-3fpk(λ)→0. By repeating this procedure, we finally achieve
(3.49)fpk(λ)⟶0,
uniformly on K. Then we obtain the following equation: (Tk-1k-1)λfpk-1→0 uniformly on every compact K. Since Tk-1k-1 has Bishop’s property (β) from Corollary 3.29, fpk-1(λ)→0 uniformly on K. By repeating this process we prove that fp1(λ)→0 uniformly on K.
Hence{fp=fp1⊕fp2⊕⋯⊕fpk} converge uniformly to 0 on any compact subset K of D, and so T has the Bishop’s property (β).Proposition 3.31.
LetT be as in Proposition 3.30. Then if Tii has Bishop’s property (β) for i=1,…,k, then T has Bishop’s property (β).Proof.
The proof is identical to the proof of Proposition3.30.Proposition 3.32.
LetT∈ℒ(⊕i=1kH) be the following k×k triangular operator matrix:
(3.50)T=(T11T12T13……T1k0T22T23……T2k00T33……T3k000………0…………Tkk).
Assume that T is of class [2QN] and σ(T)∩(-σ(T))=∅. If T11Tij=TijTjj for i=1,…j and j=1,2,…,k, then Tjj has Bishop’s property (β) for j=1,2,…,k.Proof.
Letfpj:D→H be a sequence of analytic functions such that (Tjj)λfpj→0 uniformly on every compact subset K of D, then we have for j=1,2,…,k(3.51)Tλ(Tijfpj(λ)⊕⋯⊕0)=(T11)λ(Tijfpj(λ))⊕0⊕⋯⊕0=Tij(Tjj)λfpj(λ)⊕⋯⊕0⟶0,
for i=1,2,…,j. Since T has Bishop’s property (β), we get that Tijfpj(λ)→0 uniformly on K for j=1,2,…,k. We have
(3.52)Tλ(fp1⊕⋯⊕0)=(T11)λfp1(λ)⊕⋯⊕0⟶0,
and for j=2,3,…,k,
(3.53)Tλ(0⊕⋯⊕fpj⊕⋯⊕0)=(-T1jfpj(λ))⊕⋯⊕(-Tj-1jfpj(λ))⊕(Tjj)λ(fpj(λ))⊕0⊕⋯⊕0⟶0.
Since T has Bishop’s property (β),fpj(λ)→0 uniformly on K for j=1,2,…,k. Thus, Tjj has Bishop’s property (β).Proposition 3.33.
LetT be as in Proposition 3.31. Then if T has Bishop’s property (β) and T11Tij=TijTjj for j=1,2,…,k and j=1,2,…,k, then Tjj has Bishop's property (β) for j=1,2,…,k.Proof.
The proof is identical to the proof of Proposition3.32.


## 4. Berberian Extension

Denote był∞(H) the space of all sequences (xn)n, with xn∈H,n=1,2,3,… such that ∥xn∥ is bounded. Let c0(H) denote the subspace of all null sequences of H (those such that ∥xn∥→0). If we set ∥(xn)∥=sup∥xn∥ for every sequence (xn), this defines a seminorm on l∞(H), which is zero exactly on the elements of c0(H). By means of the space ł∞(H) and the Banach limits, Berberian [20] constructed an extension H∘ of H and obtained a homomorphism form operators T∈ℒ(H) to operators T∘∈ℒ(H∘) such that T∘ is an extension of T.Theorem 4.1 (Berberian extention [20]).
LetH be a complex Hilbert space. Then there exists a Hilbert space H∘⊃H and a map
(4.1)Φ:L(H)⟶L(H∘):T⟼T∘,
satisfying: Φ which is an *-isometric isomorphism preserving the order such that(1)
(T*)∘=(T∘)*,(2)
(λT+μS)∘=λT∘+μS0,(3)
(IH)∘=IH∘,(4)
(TS)∘=T∘S∘,(5)
∥T∘∥=∥T∥,(6)
T∘≤S∘ifT≤S,(7)
σ(T∘)=σ(T),σap(T)=σap(T∘)=σp(T∘),(8)
ifT is a positive operator, then (Tα)∘=|T∘|αforallα>0.An operator is said to be reducible if it has a nontrivial reducing subspace. If an operator is not reducible, then it is called irreducible.Proposition 4.2 (see [21]).
IfT is an irreducible operator, then T∘ is an irreducible operator.Lemma 4.3.
LetD be a subset of ℂ, z0,R>0, such that B(z0,R)¯={z∈ℂ;|z-z0|≤R}⊂D, let gn:D→H be a sequence of analytic functions, and let the Taylor expansion of gn be
(4.2)gn(z)=∑k=0∞ank(z-z0)k,|z-z0|<R.
If gn is uniformly bounded on B(z0,R)¯(i.e.,M=supn≥1∥gn(z)∥B(z0,R)¯<∞), then
(4.3)‖gn(z)-gn(z0)‖≤MrR-r,z∈B(z0,r)¯,0<r<R.Proof.
For alln and z∈B(z0,r)¯ with 0<r<R, by Cauchy’s integral formula, we get the following inequality:
(4.4)‖gn(z)-gn(z0)‖=‖12iπ∫|u-z0|=Rgn(u)u-zdu-12iπ∫|u-z0|=Rgn(u)u-z0du‖≤12π∫|u-z0|=R|z-z0|‖gn(u)‖|u-z∥u-z0||du|≤MrR-r.Remark 4.4.
LetD be an open subset of ℂ. A sequence of analytic functions gn:D→H converges uniformly to 0 on every compact subset K of D if and only if for any ϵ>0 and any z0∈D there exists r>0 and n0∈N such that B(z0;r)¯⊂D and ∥gn∥B(z0,r)¯<ϵ for all n>n0.


## 5. Single-valued Extension Property form-Partial Isometries

In this section we examine the properties of SVEP and Bishop’s property(β) for some m-partial isometries operators by using an approach which is different from that used in Section 3. We recall the definition of an m-partial isometry given by (1.4) and the operator Bm(T).Definition 5.1 (see [8]).
An operatorT∈ℒ(H) is called an m-isometry if
(5.1)Bm(T)=∑k=0m(-1)k(mk)T*m-kTm-k=0.Remark 5.2.
It is easy to see thatT∈ℒ(H) is an m-partial isometry if and only if
(5.2)Bm(T)x=∑k=0m(-1)k(mk)T*m-kTm-k(x)=0,∀x∈N(T)⊥,
which shows that the class of m-partial isometries generalizes those of m-isometries and partial isometries.Theorem 5.3 (see [10]).
IfT∈ℒ(H) is reducible, that is, if it has a nontrivial reducing subspace N(T), then the following properties are equivalent.(1)
T is an m-partial isometry.(2)
T|N(T)⊥ is an m-isometry.Proposition 5.4 (see [10]).
LetT be a reducible m-partial isometry. Then(1)
λ∈σap(T)∖{0} implies λ¯∈σap(T*), that is, if Tλxn→0 for some sequence of bounded vectors {xn}⊂H, then (Tλ)*xn→0,(2)
λ∈σp(T)∖{0} implies λ¯∈σp(T*),(3)
eigenvectors ofT corresponding to distinct eigenvalues are orthogonal, that is, N(Tλ)⊥N(Tμ) if λ,μ∈σp(T),λ≠μ.Lemma 5.5.
LetT be a reducible m-partial isometry and let λ,μ∈σap(T) and (xn), (yn) be sequences of bounded vectors in H such that λ≠μ and
(5.3)‖Tλxn‖⟶0,‖Tμyn‖⟶0(asn⟶∞).
Then we have
(5.4)〈xn∣yn〉⟶0(asn⟶∞).Proof.
We may assume thatμ≠0. Then from Proposition 5.4(1) we have ∥(Tμ)*yn∥→0 as n→∞. Hence,
(5.5)(λ-μ)〈xn∣yn〉=-〈Tλxn∣yn〉+〈xn∣(Tμ)*yn〉⟶0,n⟶∞,
which implies (5.4) in view of λ≠μ and the proof is complete.Theorem 5.6.
Any reduciblem-partial isometry has SVEP.Proof.
LetU be a bounded subset of ℂ and let f:U→H be an analytic function such that
(5.6)Tλf(λ)=0forλ∈U.
Since N(Tλ)⊥N(Tμ),λ≠μ (Proposition 5.4(3)), we have
(5.7)‖f(λ)‖2=lim⁡μ→λ〈f(λ)∣f(μ)〉=0.
This shows that f(λ)=0.Lemma 5.7.
IfT is an m-partial isometry, then To is also an m-partial isometry.Proof.
It is a consequence of the properties ofTo (see Theorem 4.1).Theorem 5.8.
IfT is an m-partial isometry with a nontrivial reducing space N(T), then To has the single-valued extension property (SVEP).Proof.
To prove thatT∘ has SVEP, let λ∈σap(T∘)-{0}. Since σap(T∘)=σap(T) by Theorem 4.1, λ¯∈σap(T*)=σap((T∘)*). In particular if λ∈σp(T∘), then λ¯∈σp((T∘)*). Hence, N((Tλ)∘)⊥N((Tμ)∘) for λ and μ∈σp(T∘) with λ≠μ. In a similar way as in the proof of Theorem 5.6, we can see that T∘ has SVEP.



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*Source: 975745-2012-06-03.xml*