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

Abstract

We prove some further properties of the operator 𝑇∈[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.

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.

Let 𝐻 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),𝜎_𝑝(𝑇),𝜎_𝑎𝑝(𝑇), respectively. 𝑇^∗ means the adjoint of 𝑇.

An operator 𝑇∈ℒ(𝐻) is(1)hyponormal if and only if 𝑇𝑇^∗≤𝑇^∗𝑇⇔‖𝑇^∗𝑥‖≤‖𝑇𝑥‖,forall𝑥∈𝐻,(2)posinormal if and only if 𝑅(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 if 𝑇^∗2𝑇²−2𝑇^∗𝑇+𝐼=0 (see [8]), where 𝐼 indicates the identity operator that is, if and only if𝑇^∗2𝑇²=2𝑇^∗𝑇−𝐼.(1.1) An operator 𝑇∈ℒ(𝐻) is said to be 𝑛-power quasinormal (abbreviated as 𝑛QN), 𝑛=1,2,…, if𝑇^𝑛𝑇^∗𝑇=𝑇^∗𝑇𝑇^𝑛=𝑇^∗𝑇^𝑛+1.(1.2) If 𝑛=1, 𝑇 is called quasinormal. This class of operators being denoted by [𝑛QN], that is,[𝑛QN]∶=𝑇∈ℒ(𝐻)∶𝑇^𝑛𝑇^∗𝑇−𝑇^∗𝑇𝑇^𝑛=0(1.3) was studied by the author [9].

𝑇 is called an 𝑚-partial isometry if 𝑇 satisfies𝑇𝐵_𝑚(𝑇)=𝑇^𝑚_𝑘=0⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠(−1)_𝑘𝑇^{∗𝑚−𝑘}𝑇^𝑚−𝑘=0,(1.4) where 𝐵_𝑚(𝑇) is obtained formally from the binomial expansion of 𝐵_𝑚(𝑇)=(𝑇^∗𝑇−𝐼)^𝑚 by understanding (𝑇^∗𝑇)^𝑚−𝑘=𝑇^{∗𝑚−𝑘}𝑇^𝑚−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. Section 2 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 𝑊^𝑚2(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.

Let 𝜆 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‖𝑓‖_2,𝐷=_𝐷‖𝑓(𝜆)‖²𝑑𝜇(𝜆)^1/2<∞.(2.1)

Let 𝑂(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. Let𝐴²(𝐷,H)=𝐿²(𝐷,𝐻)∩𝑂(𝐷,𝐻)(2.2) denote the Bergman space for 𝐷 consisting of square measurable functions 𝑓 that are analytic on 𝐷.

We denote by 𝑃 the orthogonal projection of 𝐿²(𝐷,𝐻) onto 𝐴²(𝐷,𝐻).

Let us define now a special Sobolev type space. Let 𝑚 be a fixed nonnegative integer. The Sobolev space 𝑊^𝑚2(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,𝑊^𝑚2(𝐷,𝐻)=𝑓∈𝐿²(𝐷,𝐻)∶𝜕^𝑘𝑓∈𝐿²(𝐷,𝐻),for𝑘=0,1,…,𝑚.(2.3)

Endowed with the norm‖𝑓‖^2𝑊_𝑚2=^𝑚_𝑘=0‖‖‖𝜕^𝑘𝑓‖‖‖^22,𝐷.(2.4)𝑊^𝑚2(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‖_𝑊^𝑚2_(𝐷,𝐻)forall𝑓∈𝑊^𝑚2(𝐷,𝐻).

Let𝑇_𝜆=𝑇−𝜆=𝑇−𝜆𝐼(2.5) 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𝑇_𝜆𝑓(𝜆)=(𝑇−𝜆)𝑓(𝜆)=0(2.6) 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 operator 𝑇∈ℒ(𝐻) has SVEP if and only if, for each open 𝐷⊆ℂ.

The operator 𝑇_𝐷∶𝑂(𝐷,𝐻)→𝑂(𝐷,𝐻) defined by𝑇_𝐷(𝑓)(𝜆)=𝑇_𝜆𝑓(𝜆)∀𝑓∈𝑂(𝐷,𝐻),𝜆∈𝐷.(2.7) is one to one.

Recall that, for a bounded operator 𝑇 on 𝐻, the local resolvent set 𝜌_𝑇(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𝑇_𝜆𝑓(𝜆)=𝑥∀𝜆∈𝐷.(2.8) The local spectrum 𝜎_𝑇(𝑥) of 𝑇 at 𝑥∈𝐻 is the set defined by 𝜎_𝑇(𝑥)∶=ℂ⧵𝜌_𝑇(𝑥) and obviously 𝜎_𝑇(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 𝜎_𝑇(𝑥)=∅ (see for more details [11]).

Recall that a bounded operator 𝑇∈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 𝑓_𝑛∶𝐷→𝐻 with the property that𝑇_𝜆𝑓_𝑛(𝜆)⟶0as𝑛⟶∞,(2.9) uniformly on all compact subsets of 𝐷, 𝑓_𝑛(𝜆)→0 as 𝑛→∞ locally uniformly on 𝐷 or equivalently, for every open subset 𝐷 of ℂ, the operator 𝑇_𝐷 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 operator 𝑇 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,Φ∶𝐶^𝑚0(ℂ)⟶ℒ(𝐻),(2.10) such that Φ(𝑧)=𝑇, where 𝑧 stands for the identity function on ℂ and 𝐶^𝑚0(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.

Let 𝑀_𝑧 be the operator on 𝑊^𝑚2(𝐷;𝐻) such that (𝑀_𝑧𝑓)(𝑧)=𝑧𝑓(𝑧) for 𝑓∈𝑊^𝑚2(𝐷;𝐻). This has a spectral distribution of order 𝑚, defined by the functional calculus Φ_𝑀∶𝐶^𝑚0(ℂ)→ℒ(𝑊^𝑚2(𝐷,𝐻));Φ_𝑀(𝑓)=𝑀_𝑓. Therefore 𝑀_𝑧 is a scalar operator of order 𝑚. Consider a bounded open disk 𝐷 which contains 𝜎(𝑇) and the quotient space𝐻(𝐷)=𝑊^𝑚2(𝐷,𝐻)𝑇_𝑧𝑊^𝑚2(𝐷,𝐻)(2.11) endowed with the Hilbert space norm. We denote the class containing a vector 𝑓 or an operator A on 𝐻(𝐷) by 𝑓 or 𝐴, respectively. Let 𝑀_𝑧 be the operator of multiplication by 𝑧 on 𝑊^𝑚2(𝐷;𝐻). As noted above, 𝑀_𝑧 is a scalar of order 𝑚 and has a spectral distribution Φ. Let 𝑆≡𝑀_𝑧. Since 𝑇_𝑧𝑊^𝑚2(𝐷,𝐻) is invariant under every operator 𝑀_𝑓; 𝑓∈𝐶^𝑚0(ℂ), we infer that 𝑆 is a scalar operator of order 𝑚 with spectral distribution Φ. 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 of 𝑛-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)If 𝑇 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) Let 𝑆∈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 𝑇^𝑛=𝑉^∗𝑆^𝑛𝑉 and inserting 𝐼=𝑉𝑉^∗ suitably, we deduce from (1.2) that 𝑉^∗𝑆^𝑛𝑆^∗𝑆𝑉=𝑇^𝑛𝑇^∗𝑇=𝑇^∗𝑇^𝑛+1=𝑉^∗𝑆^∗𝑆^𝑛+1𝑉,(3.1) and (1.2) follows for 𝑆. Since (𝑇∣𝑀)^Δ=𝑇^Δ∣𝑀(3.2) for Δ as the 𝑛-th power or the adjoint, it follows that the left-hand side of (1.2) for (𝑇∣𝑀) reads 𝑇^𝑛𝑇^∗𝑇∣𝑀,(3.3) which is (𝑇^∗𝑇^𝑛+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. For 𝑛≥2 one has 𝑇=⎛⎜⎜⎝𝑥0𝑧𝑦⎞⎟⎟⎠∈[𝑛QN]⟺⎧⎪⎨⎪⎩𝑦𝑥𝑥𝑥^𝑛−1+𝑧𝑥^𝑛−2+⋯+𝑧^𝑛−1=0,𝑦(𝑧^𝑛−𝑥^𝑛)=0,𝑥𝑦(𝑧^𝑛−𝑥^𝑛)−𝑥^𝑛−1+𝑧𝑥^𝑛−2+⋯+𝑧^𝑛−1||𝑦||²+|𝑧|²−|𝑥|²=0.(3.4)

We remark here that Proposition 3.2 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]). If 𝑇 and 𝑇−𝐼 are of class [2QN], then 𝑇 is normal.

Theorem 3.4 (see [9]). If 𝑇 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. 𝑇_𝜆 is of class [nQN] for every 𝜆∈ℂ if and only if 𝑇 is a normal operator.

Proof. Assume that (𝑇_𝜆) 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 (𝑇−𝜆)^𝑛𝑇^∗𝑇−𝜆𝑇^∗=𝑇^∗𝑇−𝜆𝑇^∗(𝑇−𝜆)^𝑛.(3.5) By the binomial expansion, ^𝑛_𝑘=0(−1)^𝑘⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝜆_𝑘𝑇^𝑛−𝑘𝑇^∗𝑇−𝜆𝑇^∗=𝑇^∗𝑇−𝜆T^∗^𝑛_𝑘=0(−1)^𝑘⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝜆_𝑘𝑇^𝑛−𝑘,(3.6) whence by arranging terms suitably, the extremal terms vanishing in view of (1.2), ^𝑛−1_𝑘=1(−1)^𝑘⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝜆_𝑘𝑇^𝑛−𝑘𝑇^∗𝑇−𝑇^∗𝑇𝑇^𝑛−𝑘−^𝑛−1_𝑘=1(−1)^𝑘⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝜆_𝑘+1𝑇^𝑛−k𝑇^∗−𝑇^∗𝑇^𝑛−𝑘=0.(3.7)
Now note that from the second summand in (3.7), we may extract the extremal term (−1)^𝑛𝑛𝜆^𝑛(𝑇^∗𝑇−𝑇𝑇^∗) and express it in terms of the remaining terms which contain 𝜆 to the power <𝑛. Hence dividing (3.7) by 𝜆^𝑛 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, if 𝑇 is normal, then (𝑇_𝜆) is normal for every 𝜆∈ℂ, and hence (𝑇_𝜆) is of class [𝑛QN].

The following proposition gives a characterization of an 𝑛-power quasinormal operator.

Proposition 3.6. Let 𝑇∈ℒ(𝐻), 𝐴=𝑇^𝑛+𝑇^∗𝑇, and 𝐵=𝑇^𝑛−𝑇^∗𝑇. Then 𝑇 is of class [𝑛QN] if and only if 𝐴 commutes with 𝐵.

Proof. Commutativity of 𝐴 and 𝐵 is equivalent to (1.2).

Proposition 3.7. Let 𝑇,𝐴,𝐵 be as in Proposition 3.6. Then if 𝑇 is of class [𝑛QN], then 𝑇^𝑛𝑇^∗𝑇 commutes with 𝐴 and 𝐵.

Proof. By (1.2), 𝑇^𝑛𝑇^∗𝑇𝑇^𝑛±𝑇^∗𝑇=𝑇^𝑛±𝑇^∗𝑇𝑇^𝑛𝑇^∗𝑇.(3.8)
In general, the two classes [𝑛QN] and [(𝑛+1)QN] are not the same (see [9]).

Proposition 3.8. If 𝑇 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), 𝑇^𝑛+1𝑇^∗𝑇=𝑇^𝑛𝑇^∗𝑇²(3.9) so that 𝑇^𝑛+2𝑇^∗𝑇 may be transformed into 𝑇^∗𝑇𝑇^𝑛+2.
It is known that if 𝑇 belongs to [𝑛QN] some 𝑛>0, 𝑇² does not necessarily belong to the same class.

Theorem 3.9 (see [9]). If 𝑇 and 𝑇^∗ are of class [𝑛QN], then 𝑇^𝑛 is normal.

Proposition 3.10. If an operator 𝑇 of class [2QN] is a 2-isometry, then 𝑇² is of class [𝑛QN] for all integers 𝑛≥2.

Proof. From Proposition 3.8 it suffices to prove that 𝑇² is of class [2QN] and of class [3QN] because we may then proceed inductively.
Since 𝑇 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 that 𝑇⁶(𝑇^∗2𝑇²)=𝑇^∗2𝑇⁸ whence 𝑇² is of class [3QN].

Lemma 3.11 (see [9]). If 𝑇 is of class [𝑛QN], then 𝑁(𝑇^𝑛)⊂𝑁(𝑇^∗𝑛).

Proposition 3.12. If 𝑇 is both of class [𝑛QN] and [(𝑛+1)QN] such that 𝑇 is injective or 𝑇^∗ is injective, then 𝑇 is quasinormal.

Proof. Since 𝑇 is of class [𝑛QN]∩[(𝑛+1)QN], we have (3.9), which reads 𝑇^𝑛(𝑇𝑇^∗𝑇−𝑇^∗𝑇²)=0. If 𝑇 is injective, then so is 𝑇^𝑛 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 Theorem 3.1, (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 (𝑇_𝑝) in [𝑛QN] also belongs to [𝑛QN]; that is, we let (𝑇_𝑝) be a sequence of operators in [𝑛QN] converging to 𝑇∈ℒ(𝐻) in norm: ‖‖𝑇_𝑝𝑥−𝑇𝑥‖‖⟶0as𝑝⟶∞,foreach𝑥∈𝐻.(3.10) Hence it follows that ‖‖𝑇^∗𝑝𝑥−𝑇^∗𝑥‖‖=‖‖𝑇_𝑝−𝑇^∗𝑥‖‖≤‖‖𝑇_𝑝−𝑇^∗‖‖‖𝑥‖=‖‖𝑇_𝑝−𝑇‖‖‖𝑥‖⟶0,(3.11) whence (𝑇^∗𝑝) converges strongly to 𝑇^∗.
Since the product of operators is sequentially continuous in the strong topology (see [12, page 62]), one concludes that𝑇^𝑛𝑝𝑇^∗𝑝𝑇_𝑝 converge strongly to 𝑇^𝑛𝑇^∗𝑇. Similarly 𝑇^∗𝑝𝑇^𝑝𝑛+1 converges strongly to 𝑇^∗𝑇^𝑛+1. Hence the limiting case of (1.2) shows that 𝑇 belongs to [𝑛QN], completing the proof.

Proposition 3.14. If 𝑇₁,𝑇₂,…,𝑇_𝑝 are of class [𝑛QN], then both the direct sum 𝑇₁⊕𝑇₂⊕⋯⊕𝑇_𝑝 and the tensor product 𝑇₁⊗𝑇₂⊗⋯⊗𝑇_𝑝 are of class [𝑛QN].

Proof. By the compatibility principle similar to (3.2), 𝑇₁⊕𝑇₂^Δ=𝑇^Δ1⊕𝑇^Δ2,𝑇₁⊗𝑇₂^Δ=𝑇^Δ1⊗𝑇^Δ2,(3.12) where Δ indicates either the 𝑛th power or adjoint, the proof follows.

A linear operator 𝑇 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 operator 𝑇 on 𝐻 is supercyclic if there exists a vector whose scaled orbit is dense; that is, if there exists an 𝑥∈𝐻 such that {𝜆𝑇^𝑛𝑥,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. If 𝑇 is of class [nQN] with 𝜎_𝑝(𝑇^𝑛)≠∅, then 𝑇 is not hypercyclic.

Proof. If 𝑇 is hypercyclic, 𝑇^𝑛 is hypercyclic, and hence 𝜎_𝑝(𝑇^∗𝑛)=∅ by [13, corollary 2.4]. From Lemma 3.11 we have 𝜎_𝑝(𝑇^𝑛)⊂𝜎_𝑝(𝑇^∗𝑛) and, hence, 𝜎_𝑝(𝑇^𝑛)=∅, a contradiction.

Theorem 3.16 (see [14, Theorem 2]). If 𝑇 is not hypercyclic, then 𝑇 and 𝑇⁻¹ have a common non-trivial invariant closed subset.

Proposition 3.17. If 𝑇 is of class [𝑛QN] and 0∉𝜎(𝑇), then 𝑇 and 𝑇⁻¹ have a common nontrivial closed invariant subset.

Proof . Since 𝑇 is of class [𝑛QN] and 0∉𝜎(𝑇), it follows that 𝑇^𝑛 is normal, and hence (𝑇⁻¹)^𝑛 is normal. By [13, Corollary 4.5] 𝑇^𝑛 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. Operators 𝑇 that are of class [𝑛QN] such that 𝑇 is not invertible are not supercyclic.

Proof. Assume that 𝑇 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 operator 𝑇∈ℒ(𝐻) is algebraic if there is non-zero-polynomial 𝑝 such that 𝑝(𝑇)=0.

The following proposition shows that some quasinilpotent 𝑛-power quasi-normal operators are subscalar.

Proposition 3.20. If both 𝑇 and 𝑇^∗ are of class [𝑛QN] such that 𝑇 is quasinilpotent, then 𝑇 is nilpotent, and hence 𝑇 is subscalar.

Proof. Since 𝑇 is quasinilpotent, 𝜎(𝑇)={0}. Hence by the spectral mapping theorem we get 𝜎(𝑇^𝑛)=𝜎(𝑇)^𝑛={0}. Thus 𝑇^𝑛 is quasinilpotent and normal. So 𝑇^𝑛=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 disk 𝐷 in ℂ, there is a constant 𝐶_𝐷 such that for an arbitrary operator 𝑇∈ℒ(𝐻) and 𝑓∈𝑊^𝑚2(𝐷,𝐻) we have ‖‖(𝐼−𝑃)𝜕^𝑗𝑓‖‖_2,𝐷≤𝐶_𝐷‖‖‖(𝑇^𝜆)^∗𝜕^𝑗+1𝑓‖‖‖_2,𝐷+‖‖‖(𝑇_𝜆)^∗𝜕^𝑗+2𝑓‖‖‖_2,𝐷,𝑗=0,1,…,𝑚−2,(3.13) 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, Theorem 3.27.

Theorem 3.22. Let 𝐷 be an arbitrary bounded disk in ℂ. If 𝑇 is of class [2QN] and 𝜎(𝑇)∩(−𝜎(𝑇))=∅, then the operator 𝑇_𝜆∶𝑊^𝑚2(𝐷,𝐻)⟶𝑊^𝑚2(𝐷,𝐻)(3.14) is one to one.

Proof. Let 𝑔∈𝑊^𝑚2(𝐷,𝐻) such that 𝑇_𝜆𝑔=0, that is, ‖‖𝑇_𝜆𝑔‖‖_𝑊𝑚2=0.(3.15) Then for 𝑗=0,1,…,𝑚 we have ‖‖𝑇_𝜆𝜕^𝑗𝑔‖‖_2,𝐷=0.(3.16) Hence for 𝑗=1,…,𝑚 we get ‖𝑇^2𝜆₂𝜕^𝑗𝑔‖_2,𝐷=0. Since 𝜎(𝑇)∩(−𝜎(𝑇)) is empty then 𝑇² is normal [9, Theorem 2.2]. Hence, ‖‖(𝑇^2𝜆₂)^∗𝜕^𝑗𝑔‖‖_2,𝐷=0.(3.17) Now we claim that ‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑔‖‖_2,𝐷=0.(3.18) Indeed, since 𝑇_𝜆 is invertible for 𝜆∈𝐷⧵𝜎(𝑇), (3.16) implies that ‖‖𝜕^𝑗𝑔‖‖_{2,𝐷⧵𝜎(𝑇)}=0.(3.19) Therefore ‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑔‖‖_{2,𝐷⧵𝜎(𝑇)}=0.(3.20)
Since 𝜎(𝑇)∩(−𝜎(𝑇))=∅ and 𝜎(𝑇^∗)=𝜎(𝑇)^∗,(𝑇_−𝜆)^∗ is invertible for 𝜆∈𝜎(𝑇), therefore; from (3.17) we have ‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑔‖‖_2,𝜎(𝑇)=0.(3.21) It is clear form (3.20); and (3.21) that ‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑔‖‖_2,𝐷=0,for𝑗=0,1,…,𝑚.(3.22) Thus Proposition 3.21 and (3.21) imply ‖‖(𝐼−𝑃)𝜕^𝑗𝑔‖‖_2,𝐷=0for𝑗=0,1,…,𝑚−2,(3.23) where 𝑃 denotes the orthogonal projection of 𝐿²(𝐷,𝐻) onto 𝐴²(𝐷,𝐻).
Hence (𝑇^2𝜆₂)𝑃𝑔=𝑇^2𝜆₂𝑔=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, 𝑇_𝜆 is one to one.

Corollary 3.23. If 𝑇₁ and 𝑇₂ are of class [2QN] with 𝜎(𝑇_𝑖)∩(−𝜎(𝑇_𝑖))=∅, for 𝑖=1,2 and 𝑇₂𝑇₁=0. Then 𝑇₁+𝑇₂_𝜆∶𝑊^𝑚2(𝐷,𝐻)⟶𝑊^𝑚2(𝐷,𝐻)(3.24) is one to one.

Proof. If 𝑓∈𝑊^𝑚2(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 operator 𝑇∈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 𝑁^𝑚=0. If 𝜎(𝑆)∩(−𝜎(𝑆))=∅, then 𝑇_𝜆 is one-to-one.

Proof. If 𝑔∈𝑊^𝑚2(𝐷,𝐻) is such that 𝑇_𝜆𝑔=0, then 𝑆_𝜆𝑔=−𝑁𝑔.(3.25) Hence 𝑆_𝜆𝑁^𝑗−1𝑔=−𝑁^𝑗𝑔 for 𝑗=1,2,…,𝑚. We prove that 𝑁^𝑗𝑔=0 for 𝑗=0,1,…,𝑚−1 by indication. Since 𝑁^𝑚=0, 𝑆_𝜆𝑁^𝑚−1𝑔=−𝑁^𝑚𝑔=0. Since 𝑆_𝜆 is one-to-one by Theorem 3.22𝑁^𝑚−1𝑔=0. Assume it is true when 𝑗=𝑘, that is, 𝑁^𝑘𝑔=0. From (3.25), we get 𝑆_𝜆𝑁^𝑘−1𝑔=−𝑁^𝑘𝑔=0.(3.26) Since 𝑆_𝜆 is one-to-one from Theorem 3.22, 𝑁^𝑘−1𝑔=0. By indication we have 𝑔=0. Hence 𝑇_𝜆 is one-to-one.

An operator 𝑇∈ℒ(𝐻) is said to be the following.(1)It is left invertible if there is an operator 𝑆∈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. If 𝑇 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∶𝑊^𝑚2(𝐷,𝐻)→𝑊^𝑚2(𝐷,𝐻) is one-to-one.

Proof. If 𝑔∈𝑊^𝑚2(𝐷,𝐻) is such that (STR)_𝜆𝑔=0, then 𝑇_𝜆𝑅𝑔=0.(3.27) Hence for 𝑗=0,1,…,𝑚 we have 𝑇_𝜆𝑅𝜕^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 ‖(𝐼−𝑃)𝑔‖_2,𝐷=0,(3.28) 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 𝑐‖𝑃𝑔‖_2,𝐷≤‖𝜆𝑃𝑔‖_2,𝐷.(3.29) So 𝑔=𝑃𝑔=0. Thus, (STR)_𝜆 is one-to-one.

Corollary 3.26. If 𝑇 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∶𝑊^𝑚2(𝐷,𝐻)→𝑊^𝑚2(𝐷,𝐻), is one-to-one.

Now we are ready to prove our main theorem.

Theorem 3.27. If 𝑇 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 𝑉𝑔=1⊗𝑔+𝑇_𝜆𝑊^𝑚2(𝐷,𝐻)=[1⊗𝑔],(3.30) is one to one and has closed range, where 𝐻(𝐷) is as in (2.11).

Proof. First, we will prove that if {𝑔_𝑘}^∞1⊂𝐻 and {𝑓_𝑘}^∞1⊂𝑊^𝑚2(𝐷,𝐻) are sequences such that lim_𝑘→∞‖‖1⊗𝑔_𝑘+𝑇_𝜆𝑓_𝑘‖‖_𝑊𝑚2=0,(3.31) then lim_𝑘→∞𝑔_𝑘=0.
By the definition of the norm of a Sobolev space, (3.31) implies that lim_𝑘→∞‖‖𝑇_𝜆𝜕^𝑗𝑓_𝑘‖‖_2,𝐷=0for𝑗=1,…,𝑚.(3.32) From (3.32) we get lim_𝑘→∞‖‖𝑇^2𝜆₂𝜕^𝑗𝑓_𝑘‖‖_2,𝐷=0for𝑗=1,…,𝑚.(3.33)
Since 𝑇² is normal, lim_𝑘→∞‖‖(𝑇^2𝜆₂)^∗𝜕^𝑗𝑓_𝑘‖‖_2,𝐷=0for𝑗=1,…,𝑚.(3.34) Since 𝑇_𝜆 is invertible for 𝜆∈𝐷⧵𝜎(𝑇), (3.32) implies that lim_𝑘→∞‖‖𝜕^𝑗𝑓_𝑘‖‖_{2,𝐷⧵𝜎(𝑇)}=0.(3.35) Therefore lim_𝑘→∞‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑓_𝑘‖‖_{2,𝐷⧵𝜎(𝑇)}=0for𝑗=1,…,𝑚.(3.36)
Since 𝜎(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 lim_𝑘→∞‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑓_𝑘‖‖_2,𝜎(𝑇)=0.(3.37) Hence, from (3.36) and (3.37) we get lim_𝑘→∞‖‖(𝑇_𝜆)^∗𝜕^𝑗𝑓_𝑘‖‖_2,𝐷=0,𝑗=1,…,𝑚.(3.38) Then by Proposition 3.21, we have lim_𝑘→∞‖‖(𝐼−𝑃)𝜕^𝑗𝑓_𝑘‖‖_2,𝐷=0,𝑗=1,…,𝑚−2.(3.39) By (3.31) and (3.39), we have lim_𝑘→∞‖‖1⊗𝑔_𝑘+𝑇_𝜆𝑃𝑓_𝑘‖‖_2,𝐷=0.(3.40) Let Γ be a curve in 𝐷 surrounding 𝜎(𝑇). Then lim_𝑘→∞‖‖𝑃𝑓_𝑘+𝑇_𝜆⁻¹1⊗𝑔_𝑘‖‖=0(3.41) uniformly for 𝜆∈Γ by (3.40). Hence by Riesz-Dunford functional calculus lim_𝑘→∞‖‖‖12𝜋𝑖_Γ𝑃𝑓_𝑘(𝑧)𝑑𝑧+𝑔_𝑘‖‖‖=0.(3.42) But by Cauchy’s theorem _Γ𝑃𝑓_𝑘(𝑧)𝑑𝑧=0.(3.43) Hence lim_𝑘→∞𝑔_𝑘=0. Thus the map 𝑉 is one-to-one and has closed range.

Corollary 3.28. If 𝑇 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 disk 𝐷 in ℂ that contains 𝜎(𝑇) and the quotient space given in (2.11).
Let 𝑀_𝜆 be the multiplication operator by 𝜆 on 𝑊^𝑚2(𝐷,𝐻). Then 𝑀_𝜆 is a scalar operator of order 𝑚, and its spectral distribution is Φ_𝑀∶𝐶^𝑚0(ℂ)⟶ℒ𝑊^𝑚2(𝐷;𝐻),Φ_𝑀(𝑓)=𝑀_𝑓,(3.44) where 𝑀_𝑓 is the multiplication operator by 𝑓∈𝐶^𝑚0(ℂ). Let 𝑆≡𝑀_𝜆. Since 𝑇_𝜆𝑊^𝑚2(𝐷;𝐻) is invariant under every operator 𝑀_𝑓, we infer that 𝑆 is a scalar operator of order 𝑚 with spectral distribution Φ.
Let 𝑉 be the operator 𝑉𝑔=1⊗𝑔+𝑇_𝜆𝑊^𝑚2(𝐷,𝐻)(3.45) 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. If 𝑇 is of class [2QN] with the property that 𝜎(𝑇)∩(−𝜎(𝑇))=∅, then 𝑇 has Bishop’s property (𝛽).

Proof. It follows from Corollary 3.28 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. Let 𝑇∈ℒ(⊕^𝑘𝑖=1𝐻) be the following 𝑘×𝑘 triangular operator matrix 𝑇=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑇¹¹𝑇₁₂𝑇₁₃……𝑇_1𝑘0𝑇₂₂𝑇₂₃……𝑇_2𝑘00𝑇₃₃……𝑇_3𝑘00…0…0…………𝑇…_𝑘𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.47) Assume that 𝑇_𝑖𝑖 is of class [2QN] and satisfies 𝜎(𝑇_𝑖𝑖)∩(−𝜎(𝑇_𝑖𝑖))=∅ for 𝑖=1,2,…,𝑘−1 and 𝑇_𝑘𝑘 is nilpotent. Then 𝑇 has Bishop’s property (𝛽).

Proof. Let 𝑓_𝑝=⊕^𝑘𝑖=1𝑓^𝑖𝑝∶𝐷→⊕^𝑘𝑖=1𝐻 be a sequence of analytic functions such that 𝑇_𝜆𝑓_𝑝(𝜆)→0 uniformly on every compact subset 𝐾 of an open set 𝐷 of ℂ; then we have 𝑇₁₁_𝜆𝑓𝑇^1𝑝(𝜆)+𝑇₁₂𝑓^2𝑝(𝜆)+𝑇₁₃𝑓^3𝑝(𝜆)⋯+𝑇_1𝑘𝑓^𝑘𝑝(𝜆)⟶0,₂₂𝑇_𝜆𝑓^2𝑝(𝜆)+𝑇₂₃𝑓^3𝑝(𝜆)+⋯+𝑇_2𝑘𝑓^𝑘𝑝⟶0,₃₃𝑇_𝜆𝑓^3𝑝(𝜆)+⋮⋮⋯+𝑇⋮⋮_{3𝑘𝑘−1𝑘−1}𝑓^𝑘𝑝(𝜆)⟶0,_𝜆𝑓𝑇^𝑝𝑘−1(𝜆)+𝑇_𝑘−1𝑘𝑓^𝑘𝑝(𝜆)⟶0,_𝑘𝑘_𝜆𝑓^𝑘𝑝(𝜆)⟶0.(3.48) Since 𝑇^𝑚𝑘𝑘=0,𝜆𝑇^{𝑚−1𝑘𝑘}𝑓^𝑘𝑝(𝜆)→0 and hence 𝑇^{𝑚−1𝑘𝑘}𝑓^𝑘𝑝(𝜆)→0 if 𝜆≠0. Since (𝑇_𝑘𝑘)_𝜆𝑓^𝑘𝑝(𝜆)→0 from (3.48) 𝜆𝑇^{𝑚−2𝑘𝑘}𝑓^𝑘𝑝(𝜆)→0. By the same reason, 𝑇^{𝑚−3𝑘𝑘}𝑓^𝑘𝑝(𝜆)→0. By repeating this procedure, we finally achieve 𝑓^𝑘𝑝(𝜆)⟶0,(3.49) uniformly on 𝐾. Then we obtain the following equation: (𝑇_{𝑘−1𝑘−1})_𝜆𝑓^𝑝𝑘−1→0 uniformly on every compact 𝐾. Since 𝑇_{𝑘−1𝑘−1} has Bishop’s property (𝛽) from Corollary 3.29, 𝑓^𝑝𝑘−1(𝜆)→0 uniformly on 𝐾. By repeating this process we prove that 𝑓^1𝑝(𝜆)→0 uniformly on 𝐾.
Hence {𝑓_𝑝=𝑓^1𝑝⊕𝑓^2𝑝⊕⋯⊕𝑓^𝑘𝑝} converge uniformly to 0 on any compact subset 𝐾 of 𝐷, and so 𝑇 has the Bishop’s property (𝛽).

Proposition 3.31. Let 𝑇 be as in Proposition 3.30. Then if 𝑇_𝑖𝑖 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 Proposition 3.30.

Proposition 3.32. Let 𝑇∈ℒ(⊕^𝑘𝑖=1𝐻) be the following 𝑘×𝑘 triangular operator matrix: 𝑇=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑇¹¹𝑇₁₂𝑇₁₃……𝑇_1𝑘0𝑇₂₂𝑇₂₃……𝑇_2𝑘00𝑇₃₃……𝑇_3𝑘00…0…0…………𝑇…_𝑘𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.50) Assume that 𝑇 is of class [2QN] and 𝜎(𝑇)∩(−𝜎(𝑇))=∅. If 𝑇₁₁𝑇_𝑖𝑗=𝑇_𝑖𝑗𝑇_𝑗𝑗 for 𝑖=1,…𝑗 and 𝑗=1,2,…,𝑘, then 𝑇_𝑗𝑗 has Bishop’s property (𝛽) for 𝑗=1,2,…,𝑘.

Proof. Let 𝑓^𝑗𝑝∶𝐷→𝐻 be a sequence of analytic functions such that (𝑇_𝑗𝑗)_𝜆𝑓^𝑗𝑝→0 uniformly on every compact subset 𝐾 of 𝐷, then we have for 𝑗=1,2,…,𝑘𝑇_𝜆𝑇_𝑖𝑗𝑓^𝑗𝑝(𝜆)⊕⋯⊕0==𝑇𝑇_11𝑖𝑗_𝜆𝑇_𝑖𝑗𝑓^𝑗𝑝(𝜆)⊕0⊕⋯⊕0⟶0,𝑇_𝑗𝑗_𝜆𝑓^𝑗𝑝(𝜆)⊕⋯⊕0(3.51) for 𝑖=1,2,…,𝑗. Since 𝑇 has Bishop’s property (𝛽), we get that 𝑇_𝑖𝑗𝑓^𝑗𝑝(𝜆)→0 uniformly on 𝐾 for 𝑗=1,2,…,𝑘. We have 𝑇_𝜆𝑓^1𝑝⊕⋯⊕0=𝑇₁₁_𝜆𝑓^1𝑝(𝜆)⊕⋯⊕0⟶0,(3.52) and for 𝑗=2,3,…,𝑘, 𝑇_𝜆0⊕⋯⊕𝑓^𝑗𝑝⊕⋯⊕0=⟶−0𝑇._1𝑗𝑓^𝑗𝑝(𝜆)⊕⋯⊕−𝑇_𝑗−1𝑗𝑓^𝑗𝑝(𝜆)⊕𝑇_𝑗𝑗_𝜆𝑓^𝑗𝑝(𝜆)⊕0⊕⋯⊕0(3.53) Since 𝑇 has Bishop’s property (𝛽),𝑓^𝑗𝑝(𝜆)→0 uniformly on 𝐾 for 𝑗=1,2,…,𝑘. Thus, 𝑇_𝑗𝑗 has Bishop’s property (𝛽).

Proposition 3.33. Let 𝑇 be as in Proposition 3.31. Then if 𝑇 has Bishop’s property (𝛽) and 𝑇₁₁𝑇_𝑖𝑗=𝑇_𝑖𝑗𝑇_𝑗𝑗 for 𝑗=1,2,…,𝑘 and 𝑗=1,2,…,𝑘, then 𝑇_𝑗𝑗 has Bishop's property (𝛽) for 𝑗=1,2,…,𝑘.

Proof. The proof is identical to the proof of Proposition 3.32.

4. Berberian Extension

Denote by ^∞(𝐻) the space of all sequences (𝑥_𝑛)_𝑛, with 𝑥_𝑛∈𝐻,𝑛=1,2,3,… such that ‖𝑥_𝑛‖ is bounded. Let 𝑐₀(𝐻) denote the subspace of all null sequences of 𝐻 (those such that ‖𝑥_𝑛‖→0). If we set ‖(𝑥_𝑛)‖=sup‖𝑥_𝑛‖ for every sequence (𝑥_𝑛), this defines a seminorm on 𝑙^∞(𝐻), which is zero exactly on the elements of 𝑐₀(𝐻). By means of the space ^∞(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]). Let 𝐻 be a complex Hilbert space. Then there exists a Hilbert space 𝐻^∘⊃𝐻 and a map Φ∶ℒ(𝐻)⟶ℒ(𝐻^∘)∶𝑇⟼𝑇^∘,(4.1) satisfying: Φ which is an *-isometric isomorphism preserving the order such that(1)(𝑇^∗)^∘=(𝑇^∘)^∗, (2)(𝜆𝑇+𝜇𝑆)^∘=𝜆𝑇^∘+𝜇𝑆⁰, (3)(𝐼_𝐻)^∘=𝐼_𝐻^∘, (4)(𝑇𝑆)^∘=𝑇^∘𝑆^∘, (5)‖𝑇^∘‖=‖𝑇‖, (6)𝑇^∘≤𝑆^∘𝑖𝑓𝑇≤𝑆, (7)𝜎(𝑇^∘)=𝜎(𝑇),𝜎_𝑎𝑝(𝑇)=𝜎_𝑎𝑝(𝑇^∘)=𝜎_𝑝(𝑇^∘), (8)if 𝑇 is a positive operator, then (𝑇^𝛼)^∘=|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]). If 𝑇 is an irreducible operator, then 𝑇^∘ is an irreducible operator.

Lemma 4.3. Let 𝐷 be a subset of ℂ, 𝑧₀,𝑅>0, such that 𝐵(𝑧₀,𝑅)={𝑧∈ℂ;|𝑧−𝑧₀|≤𝑅}⊂𝐷, let 𝑔_𝑛∶𝐷→𝐻 be a sequence of analytic functions, and let the Taylor expansion of 𝑔_𝑛 be 𝑔_𝑛(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘𝑧−𝑧₀^𝑘,||𝑧−𝑧₀||<𝑅.(4.2) If 𝑔_𝑛 is uniformly bounded on 𝐵(𝑧₀,𝑅)(i.e.,𝑀=sup_𝑛≥1‖𝑔_𝑛(𝑧)‖_{𝐵(𝑧0,𝑅)}<∞), then ‖‖𝑔_𝑛(𝑧)−𝑔_𝑛𝑧₀‖‖≤𝑀𝑟𝑅−𝑟,𝑧∈𝐵(𝑧₀,𝑟),0<𝑟<𝑅.(4.3)

Proof. For all 𝑛 and 𝑧∈𝐵(𝑧₀,𝑟) with 0<𝑟<𝑅, by Cauchy’s integral formula, we get the following inequality: ‖‖𝑔_𝑛(𝑧)−𝑔_𝑛𝑧₀‖‖=‖‖‖12𝑖𝜋_{|𝑢−𝑧0|=𝑅}𝑔_𝑛(𝑢)𝑢−𝑧𝑑𝑢−21𝑖𝜋_{|𝑢−𝑧0|=𝑅}𝑔_𝑛(𝑢)𝑢−𝑧₀≤1𝑑𝑢‖‖‖2𝜋_{|𝑢−𝑧0|=𝑅}||𝑧−𝑧₀||‖‖𝑔_𝑛(𝑢)‖‖≤𝑀𝑟||𝑢−𝑧‖𝑢−𝑧₀||||𝑑𝑢||𝑅−𝑟.(4.4)

Remark 4.4. Let 𝐷 be an open subset of ℂ. A sequence of analytic functions 𝑔_n∶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 ‖𝑔_𝑛‖_{𝐵(𝑧0,𝑟)}<𝜖 for all 𝑛>𝑛₀.

5. Single-valued Extension Property for 𝑚-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 𝐵_𝑚(𝑇).

Definition 5.1 (see [8]). An operator 𝑇∈ℒ(𝐻) is called an 𝑚-isometry if 𝐵_𝑚(𝑇)=^𝑚_𝑘=0(−1)^𝑘⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝑇_{∗𝑚−𝑘}𝑇^𝑚−𝑘=0.(5.1)

Remark 5.2. It is easy to see that 𝑇∈ℒ(𝐻) is an 𝑚-partial isometry if and only if 𝐵_𝑚(𝑇)𝑥=^𝑚_𝑘=0(−1)^𝑘⎛⎜⎜⎝𝑚𝑘⎞⎟⎟⎠𝑇_{∗𝑚−𝑘}𝑇^𝑚−𝑘(𝑥)=0,∀𝑥∈𝑁(𝑇)^⟂,(5.2) which shows that the class of 𝑚-partial isometries generalizes those of 𝑚-isometries and partial isometries.

Theorem 5.3 (see [10]). If 𝑇∈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]). Let 𝑇 be a reducible 𝑚-partial isometry. Then(1)𝜆∈𝜎_ap(𝑇)⧵{0} implies 𝜆∈𝜎_ap(𝑇^∗), that is, if 𝑇_𝜆𝑥_𝑛→0 for some sequence of bounded vectors {𝑥_𝑛}⊂𝐻, then (𝑇_𝜆)^∗𝑥_𝑛→0,(2)𝜆∈𝜎_𝑝(𝑇)⧵{0} implies 𝜆∈𝜎_𝑝(𝑇^∗),(3)eigenvectors of 𝑇 corresponding to distinct eigenvalues are orthogonal, that is, 𝑁(𝑇_𝜆)⟂𝑁(𝑇_𝜇) if 𝜆,𝜇∈𝜎_𝑝(𝑇),𝜆≠𝜇.

Lemma 5.5. Let 𝑇 be a reducible 𝑚-partial isometry and let 𝜆,𝜇∈𝜎_𝑎𝑝(𝑇) and (𝑥_𝑛), (𝑦_𝑛) be sequences of bounded vectors in 𝐻 such that 𝜆≠𝜇 and ‖‖𝑇_𝜆𝑥_𝑛‖‖⟶0,‖‖𝑇_𝜇𝑦_𝑛‖‖⟶0(𝑎𝑠𝑛⟶∞).(5.3) Then we have ⟨𝑥_𝑛∣𝑦_𝑛⟩⟶0(as𝑛⟶∞).(5.4)

Proof. We may assume that 𝜇≠0. Then from Proposition 5.4(1) we have ‖(𝑇_𝜇)^∗𝑦_𝑛‖→0 as 𝑛→∞. Hence, (𝜆−𝜇)⟨𝑥_𝑛∣𝑦_𝑛⟩=−⟨𝑇_𝜆𝑥_𝑛∣𝑦_𝑛⟩+⟨𝑥_𝑛∣𝑇_𝜇^∗𝑦_𝑛⟩⟶0,𝑛⟶∞,(5.5) which implies (5.4) in view of 𝜆≠𝜇 and the proof is complete.

Theorem 5.6. Any reducible 𝑚-partial isometry has SVEP.

Proof. Let 𝑈 be a bounded subset of ℂ and let 𝑓∶𝑈→𝐻 be an analytic function such that 𝑇_𝜆𝑓(𝜆)=0for𝜆∈𝑈.(5.6) Since 𝑁(𝑇_𝜆)⟂𝑁(𝑇_𝜇),𝜆≠𝜇 (Proposition 5.4(3)), we have ‖𝑓(𝜆)‖²=lim_𝜇→𝜆⟨𝑓(𝜆)∣𝑓(𝜇)⟩=0.(5.7) This shows that 𝑓(𝜆)=0.

Lemma 5.7. If 𝑇 is an 𝑚-partial isometry, then 𝑇^𝑜 is also an 𝑚-partial isometry.

Proof. It is a consequence of the properties of 𝑇^𝑜 (see Theorem 4.1).

Theorem 5.8. If 𝑇 is an 𝑚-partial isometry with a nontrivial reducing space 𝑁(𝑇), then 𝑇^𝑜 has the single-valued extension property (SVEP).

Proof. To prove that 𝑇^∘ has SVEP, let 𝜆∈𝜎_𝑎𝑝(𝑇^∘)−{0}. Since 𝜎_𝑎𝑝(𝑇^∘)=𝜎_𝑎𝑝(𝑇) by Theorem 4.1, 𝜆∈𝜎_𝑎𝑝(𝑇^∗)=𝜎_𝑎𝑝((𝑇^∘)^∗). In particular if 𝜆∈𝜎_𝑝(𝑇^∘), then 𝜆∈𝜎_𝑝((𝑇^∘)^∗). Hence, 𝑁((𝑇_𝜆)^∘)⟂𝑁((𝑇_𝜇)^∘) for 𝜆 and 𝜇∈𝜎_𝑝(𝑇^∘) with 𝜆≠𝜇. In a similar way as in the proof of Theorem 5.6, we can see that 𝑇^∘ has SVEP.

Acknowledgment

The author would like to express their gratitude to the referees for their helpful and many valuable suggestions.