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International Journal of Mathematics and Mathematical Sciences
Volumeย 2012, Article IDย 975745, 20 pages
http://dx.doi.org/10.1155/2012/975745
Research Article

On Some Normality-Like Properties and Bishop's Property (๐›ฝ) for a Class of Operators on Hilbert Spaces

Mathematics Department, College of Science, Al Jouf University, Al Jouf 2014, Saudi Arabia

Received 11 December 2011; Accepted 19 February 2012

Academic Editor: Shigeruย Kanemitsu

Copyright ยฉ 2012 Sid Ahmed Ould Ahmed Mahmoud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove some further properties of the operator ๐‘‡โˆˆ[๐‘›QN] (๐‘›-power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator ๐‘‡โˆˆ[๐‘›QN] satisfying the translation invariant property is normal and that the operator ๐‘‡โˆˆ[๐‘›QN] is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator ๐‘‡โˆˆ[2QN] is subscalar of order ๐‘š; that is, it is similar to the restriction of a scalar operator of order ๐‘š to an invariant subspace.

1. Introduction

Although normality of operators (๐‘‡โˆ—๐‘‡=๐‘‡๐‘‡โˆ—) 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 ๐‘-hyponormal, log-hyponormal, posinormal, k-quasihyponormal classes, and so forth (see [2โ€“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 ๐ป be an infinite dimensional separable complex Hilbert space, let ๐พ be a complex Hilbert space, and let โ„’(๐ป,๐พ) be the algebra of all bounded linear operators from ๐ป to ๐พ. We write โ„’(๐ป) for โ„’(๐ป,๐ป). If ๐‘‡โˆˆโ„’(๐ป,๐พ), we will write ๐‘(๐‘‡) and ๐‘…(๐‘‡) for the null space (also referred to as the reducing subspace) and the range of ๐‘‡, respectively. The spectrum, the point spectrum, and the approximate point spectrum of an operator ๐‘‡ are denoted by ๐œŽ(๐‘‡),๐œŽ๐‘(๐‘‡),๐œŽ๐‘Ž๐‘(๐‘‡), respectively. ๐‘‡โˆ— means the adjoint of ๐‘‡.

An operator ๐‘‡โˆˆโ„’(๐ป) is(1)hyponormal if and only if ๐‘‡๐‘‡โˆ—โ‰ค๐‘‡โˆ—๐‘‡โ‡”โ€–๐‘‡โˆ—๐‘ฅโ€–โ‰คโ€–๐‘‡๐‘ฅโ€–,forall๐‘ฅโˆˆ๐ป,(2)posinormal if and only if ๐‘…(๐‘‡)โŠ‚๐‘…(๐‘‡โˆ—), or equivalently ๐‘‡๐‘‡โˆ—โ‰ค๐œ†2๐‘‡โˆ—๐‘‡ for some ๐œ†>0,(3)๐‘-hyponormal if and only if (๐‘‡๐‘‡โˆ—)๐‘โ‰ค(๐‘‡โˆ—๐‘‡)๐‘,0<๐‘โ‰ค1,(4)๐‘-quasihyponormal if and only if ๐‘‡โˆ—[(๐‘‡๐‘‡โˆ—)๐‘โˆ’(๐‘‡โˆ—๐‘‡)๐‘]๐‘‡โ‰ค0,0<๐‘<1,(5)log-hyponormal if ๐‘‡ is invertible and log(๐‘‡๐‘‡โˆ—)โ‰คlog(๐‘‡โˆ—๐‘‡),(6)2-isometry if and only if ๐‘‡โˆ—2๐‘‡2โˆ’2๐‘‡โˆ—๐‘‡+๐ผ=0 (see [8]), where ๐ผ indicates the identity operator that is, if and only if๐‘‡โˆ—2๐‘‡2=2๐‘‡โˆ—๐‘‡โˆ’๐ผ.(1.1) An operator ๐‘‡โˆˆโ„’(๐ป) is said to be ๐‘›-power quasinormal (abbreviated as ๐‘›QN), ๐‘›=1,2,โ€ฆ, if๐‘‡๐‘›๐‘‡โˆ—๐‘‡=๐‘‡โˆ—๐‘‡๐‘‡๐‘›=๐‘‡โˆ—๐‘‡๐‘›+1.(1.2) If ๐‘›=1, ๐‘‡ is called quasinormal. This class of operators being denoted by [๐‘›QN], that is,[๐‘›QN]๎€ฝโˆถ=๐‘‡โˆˆโ„’(๐ป)โˆถ๐‘‡๐‘›๐‘‡โˆ—๐‘‡โˆ’๐‘‡โˆ—๐‘‡๐‘‡๐‘›๎€พ=0(1.3) was studied by the author [9].

๐‘‡ is called an ๐‘š-partial isometry if ๐‘‡ satisfies๐‘‡๐ต๐‘š(๐‘‡)=๐‘‡๐‘š๎“๐‘˜=0โŽ›โŽœโŽœโŽ๐‘š๐‘˜โŽžโŽŸโŽŸโŽ (โˆ’1)๐‘˜๐‘‡โˆ—๐‘šโˆ’๐‘˜๐‘‡๐‘šโˆ’๐‘˜=0,(1.4) where ๐ต๐‘š(๐‘‡) is obtained formally from the binomial expansion of ๐ต๐‘š(๐‘‡)=(๐‘‡โˆ—๐‘‡โˆ’๐ผ)๐‘š by understanding (๐‘‡โˆ—๐‘‡)๐‘šโˆ’๐‘˜=๐‘‡โˆ—๐‘šโˆ’๐‘˜๐‘‡๐‘šโˆ’๐‘˜. The case when ๐‘š=1 is called the partial isometries class. The class of ๐‘š-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 [๐‘›QN]. In particular we show that an operator ๐‘‡โˆˆ[๐‘›QN] satisfying the translation invariant property is normal, and an invertible operator ๐‘‡โˆˆ[๐‘›QN] and its inverse ๐‘‡โˆ’1 have a common nontrivial invariant closed set provided that ๐‘‡โˆ—โˆˆ[๐‘›QN]. Also we show that some of class [2QN] satisfy an analogue of the single-valued extension for ๐‘Š๐‘š2(๐ท,๐ป) 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 ๐‘š-partial isometries operators.

2. Spaces of Vector-Valued Functions

We will need the following function spaces.

Let ๐œ† be the coordinate in โ„‚ and let ๐‘‘๐œ‡(๐œ†) denote the planar Lebesgue measure. Let ๐ท be a bounded open subset of โ„‚. We will denote by ๐ฟ2(๐ท,๐ป) the Hilbert space of square measurable (or summable) functions ๐‘“โˆถ๐ทโ†’๐ป such thatโ€–๐‘“โ€–2,๐ท=๎‚ป๎€œ๐ทโ€–๐‘“(๐œ†)โ€–2๎‚ผ๐‘‘๐œ‡(๐œ†)1/2<โˆž.(2.1)

Let ๐‘‚(๐ท,๐ป) denote the space of ๐ป-valued functions analytic on ๐ท, that is, ๐œ•๐‘“=๐œ•๐‘“/๐œ•๐‘ง=0. Equipped with the topology of uniform convergence on compact subsets of ๐ท, ๐‘‚(๐ท,๐ป) is a Frechet space. Let๐ด2(๐ท,H)=๐ฟ2(๐ท,๐ป)โˆฉ๐‘‚(๐ท,๐ป)(2.2) denote the Bergman space for ๐ท consisting of square measurable functions ๐‘“ that are analytic on ๐ท.

We denote by ๐‘ƒ the orthogonal projection of ๐ฟ2(๐ท,๐ป) onto ๐ด2(๐ท,๐ป).

Let us define now a special Sobolev type space. Let ๐‘š be a fixed nonnegative integer. The Sobolev space ๐‘Š๐‘š2(๐ท,๐ป) of order ๐‘š of vector-valued functions with respect to ๐œ• will be the space of those functions ๐‘“โˆˆ๐ฟ2(๐ท,๐ป) whose derivatives ๐œ•๐‘“,โ€ฆ,๐œ•๐‘š๐‘“ in the sense of distributions also belong to ๐ฟ2(๐ท,๐ป), that is,๐‘Š๐‘š2๎‚†(๐ท,๐ป)=๐‘“โˆˆ๐ฟ2(๐ท,๐ป)โˆถ๐œ•๐‘˜๐‘“โˆˆ๐ฟ2(๐ท,๐ป),for๎‚‡.๐‘˜=0,1,โ€ฆ,๐‘š(2.3)

Endowed with the normโ€–๐‘“โ€–2๐‘Š๐‘š2=๐‘š๎“๐‘˜=0โ€–โ€–โ€–๐œ•๐‘˜๐‘“โ€–โ€–โ€–22,๐ท.(2.4)๐‘Š๐‘š2(๐ท,๐ป) becomes a Hilbert space contained continuously in ๐ฟ2(D,๐ป); that is, there is a constant 0<๐ถ<โˆž such that โ€–๐‘“โ€–๐ฟ2(๐ท,๐ป)โ‰ค๐ถโ€–๐‘“โ€–๐‘Š๐‘š2(๐ท,๐ป)forall๐‘“โˆˆ๐‘Š๐‘š2(๐ท,๐ป).

Let๐‘‡๐œ†=๐‘‡โˆ’๐œ†=๐‘‡โˆ’๐œ†๐ผ(2.5) for ๐œ†โˆˆโ„‚ once and for all, whenever the definition is meaningful. We say that ๐‘‡ has the single-valued extension property at ๐œ†0โˆˆโ„‚ (abbreviated SVEP at ๐œ†0) if, for every open neighborhood ๐‘ˆ of ๐œ†0, the only analytic solution ๐‘“ to the equation๎€ท๐‘‡๐œ†๐‘“๎€ธ(๐œ†)=(๐‘‡โˆ’๐œ†)๐‘“(๐œ†)=0(2.6) for all ๐œ† in ๐‘ˆ is the constant function ๐‘“โ‰ก0. We say that ๐‘‡ has SVEP if ๐‘‡ has a SVEP at every ๐œ†โˆˆโ„‚.

It is easily seen that an operator ๐‘‡โˆˆโ„’(๐ป) has SVEP if and only if, for each open ๐ทโŠ†โ„‚.

The operator ๐‘‡๐ทโˆถ๐‘‚(๐ท,๐ป)โ†’๐‘‚(๐ท,๐ป) defined by๐‘‡๐ท(๐‘“)(๐œ†)=๐‘‡๐œ†๐‘“(๐œ†)โˆ€๐‘“โˆˆ๐‘‚(๐ท,๐ป),๐œ†โˆˆ๐ท.(2.7) is one to one.

Recall that, for a bounded operator ๐‘‡ on ๐ป, the local resolvent set ๐œŒ๐‘‡(๐‘ฅ) of ๐‘‡ at the point ๐‘ฅโˆˆ๐ป is defined as the union of all open subsets ๐ท of โ„‚ such that there exists an analytic function ๐‘“โˆถ๐ทโ†’๐ป which satisfies๎€ท๐‘‡๐œ†๎€ธ๐‘“(๐œ†)=๐‘ฅโˆ€๐œ†โˆˆ๐ท.(2.8) The local spectrum ๐œŽ๐‘‡(๐‘ฅ) of ๐‘‡ at ๐‘ฅโˆˆ๐ป is the set defined by ๐œŽ๐‘‡(๐‘ฅ)โˆถ=โ„‚โงต๐œŒ๐‘‡(๐‘ฅ) and obviously ๐œŽ๐‘‡(๐‘ฅ)โŠ‚๐œŽ(๐‘‡). It is clear from the definition that, ๐‘‡ has SVEP if and only if zero is the unique vector ๐‘ฅโˆˆ๐ป such that ๐œŽ๐‘‡(๐‘ฅ)=โˆ… (see for more details [11]).

Recall that a bounded operator ๐‘‡โˆˆโ„’(๐ป) is said to have the Bishopโ€™s property (๐›ฝ) if for every open subset ๐ท of the complex plane โ„‚ and every sequence of analytic functions ๐‘“๐‘›โˆถ๐ทโ†’๐ป with the property that๐‘‡๐œ†๐‘“๐‘›(๐œ†)โŸถ0as๐‘›โŸถโˆž,(2.9) uniformly on all compact subsets of ๐ท, ๐‘“๐‘›(๐œ†)โ†’0 as ๐‘›โ†’โˆž locally uniformly on ๐ท or equivalently, for every open subset ๐ท of โ„‚, the operator ๐‘‡๐ท 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 Bishopโ€™s property (๐›ฝ).

A bounded operator ๐‘‡ on ๐ป is called scalar of order ๐‘š if it possesses a spectral distribution of order ๐‘š, that is, if there is a continuous unital morphism,ฮฆโˆถ๐ถ๐‘š0(โ„‚)โŸถโ„’(๐ป),(2.10) such that ฮฆ(๐‘ง)=๐‘‡, where ๐‘ง stands for the identity function on โ„‚ and ๐ถ๐‘š0(โ„‚) for the space of compactly supported functions on โ„‚, continuously differentiable of order ๐‘š,0โ‰ค๐‘šโ‰คโˆž. An operator is subscalar if it is similar to the restriction of a scalar operator to an invariant subspace.

Let ๐‘€๐‘ง be the operator on ๐‘Š๐‘š2(๐ท;๐ป) such that (๐‘€๐‘ง๐‘“)(๐‘ง)=๐‘ง๐‘“(๐‘ง) for ๐‘“โˆˆ๐‘Š๐‘š2(๐ท;๐ป). This has a spectral distribution of order ๐‘š, defined by the functional calculus ฮฆ๐‘€โˆถ๐ถ๐‘š0(โ„‚)โ†’โ„’(๐‘Š๐‘š2(๐ท,๐ป));ฮฆ๐‘€(๐‘“)=๐‘€๐‘“. Therefore ๐‘€๐‘ง is a scalar operator of order ๐‘š. Consider a bounded open disk ๐ท which contains ๐œŽ(๐‘‡) and the quotient space๐‘Š๐ป(๐ท)=๐‘š2(๐ท,๐ป)๐‘‡๐‘ง๐‘Š๐‘š2(๐ท,๐ป)(2.11) endowed with the Hilbert space norm. We denote the class containing a vector ๐‘“ or an operator A on ๐ป(๐ท) by ๎๐‘“ or ๎๐ด, respectively. Let ๐‘€๐‘ง be the operator of multiplication by ๐‘ง on ๐‘Š๐‘š2(๐ท;๐ป). As noted above, ๐‘€๐‘ง is a scalar of order ๐‘š and has a spectral distribution ฮฆ. Let ๎‚Š๐‘€๐‘†โ‰ก๐‘ง. Since ๐‘‡๐‘ง๐‘Š๐‘š2(๐ท,๐ป) is invariant under every operator ๐‘€๐‘“; ๐‘“โˆˆ๐ถ๐‘š0(โ„‚), we infer that ๐‘† is a scalar operator of order ๐‘š with spectral distribution ๎ฮฆ. Consider the natural map ๐‘‰โˆถ๐ปโ†’๐ป(๐ท) defined by ๎„ž๐‘‰โ„Ž=[1โŠ—โ„Ž], for โ„Žโˆˆ๐ป, where 1โŠ—โ„Ž denotes the constant function identically equal to โ„Ž. In [7], Putinar showed that if ๐‘‡โˆˆโ„’(๐ป) is a hyponormal operator then ๐‘‰ is one to one and has closed range such that ๐‘‰๐‘‡=๐‘†๐‘‰, and so ๐‘‡ is subscalar of order ๐‘š.

3. Further Properties of the Class [๐‘›QN]

We start this section with some properties of ๐‘›-power quasinormal operators.

Theorem 3.1. The class [๐‘›QN] has the following properties.(1)The class [๐‘›QN] is closed under unitary equivalence and scalar multiplication.(2)If ๐‘‡ is of class [๐‘›QN] and ๐‘€ is a closed subspace of ๐ป that reduces ๐‘‡, then ๐‘‡โˆฃ๐‘€ (the restriction of๐‘‡ to ๐‘€) is of class [๐‘›QN].

Proof. (1) Let ๐‘†โˆˆโ„’(๐ป) be unitary equivalent to ๐‘‡. Then there is a unitary operator ๐‘‰โˆˆโ„’(๐ป) such that ๐‘‡=๐‘‰โˆ—๐‘†๐‘‰ which implies that ๐‘‡โˆ—=๐‘‰โˆ—๐‘†โˆ—๐‘‰. Noting that ๐‘‡๐‘›=๐‘‰โˆ—๐‘†๐‘›๐‘‰ and inserting ๐ผ=๐‘‰๐‘‰โˆ— suitably, we deduce from (1.2) that ๐‘‰โˆ—๐‘†๐‘›๐‘†โˆ—๐‘†๐‘‰=๐‘‡๐‘›๐‘‡โˆ—๐‘‡=๐‘‡โˆ—๐‘‡๐‘›+1=๐‘‰โˆ—๐‘†โˆ—๐‘†๐‘›+1๐‘‰,(3.1) and (1.2) follows for ๐‘†. Since (๐‘‡โˆฃ๐‘€)ฮ”=๎€ท๐‘‡ฮ”๎€ธโˆฃ๐‘€(3.2) for ฮ” as the ๐‘›-th power or the adjoint, it follows that the left-hand side of (1.2) for (๐‘‡โˆฃ๐‘€) reads ๎€ท๐‘‡๐‘›๐‘‡โˆ—๎€ธ,๐‘‡โˆฃ๐‘€(3.3) which is (๐‘‡โˆ—๐‘‡๐‘›+1โˆฃ๐‘€)=(๐‘‡โˆฃ๐‘€)โˆ—(๐‘‡โˆฃ๐‘€)๐‘›+1, which is the right-hand side of (1.2). Thus, ๐‘‡โˆฃ๐‘€ is of class [๐‘›QN].

Next we characterize a matrix on a 2-dimensional complex Hilbert space which is in [๐‘›QN]. Since every matrix on a finite dimensional complex Hilbert space is unitarily equivalent to an upper triangular matrix and an ๐‘›-power quasinormal operator is unitarily invariant, it suffices to characterize an upper triangular matrix ๐‘‡. From the direct calculation, we get the following characterization.

Proposition 3.2. For ๐‘›โ‰ฅ2 one has โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โˆˆ[๐‘›๐‘‡=๐‘ฅ๐‘ฆ0๐‘งQN]โŸบโŽงโŽชโŽจโŽชโŽฉ๎€ท๐‘ฅ๐‘ฆ๐‘ฅ๐‘›โˆ’1+๐‘ง๐‘ฅ๐‘›โˆ’2+โ‹ฏ+๐‘ง๐‘›โˆ’1๎€ธ๐‘ฅ=0,๐‘ฆ(๐‘ง๐‘›โˆ’๐‘ฅ๐‘›)=0,๐‘ฅ๐‘ฆ(๐‘ง๐‘›โˆ’๐‘ฅ๐‘›๎€ท๐‘ฅ)โˆ’๐‘›โˆ’1+๐‘ง๐‘ฅ๐‘›โˆ’2+โ‹ฏ+๐‘ง๐‘›โˆ’1๎€ธ๎‚€||๐‘ฆ||2+|๐‘ง|2โˆ’|๐‘ฅ|2๎‚=0.(3.4)

We remark here that Proposition 3.2 offers the convenient criterion to find some examples of operators in [๐‘›QN]. Also we observe that [๐‘›QN] is not necessarily normal on a finite dimensional space.

Next couple of results show that [๐‘›QN] does not have a translation invariant property.

Theorem 3.3 (see [9]). If ๐‘‡ and ๐‘‡โˆ’๐ผ are of class [2QN], then ๐‘‡ is normal.

Theorem 3.4 (see [9]). If ๐‘‡ is of class [2QN]โˆฉ[3QN] such that ๐‘‡โˆ’๐ผ is of class [๐‘›QN], then ๐‘‡ is normal.

It is natural to ask the following question: what is the operators in [๐‘›QN] satisfying the translation invariant property? The answer to this question is provided by the following theorem.

Theorem 3.5. ๐‘‡๐œ† is of class [nQN] for every ๐œ†โˆˆโ„‚ if and only if ๐‘‡ is a normal operator.

Proof. Assume that (๐‘‡๐œ†) is of class [๐‘›QN] for every ๐œ†โˆˆโ„‚. Then (1.2) reads (๐‘‡โˆ’๐œ†)๐‘›(๐‘‡โˆ’๐œ†)โˆ—(๐‘‡โˆ’๐œ†)=(๐‘‡โˆ’๐œ†)โˆ—(๐‘‡โˆ’๐œ†)(๐‘‡โˆ’๐œ†)๐‘›, which reduces on eliminating the common factor โˆ’๐œ†(๐‘‡โˆ’๐œ†)๐‘›+1 to (๐‘‡โˆ’๐œ†)๐‘›๎€ท๐‘‡โˆ—๐‘‡โˆ’๐œ†๐‘‡โˆ—๎€ธ=๎€ท๐‘‡โˆ—๐‘‡โˆ’๐œ†๐‘‡โˆ—๎€ธ(๐‘‡โˆ’๐œ†)๐‘›.(3.5) By the binomial expansion, ๐‘›๎“๐‘˜=0(โˆ’1)๐‘˜โŽ›โŽœโŽœโŽ๐‘›๐‘˜โŽžโŽŸโŽŸโŽ ๐œ†๐‘˜๐‘‡๐‘›โˆ’๐‘˜๎€ท๐‘‡โˆ—๐‘‡โˆ’๐œ†๐‘‡โˆ—๎€ธ=๎€ท๐‘‡โˆ—๐‘‡โˆ’๐œ†Tโˆ—๎€ธ๐‘›๎“๐‘˜=0(โˆ’1)๐‘˜โŽ›โŽœโŽœโŽ๐‘›๐‘˜โŽžโŽŸโŽŸโŽ ๐œ†๐‘˜๐‘‡๐‘›โˆ’๐‘˜,(3.6) whence by arranging terms suitably, the extremal terms vanishing in view of (1.2), ๐‘›โˆ’1๎“๐‘˜=1(โˆ’1)๐‘˜โŽ›โŽœโŽœโŽ๐‘›๐‘˜โŽžโŽŸโŽŸโŽ ๐œ†๐‘˜๎€ท๐‘‡๐‘›โˆ’๐‘˜๐‘‡โˆ—๐‘‡โˆ’๐‘‡โˆ—๐‘‡๐‘‡๐‘›โˆ’๐‘˜๎€ธโˆ’๐‘›โˆ’1๎“๐‘˜=1(โˆ’1)๐‘˜โŽ›โŽœโŽœโŽ๐‘›๐‘˜โŽžโŽŸโŽŸโŽ ๐œ†๐‘˜+1๎€ท๐‘‡๐‘›โˆ’k๐‘‡โˆ—โˆ’๐‘‡โˆ—๐‘‡๐‘›โˆ’๐‘˜๎€ธ=0.(3.7)
Now note that from the second summand in (3.7), we may extract the extremal term (โˆ’1)๐‘›๐‘›๐œ†๐‘›(๐‘‡โˆ—๐‘‡โˆ’๐‘‡๐‘‡โˆ—) and express it in terms of the remaining terms which contain ๐œ† to the power <๐‘›. Hence dividing (3.7) by ๐œ†๐‘› and letting ๐œ†โ†’โˆž, we conclude that ๐‘‡โˆ—๐‘‡โˆ’๐‘‡๐‘‡โˆ—โ†’0, whence the normality of ๐‘‡ follows.

Conversely it is known that normality is a translation invariant property; that is, if ๐‘‡ is normal, then (๐‘‡๐œ†) is normal for every ๐œ†โˆˆโ„‚, and hence (๐‘‡๐œ†) is of class [๐‘›QN].

The following proposition gives a characterization of an ๐‘›-power quasinormal operator.

Proposition 3.6. Let ๐‘‡โˆˆโ„’(๐ป), ๐ด=๐‘‡๐‘›+๐‘‡โˆ—๐‘‡, and ๐ต=๐‘‡๐‘›โˆ’๐‘‡โˆ—๐‘‡. Then ๐‘‡ is of class [๐‘›QN] if and only if ๐ด commutes with ๐ต.

Proof. Commutativity of ๐ด and ๐ต is equivalent to (1.2).

Proposition 3.7. Let ๐‘‡,๐ด,๐ต be as in Proposition 3.6. Then if ๐‘‡ is of class [๐‘›QN], then ๐‘‡๐‘›๐‘‡โˆ—๐‘‡ commutes with ๐ด and ๐ต.

Proof. By (1.2), ๐‘‡๐‘›๐‘‡โˆ—๐‘‡๎€ท๐‘‡๐‘›ยฑ๐‘‡โˆ—๐‘‡๎€ธ=๎€ท๐‘‡๐‘›ยฑ๐‘‡โˆ—๐‘‡๎€ธ๐‘‡๐‘›๐‘‡โˆ—๐‘‡.(3.8)
In general, the two classes [๐‘›QN] and [(๐‘›+1)QN] are not the same (see [9]).

Proposition 3.8. If ๐‘‡ is both of class [๐‘›QN] and [(๐‘›+1)QN], then it is of class [(๐‘›+2)QN], that is, [nQN]โˆฉ[(๐‘›+1)QN]โŠ‚[(๐‘›+2)QN].

Proof. By (1.2), ๐‘‡๐‘›+1๐‘‡โˆ—๐‘‡=๐‘‡๐‘›๐‘‡โˆ—๐‘‡2(3.9) so that ๐‘‡๐‘›+2๐‘‡โˆ—๐‘‡ may be transformed into ๐‘‡โˆ—๐‘‡๐‘‡๐‘›+2.
It is known that if ๐‘‡ belongs to [๐‘›QN] some ๐‘›>0, ๐‘‡2 does not necessarily belong to the same class.

Theorem 3.9 (see [9]). If ๐‘‡ and ๐‘‡โˆ— are of class [๐‘›QN], then ๐‘‡๐‘› is normal.

Proposition 3.10. If an operator ๐‘‡ of class [2QN] is a 2-isometry, then ๐‘‡2 is of class [๐‘›QN] for all integers ๐‘›โ‰ฅ2.

Proof. From Proposition 3.8 it suffices to prove that ๐‘‡2 is of class [2QN] and of class [3QN] because we may then proceed inductively.
Since ๐‘‡ is a 2-isometry, we have ๐‘‡4(๐‘‡โˆ—2๐‘‡2)=๐‘‡4(2๐‘‡โˆ—๐‘‡โˆ’๐ผ). Using ๐‘‡2๐‘‡โˆ—๐‘‡=๐‘‡โˆ—๐‘‡3, we may shift the power of ๐‘‡2 to the left, arriving at ๐‘‡4(๐‘‡โˆ—2๐‘‡2)=๐‘‡โˆ—2๐‘‡6; that is, ๐‘‡2 is of class [2QN].
In the same way, we may deduce that ๐‘‡6(๐‘‡โˆ—2๐‘‡2)=๐‘‡โˆ—2๐‘‡8 whence ๐‘‡2 is of class [3QN].

Lemma 3.11 (see [9]). If ๐‘‡ is of class [๐‘›QN], then ๐‘(๐‘‡๐‘›)โŠ‚๐‘(๐‘‡โˆ—๐‘›).

Proposition 3.12. If ๐‘‡ is both of class [๐‘›QN] and [(๐‘›+1)QN] such that ๐‘‡ is injective or ๐‘‡โˆ— is injective, then ๐‘‡ is quasinormal.

Proof. Since ๐‘‡ is of class [๐‘›QN]โˆฉ[(๐‘›+1)QN], we have (3.9), which reads ๐‘‡๐‘›(๐‘‡๐‘‡โˆ—๐‘‡โˆ’๐‘‡โˆ—๐‘‡2)=0. If ๐‘‡ is injective, then so is ๐‘‡๐‘› and we have ๐‘‡๐‘‡โˆ—๐‘‡โˆ’๐‘‡โˆ—๐‘‡2=0, whence ๐‘‡ is quasi-normal. If ๐‘‡โˆ— is injective, we may appeal to Lemma 3.11.

In the following theorem we prove some topological properties of the class [๐‘›QN].

Theorem 3.13. The class [๐‘›QN] is an arcwise-connected, closed subset of โ„’(๐ป) equipped with the uniform operator (norm) topology.

Proof. By Theorem 3.1, (1), we see that the ray ๐‘Ž๐‘‡ in โ„’(๐ป) through ๐‘‡ is contained in [๐‘›QN] for every complex number ๐‘Ž, and therefore [๐‘›QN] is arcwise-connected.
To see that [๐‘›QN] is closed, we prove that any strong limit ๐‘‡โˆˆโ„’(๐ป) of a sequence (๐‘‡๐‘) in [๐‘›QN] also belongs to [๐‘›QN]; that is, we let (๐‘‡๐‘) be a sequence of operators in [๐‘›QN] converging to ๐‘‡โˆˆโ„’(๐ป) in norm: โ€–โ€–๐‘‡๐‘โ€–โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโŸถ0as๐‘โŸถโˆž,foreach๐‘ฅโˆˆ๐ป.(3.10) Hence it follows that โ€–โ€–๐‘‡โˆ—๐‘๐‘ฅโˆ’๐‘‡โˆ—๐‘ฅโ€–โ€–=โ€–โ€–๎€ท๐‘‡๐‘๎€ธโˆ’๐‘‡โˆ—๐‘ฅโ€–โ€–โ‰คโ€–โ€–๎€ท๐‘‡๐‘๎€ธโˆ’๐‘‡โˆ—โ€–โ€–โ€–โ€–๐‘‡โ€–๐‘ฅโ€–=๐‘โ€–โ€–โˆ’๐‘‡โ€–๐‘ฅโ€–โŸถ0,(3.11) whence (๐‘‡โˆ—๐‘) converges strongly to ๐‘‡โˆ—.
Since the product of operators is sequentially continuous in the strong topology (see [12, page 62]), one concludes that๐‘‡๐‘›๐‘๐‘‡โˆ—๐‘๐‘‡๐‘ converge strongly to ๐‘‡๐‘›๐‘‡โˆ—๐‘‡. Similarly ๐‘‡โˆ—๐‘๐‘‡๐‘๐‘›+1 converges strongly to ๐‘‡โˆ—๐‘‡๐‘›+1. Hence the limiting case of (1.2) shows that ๐‘‡ belongs to [๐‘›QN], completing the proof.

Proposition 3.14. If ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘ are of class [๐‘›QN], then both the direct sum ๐‘‡1โŠ•๐‘‡2โŠ•โ‹ฏโŠ•๐‘‡๐‘ and the tensor product ๐‘‡1โŠ—๐‘‡2โŠ—โ‹ฏโŠ—๐‘‡๐‘ are of class [๐‘›QN].

Proof. By the compatibility principle similar to (3.2), ๎€ท๐‘‡1โŠ•๐‘‡2๎€ธฮ”=๎€ท๐‘‡ฮ”1โŠ•๐‘‡ฮ”2๎€ธ,๎€ท๐‘‡1โŠ—๐‘‡2๎€ธฮ”=๎€ท๐‘‡ฮ”1โŠ—๐‘‡ฮ”2๎€ธ,(3.12) where ฮ” indicates either the ๐‘›th power or adjoint, the proof follows.

A linear operator ๐‘‡ on ๐ป is hypercyclic if there is a vector with dense orbit; that is, if there exists an ๐‘ฅโˆˆ๐ป such that orbit Orb(๐‘‡,๐‘ฅ)โˆถ={๐‘ฅ,๐‘‡๐‘ฅ,๐‘‡2๐‘ฅ,โ€ฆ} is dense in ๐ป, and in this case ๐‘ฅ is called a hypercyclic vector for ๐‘‡.

An operator ๐‘‡ on ๐ป is supercyclic if there exists a vector whose scaled orbit is dense; that is, if there exists an ๐‘ฅโˆˆ๐ป such that {๐œ†๐‘‡๐‘›๐‘ฅ,๐‘›โ‰ฅ0,๐œ†โˆˆโ„‚} is dense in ๐ป and in this case ๐‘ฅ is called a supercyclic vector for ๐‘‡.

Kitai [13] showed that hyponormal operators are not hypercyclic. We generalize Kitaiโ€™s theorem to the class [๐‘›QN].

Proposition 3.15. If ๐‘‡ is of class [nQN] with ๐œŽ๐‘(๐‘‡๐‘›)โ‰ โˆ…, then ๐‘‡ is not hypercyclic.

Proof. If ๐‘‡ is hypercyclic, ๐‘‡๐‘› is hypercyclic, and hence ๐œŽ๐‘(๐‘‡โˆ—๐‘›)=โˆ… by [13, corollary 2.4]. From Lemma 3.11 we have ๐œŽ๐‘(๐‘‡๐‘›)โŠ‚๐œŽ๐‘(๐‘‡โˆ—๐‘›) and, hence, ๐œŽ๐‘(๐‘‡๐‘›)=โˆ…, a contradiction.

Theorem 3.16 (see [14, Theorem 2]). If ๐‘‡ is not hypercyclic, then ๐‘‡ and ๐‘‡โˆ’1 have a common non-trivial invariant closed subset.

Proposition 3.17. If ๐‘‡ is of class [๐‘›QN] and 0โˆ‰๐œŽ(๐‘‡), then ๐‘‡ and ๐‘‡โˆ’1 have a common nontrivial closed invariant subset.

Proof . Since ๐‘‡ is of class [๐‘›QN] and 0โˆ‰๐œŽ(๐‘‡), it follows that ๐‘‡๐‘› is normal, and hence (๐‘‡โˆ’1)๐‘› is normal. By [13, Corollary 4.5] ๐‘‡๐‘› and (๐‘‡โˆ’1)๐‘› have no hypercyclic vector. Thus by [15], neither ๐‘‡ or ๐‘‡โˆ’1 has a hypercyclic vector. Therefore by [15] ๐‘‡ and ๐‘‡โˆ’1 have a common nontrivial closed invariant subset. Hence Theorem 3.16 completes the proof.

Proposition 3.18. Operators ๐‘‡ that are of class [๐‘›QN] such that ๐‘‡ is not invertible are not supercyclic.

Proof. Assume that ๐‘‡ is of class [๐‘›QN] and supercyclic. Considering the class [๐‘›QN] being closed under multiplication by nonzero scalars, we may assume that โ€–๐‘‡โ€–=1. Since the supercyclic contraction ๐‘‡ satisfies property (๐›ฝ), ๐œŽ(๐‘‡) is contained in the boundary ๐œ•๐”ป of the unit disk ๐”ป [11, Proposition 3.3.18]. Thus ๐‘‡ is invertible, and we have a contradiction.

Definition 3.19. An operator ๐‘‡โˆˆโ„’(๐ป) is algebraic if there is non-zero-polynomial ๐‘ such that ๐‘(๐‘‡)=0.

The following proposition shows that some quasinilpotent ๐‘›-power quasi-normal operators are subscalar.

Proposition 3.20. If both ๐‘‡ and ๐‘‡โˆ— are of class [๐‘›QN] such that ๐‘‡ is quasinilpotent, then ๐‘‡ is nilpotent, and hence ๐‘‡ is subscalar.

Proof. Since ๐‘‡ is quasinilpotent, ๐œŽ(๐‘‡)={0}. Hence by the spectral mapping theorem we get ๐œŽ(๐‘‡๐‘›)=๐œŽ(๐‘‡)๐‘›={0}. Thus ๐‘‡๐‘› is quasinilpotent and normal. So ๐‘‡๐‘›=0;thatis,๐‘‡ is nilpotent, and ๐‘‡ is algebraic operator, and hence ๐‘‡ is subscalar.

Proposition 3.21 (see [7, Proposition 2.1]). For every bounded disk ๐ท in โ„‚, there is a constant ๐ถ๐ท such that for an arbitrary operator ๐‘‡โˆˆโ„’(๐ป) and ๐‘“โˆˆ๐‘Š๐‘š2(๐ท,๐ป) we have โ€–โ€–(๐ผโˆ’๐‘ƒ)๐œ•๐‘—๐‘“โ€–โ€–2,๐ทโ‰ค๐ถ๐ท๎‚ตโ€–โ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—+1๐‘“โ€–โ€–โ€–2,๐ท+โ€–โ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—+2๐‘“โ€–โ€–โ€–2,๐ท๎‚ถ,๐‘—=0,1,โ€ฆ,๐‘šโˆ’2,(3.13) where ๐‘ƒ is the orthogonal projection of ๐ฟ2(๐ท,๐ป) onto ๐ด2(๐ท,๐ป).

The next theorem is important for the proof of our main theorem, Theorem 3.27.

Theorem 3.22. Let ๐ท be an arbitrary bounded disk in โ„‚. If ๐‘‡ is of class [2QN] and ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ…, then the operator ๐‘‡๐œ†โˆถ๐‘Š๐‘š2(๐ท,๐ป)โŸถ๐‘Š๐‘š2(๐ท,๐ป)(3.14) is one to one.

Proof. Let ๐‘”โˆˆ๐‘Š๐‘š2(๐ท,๐ป) such that ๐‘‡๐œ†๐‘”=0, that is, โ€–โ€–๐‘‡๐œ†๐‘”โ€–โ€–๐‘Š๐‘š2=0.(3.15) Then for ๐‘—=0,1,โ€ฆ,๐‘š we have โ€–โ€–๐‘‡๐œ†๐œ•๐‘—๐‘”โ€–โ€–2,๐ท=0.(3.16) Hence for ๐‘—=1,โ€ฆ,๐‘š we get โ€–๐‘‡2๐œ†2๐œ•๐‘—๐‘”โ€–2,๐ท=0. Since ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡)) is empty then ๐‘‡2 is normal [9, Theorem 2.2]. Hence, โ€–โ€–(๐‘‡2๐œ†2)โˆ—๐œ•๐‘—๐‘”โ€–โ€–2,๐ท=0.(3.17) Now we claim that โ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘”โ€–โ€–2,๐ท=0.(3.18) Indeed, since ๐‘‡๐œ† is invertible for ๐œ†โˆˆ๐ทโงต๐œŽ(๐‘‡), (3.16) implies that โ€–โ€–๐œ•๐‘—๐‘”โ€–โ€–2,๐ทโงต๐œŽ(๐‘‡)=0.(3.19) Therefore โ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘”โ€–โ€–2,๐ทโงต๐œŽ(๐‘‡)=0.(3.20)
Since ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ… and ๐œŽ(๐‘‡โˆ—)=๐œŽ(๐‘‡)โˆ—,(๐‘‡โˆ’๐œ†)โˆ— is invertible for ๐œ†โˆˆ๐œŽ(๐‘‡), therefore; from (3.17) we have โ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘”โ€–โ€–2,๐œŽ(๐‘‡)=0.(3.21) It is clear form (3.20); and (3.21) that โ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘”โ€–โ€–2,๐ท=0,for๐‘—=0,1,โ€ฆ,๐‘š.(3.22) Thus Proposition 3.21 and (3.21) imply โ€–โ€–(๐ผโˆ’๐‘ƒ)๐œ•๐‘—๐‘”โ€–โ€–2,๐ท=0for๐‘—=0,1,โ€ฆ,๐‘šโˆ’2,(3.23) where ๐‘ƒ denotes the orthogonal projection of ๐ฟ2(๐ท,๐ป) onto ๐ด2(๐ท,๐ป).
Hence (๐‘‡2๐œ†2)๐‘ƒ๐‘”=๐‘‡2๐œ†2๐‘”=0. Since ๐‘‡2 has SVEP, ๐‘‡ has SVEP. Also ๐‘”=๐‘ƒ๐‘” is analytic and (๐‘‡โˆ’๐œ†)๐‘”(๐œ†)=0 for ๐œ†โˆˆ๐ท. Hence ๐‘”=0. Thus, ๐‘‡๐œ† is one to one.

Corollary 3.23. If ๐‘‡1 and ๐‘‡2 are of class [2QN] with ๐œŽ(๐‘‡๐‘–)โˆฉ(โˆ’๐œŽ(๐‘‡๐‘–))=โˆ…, for ๐‘–=1,2 and ๐‘‡2๐‘‡1=0. Then ๎€ท๐‘‡1+๐‘‡2๎€ธ๐œ†โˆถ๐‘Š๐‘š2(๐ท,๐ป)โŸถ๐‘Š๐‘š2(๐ท,๐ป)(3.24) is one to one.

Proof. If ๐‘“โˆˆ๐‘Š๐‘š2(๐ท,๐ป) is such that (๐‘‡1+๐‘‡2)๐œ†๐‘“=0. Since ๐‘‡2๐‘‡1=0, we get (๐‘‡2โˆ’๐œ†)๐‘‡2๐‘“=0. Since (๐‘‡2)๐œ† is one to one, ๐‘‡2๐‘“=0. Hence, (๐‘‡1)๐œ†๐‘“=0. Since (๐‘‡1)๐œ† is one to one, ๐‘“=0.

The following corollary shows that the nilpotent perturbation of operators in [2QN] satisfying SVEP satisfies SVEP.

Corollary 3.24. If an operator ๐‘‡โˆˆโ„’(๐ป) is a nilpotent perturbation of a 2-power quasi-normal operator ๐‘†, that is, ๐‘‡=๐‘†+๐‘, where ๐‘† is of class [2QN], ๐‘† and ๐‘ commute, and ๐‘๐‘š=0. If ๐œŽ(๐‘†)โˆฉ(โˆ’๐œŽ(๐‘†))=โˆ…, then ๐‘‡๐œ† is one-to-one.

Proof. If ๐‘”โˆˆ๐‘Š๐‘š2(๐ท,๐ป) is such that ๐‘‡๐œ†๐‘”=0, then ๐‘†๐œ†๐‘”=โˆ’๐‘๐‘”.(3.25) Hence ๐‘†๐œ†๐‘๐‘—โˆ’1๐‘”=โˆ’๐‘๐‘—๐‘” for ๐‘—=1,2,โ€ฆ,๐‘š. We prove that ๐‘๐‘—๐‘”=0 for ๐‘—=0,1,โ€ฆ,๐‘šโˆ’1 by indication. Since ๐‘๐‘š=0, ๐‘†๐œ†๐‘๐‘šโˆ’1๐‘”=โˆ’๐‘๐‘š๐‘”=0. Since ๐‘†๐œ† is one-to-one by Theorem 3.22๐‘๐‘šโˆ’1๐‘”=0. Assume it is true when ๐‘—=๐‘˜, that is, ๐‘๐‘˜๐‘”=0. From (3.25), we get ๐‘†๐œ†๐‘๐‘˜โˆ’1๐‘”=โˆ’๐‘๐‘˜๐‘”=0.(3.26) Since ๐‘†๐œ† is one-to-one from Theorem 3.22, ๐‘๐‘˜โˆ’1๐‘”=0. By indication we have ๐‘”=0. Hence ๐‘‡๐œ† is one-to-one.

An operator ๐‘‡โˆˆโ„’(๐ป) is said to be the following.(1)It is left invertible if there is an operator ๐‘†โˆˆโ„’(๐ป) such that ๐‘†๐‘‡=๐ผ, where ๐ผ denotes the identity operator. The operator ๐‘† is called a left inverse of ๐‘‡.(2)It is right invertible if there is an operator๐‘…โˆˆโ„’(๐‘‹) such that ๐‘‡๐‘…=๐ผ. The operator ๐‘… is called a right inverse of ๐‘‡ (see [16]).

Corollary 3.25. If ๐‘‡ is of class [2QN] with the property ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ…, and if ๐‘† is a left invertible operator with the left inverse ๐‘…, then the operator (๐‘†๐‘‡๐‘…)๐œ†โˆถ๐‘Š๐‘š2(๐ท,๐ป)โ†’๐‘Š๐‘š2(๐ท,๐ป) is one-to-one.

Proof. If ๐‘”โˆˆ๐‘Š๐‘š2(๐ท,๐ป) is such that (STR)๐œ†๐‘”=0, then ๐‘‡๐œ†๐‘…๐‘”=0.(3.27) Hence for ๐‘—=0,1,โ€ฆ,๐‘š we have ๐‘‡๐œ†๐‘…๐œ•๐‘—๐‘”=0. From Theorem 3.22, we get ๐‘…๐œ•๐‘—๐‘”=0 for ๐‘—=0,1,โ€ฆ,๐‘š.
Thus, STR๐œ•๐‘—๐‘”=0 for ๐‘—=0,1,โ€ฆ,๐‘š. It follows that ๐œ†๐œ•๐‘—๐‘”=0 for ๐‘—=0,1,โ€ฆ,๐‘š. By application of [7, Proposition 2.1] with ๐‘‡=(0), we have โ€–(๐ผโˆ’๐‘ƒ)๐‘”โ€–2,๐ท=0,(3.28) where ๐‘ƒ denotes the orthogonal projection of ๐ฟ2(๐ท,๐ป) onto the Bergman space ๐ด2(๐ท,๐ป). Hence ๐œ†๐‘”=๐œ†๐‘ƒ๐‘”=0. From [17, Corollary 10.7], there exists a constant ๐‘>0 such that ๐‘โ€–๐‘ƒ๐‘”โ€–2,๐ทโ‰คโ€–๐œ†๐‘ƒ๐‘”โ€–2,๐ท.(3.29) So ๐‘”=๐‘ƒ๐‘”=0. Thus, (STR)๐œ† is one-to-one.

Corollary 3.26. If ๐‘‡ is of class [2QN] with the property ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ…, and if ๐‘† is a right invertible operator with the right inverse ๐‘… then the operator (๐‘…๐‘‡๐‘†)๐œ†โˆถ๐‘Š๐‘š2(๐ท,๐ป)โ†’๐‘Š๐‘š2(๐ท,๐ป), is one-to-one.

Now we are ready to prove our main theorem.

Theorem 3.27. If ๐‘‡ is of class [2QN] with the property that ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ… and let ๐ท be a bounded disk which contains ๐œŽ(๐‘‡), then the operator ๐‘‰โˆถ๐ปโ†’๐ป(๐ท), defined by ๐‘‰๐‘”=1โŠ—๐‘”+๐‘‡๐œ†๐‘Š๐‘š2๎„ž(๐ท,๐ป)=[1โŠ—๐‘”],(3.30) is one to one and has closed range, where ๐ป(๐ท) is as in (2.11).

Proof. First, we will prove that if {๐‘”๐‘˜}โˆž1โŠ‚๐ป and {๐‘“๐‘˜}โˆž1โŠ‚๐‘Š๐‘š2(๐ท,๐ป) are sequences such that lim๐‘˜โ†’โˆžโ€–โ€–1โŠ—๐‘”๐‘˜+๐‘‡๐œ†๐‘“๐‘˜โ€–โ€–๐‘Š๐‘š2=0,(3.31) then lim๐‘˜โ†’โˆž๐‘”๐‘˜=0.
By the definition of the norm of a Sobolev space, (3.31) implies that lim๐‘˜โ†’โˆžโ€–โ€–๐‘‡๐œ†๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ท=0for๐‘—=1,โ€ฆ,๐‘š.(3.32) From (3.32) we get lim๐‘˜โ†’โˆžโ€–โ€–๐‘‡2๐œ†2๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ท=0for๐‘—=1,โ€ฆ,๐‘š.(3.33)
Since ๐‘‡2 is normal, lim๐‘˜โ†’โˆžโ€–โ€–(๐‘‡2๐œ†2)โˆ—๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ท=0for๐‘—=1,โ€ฆ,๐‘š.(3.34) Since ๐‘‡๐œ† is invertible for ๐œ†โˆˆ๐ทโงต๐œŽ(๐‘‡), (3.32) implies that lim๐‘˜โ†’โˆžโ€–โ€–๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ทโงต๐œŽ(๐‘‡)=0.(3.35) Therefore lim๐‘˜โ†’โˆžโ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ทโงต๐œŽ(๐‘‡)=0for๐‘—=1,โ€ฆ,๐‘š.(3.36)
Since ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(T))=โˆ… and ๐œŽ(๐‘‡โˆ—)=๐œŽ(๐‘‡)โˆ—, it is clear that (๐‘‡โˆ’๐œ†)โˆ— is invertible for ๐œ†โˆˆ๐œŽ(๐‘‡). Therefore from (3.34), we have lim๐‘˜โ†’โˆžโ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐œŽ(๐‘‡)=0.(3.37) Hence, from (3.36) and (3.37) we get lim๐‘˜โ†’โˆžโ€–โ€–(๐‘‡๐œ†)โˆ—๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ท=0,๐‘—=1,โ€ฆ,๐‘š.(3.38) Then by Proposition 3.21, we have lim๐‘˜โ†’โˆžโ€–โ€–(๐ผโˆ’๐‘ƒ)๐œ•๐‘—๐‘“๐‘˜โ€–โ€–2,๐ท=0,๐‘—=1,โ€ฆ,๐‘šโˆ’2.(3.39) By (3.31) and (3.39), we have lim๐‘˜โ†’โˆžโ€–โ€–1โŠ—๐‘”๐‘˜+๐‘‡๐œ†๐‘ƒ๐‘“๐‘˜โ€–โ€–2,๐ท=0.(3.40) Let ฮ“ be a curve in ๐ท surrounding ๐œŽ(๐‘‡). Then lim๐‘˜โ†’โˆžโ€–โ€–๐‘ƒ๐‘“๐‘˜+๎€ท๐‘‡๐œ†๎€ธโˆ’1๎€ท1โŠ—๐‘”๐‘˜๎€ธโ€–โ€–=0(3.41) uniformly for ๐œ†โˆˆฮ“ by (3.40). Hence by Riesz-Dunford functional calculus lim๐‘˜โ†’โˆžโ€–โ€–โ€–1๎€œ2๐œ‹๐‘–ฮ“๐‘ƒ๐‘“๐‘˜(๐‘ง)๐‘‘๐‘ง+๐‘”๐‘˜โ€–โ€–โ€–=0.(3.42) But by Cauchyโ€™s theorem ๎€œฮ“๐‘ƒ๐‘“๐‘˜(๐‘ง)๐‘‘๐‘ง=0.(3.43) Hence lim๐‘˜โ†’โˆž๐‘”๐‘˜=0. Thus the map ๐‘‰ is one-to-one and has closed range.

Corollary 3.28. If ๐‘‡ is of class [2QN] with the property that ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ…, then T is subscalar of order ๐‘šโ‰ฅ2.

Proof. Consider an arbitrary bounded open disk ๐ท in โ„‚ that contains ๐œŽ(๐‘‡) and the quotient space given in (2.11).
Let ๐‘€๐œ† be the multiplication operator by ๐œ† on ๐‘Š๐‘š2(๐ท,๐ป). Then ๐‘€๐œ† is a scalar operator of order ๐‘š, and its spectral distribution is ฮฆ๐‘€โˆถ๐ถ๐‘š0๎€ท๐‘Š(โ„‚)โŸถโ„’๐‘š2๎€ธ(๐ท;๐ป),ฮฆ๐‘€(๐‘“)=๐‘€๐‘“,(3.44) where ๐‘€๐‘“ is the multiplication operator by ๐‘“โˆˆ๐ถ๐‘š0(โ„‚). Let ๎‚Š๐‘€๐‘†โ‰ก๐œ†. Since ๐‘‡๐œ†๐‘Š๐‘š2(๐ท;๐ป) is invariant under every operator ๐‘€๐‘“, we infer that ๐‘† is a scalar operator of order ๐‘š with spectral distribution ๎ฮฆ.
Let ๐‘‰ be the operator ๐‘‰๐‘”=1โŠ—๐‘”+๐‘‡๐œ†๐‘Š๐‘š2(๐ท,๐ป)(3.45) from ๐ป into ๐ป(๐ท). Then we have the following commutative diagram 975745.equation.001(3.46) By the previous theorem the operator ๐‘‰ is a topological isomorphism of ๐ป into ๐‘…(๐‘‰). The relation ๐‘‰๐‘‡=๐‘†๐‘‰ shows that ๐‘…(๐‘‰) is ๐‘†-invariant. Hence ๐‘† is an extension of the operator ๐‘‰, so this operator is subscalar. Since ๐‘‰ is invertible on ๐‘…(๐‘‰), then the operator ๐‘‡ is subscalar of order ๐‘š. On the other hand from [18, Theorem 4.3] we deduce that ๐‘šโ‰ฅ2 and the theorem is proved.

Corollary 3.29. If ๐‘‡ is of class [2QN] with the property that ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ…, then ๐‘‡ has Bishopโ€™s property (๐›ฝ).

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 Bishopโ€™s property (๐›ฝ).

Proposition 3.30. Let ๐‘‡โˆˆโ„’(โŠ•๐‘˜๐‘–=1๐ป) be the following ๐‘˜ร—๐‘˜ triangular operator matrix โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘‡๐‘‡=11๐‘‡12๐‘‡13โ€ฆโ€ฆ๐‘‡1๐‘˜0๐‘‡22๐‘‡23โ€ฆโ€ฆ๐‘‡2๐‘˜00๐‘‡33โ€ฆโ€ฆ๐‘‡3๐‘˜000โ€ฆโ€ฆโ€ฆ0โ€ฆโ€ฆโ€ฆโ€ฆ๐‘‡๐‘˜๐‘˜โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(3.47) Assume that ๐‘‡๐‘–๐‘– is of class [2QN] and satisfies ๐œŽ(๐‘‡๐‘–๐‘–)โˆฉ(โˆ’๐œŽ(๐‘‡๐‘–๐‘–))=โˆ… for ๐‘–=1,2,โ€ฆ,๐‘˜โˆ’1 and ๐‘‡๐‘˜๐‘˜ is nilpotent. Then ๐‘‡ has Bishopโ€™s property (๐›ฝ).

Proof. Let ๐‘“๐‘=โŠ•๐‘˜๐‘–=1๐‘“๐‘–๐‘โˆถ๐ทโ†’โŠ•๐‘˜๐‘–=1๐ป be a sequence of analytic functions such that ๐‘‡๐œ†๐‘“๐‘(๐œ†)โ†’0 uniformly on every compact subset ๐พ of an open set ๐ท of โ„‚; then we have ๎€ท๐‘‡11๎€ธ๐œ†๐‘“1๐‘(๐œ†)+๐‘‡12๐‘“2๐‘(๐œ†)+๐‘‡13๐‘“3๐‘(๐œ†)โ‹ฏ+๐‘‡1๐‘˜๐‘“๐‘˜๐‘๎€ท๐‘‡(๐œ†)โŸถ0,22๎€ธ๐œ†๐‘“2๐‘(๐œ†)+๐‘‡23๐‘“3๐‘(๐œ†)+โ‹ฏ+๐‘‡2๐‘˜๐‘“๐‘˜๐‘๎€ท๐‘‡โŸถ0,33๎€ธ๐œ†๐‘“3๐‘(๐œ†)+โ‹ฏ+๐‘‡3๐‘˜๐‘“๐‘˜๐‘๎€ท๐‘‡(๐œ†)โŸถ0,โ‹ฎโ‹ฎโ‹ฎโ‹ฎ๐‘˜โˆ’1๐‘˜โˆ’1๎€ธ๐œ†๐‘“๐‘๐‘˜โˆ’1(๐œ†)+๐‘‡๐‘˜โˆ’1๐‘˜๐‘“๐‘˜๐‘(๎€ท๐‘‡๐œ†)โŸถ0,๐‘˜๐‘˜๎€ธ๐œ†๐‘“๐‘˜๐‘(๐œ†)โŸถ0.(3.48) Since ๐‘‡๐‘š๐‘˜๐‘˜=0,๐œ†๐‘‡๐‘šโˆ’1๐‘˜๐‘˜๐‘“๐‘˜๐‘(๐œ†)โ†’0 and hence ๐‘‡๐‘šโˆ’1๐‘˜๐‘˜๐‘“๐‘˜๐‘(๐œ†)โ†’0 if ๐œ†โ‰ 0. Since (๐‘‡๐‘˜๐‘˜)๐œ†๐‘“๐‘˜๐‘(๐œ†)โ†’0 from (3.48) ๐œ†๐‘‡๐‘šโˆ’2๐‘˜๐‘˜๐‘“๐‘˜๐‘(๐œ†)โ†’0. By the same reason, ๐‘‡๐‘šโˆ’3๐‘˜๐‘˜๐‘“๐‘˜๐‘(๐œ†)โ†’0. By repeating this procedure, we finally achieve ๐‘“๐‘˜๐‘(๐œ†)โŸถ0,(3.49) uniformly on ๐พ. Then we obtain the following equation: (๐‘‡๐‘˜โˆ’1๐‘˜โˆ’1)๐œ†๐‘“๐‘๐‘˜โˆ’1โ†’0 uniformly on every compact ๐พ. Since ๐‘‡๐‘˜โˆ’1๐‘˜โˆ’1 has Bishopโ€™s property (๐›ฝ) from Corollary 3.29, ๐‘“๐‘๐‘˜โˆ’1(๐œ†)โ†’0 uniformly on ๐พ. By repeating this process we prove that ๐‘“1๐‘(๐œ†)โ†’0 uniformly on ๐พ.
Hence {๐‘“๐‘=๐‘“1๐‘โŠ•๐‘“2๐‘โŠ•โ‹ฏโŠ•๐‘“๐‘˜๐‘} converge uniformly to 0 on any compact subset ๐พ of ๐ท, and so ๐‘‡ has the Bishopโ€™s property (๐›ฝ).

Proposition 3.31. Let ๐‘‡ be as in Proposition 3.30. Then if ๐‘‡๐‘–๐‘– has Bishopโ€™s property (๐›ฝ) for ๐‘–=1,โ€ฆ,๐‘˜, then ๐‘‡ has Bishopโ€™s property (๐›ฝ).

Proof. The proof is identical to the proof of Proposition 3.30.

Proposition 3.32. Let ๐‘‡โˆˆโ„’(โŠ•๐‘˜๐‘–=1๐ป) be the following ๐‘˜ร—๐‘˜ triangular operator matrix: โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘‡๐‘‡=11๐‘‡12๐‘‡13โ€ฆโ€ฆ๐‘‡1๐‘˜0๐‘‡22๐‘‡23โ€ฆโ€ฆ๐‘‡2๐‘˜00๐‘‡33โ€ฆโ€ฆ๐‘‡3๐‘˜000โ€ฆโ€ฆโ€ฆ0โ€ฆโ€ฆโ€ฆโ€ฆ๐‘‡๐‘˜๐‘˜โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(3.50) Assume that ๐‘‡ is of class [2QN] and ๐œŽ(๐‘‡)โˆฉ(โˆ’๐œŽ(๐‘‡))=โˆ…. If ๐‘‡11๐‘‡๐‘–๐‘—=๐‘‡๐‘–๐‘—๐‘‡๐‘—๐‘— for ๐‘–=1,โ€ฆ๐‘— and ๐‘—=1,2,โ€ฆ,๐‘˜, then ๐‘‡๐‘—๐‘— has Bishopโ€™s property (๐›ฝ) for ๐‘—=1,2,โ€ฆ,๐‘˜.

Proof. Let ๐‘“๐‘—๐‘โˆถ๐ทโ†’๐ป be a sequence of analytic functions such that (๐‘‡๐‘—๐‘—)๐œ†๐‘“๐‘—๐‘โ†’0 uniformly on every compact subset ๐พ of ๐ท, then we have for ๐‘—=1,2,โ€ฆ,๐‘˜๐‘‡๐œ†๎€ท๐‘‡๐‘–๐‘—๐‘“๐‘—๐‘๎€ธ=๎€ท๐‘‡(๐œ†)โŠ•โ‹ฏโŠ•011๎€ธ๐œ†๎€ท๐‘‡๐‘–๐‘—๐‘“๐‘—๐‘๎€ธ(๐œ†)โŠ•0โŠ•โ‹ฏโŠ•0=๐‘‡๐‘–๐‘—๎€ท๐‘‡๐‘—๐‘—๎€ธ๐œ†๐‘“๐‘—๐‘(๐œ†)โŠ•โ‹ฏโŠ•0โŸถ0,(3.51) for ๐‘–=1,2,โ€ฆ,๐‘—. Since ๐‘‡ has Bishopโ€™s property (๐›ฝ), we get that ๐‘‡๐‘–๐‘—๐‘“๐‘—๐‘(๐œ†)โ†’0 uniformly on ๐พ for ๐‘—=1,2,โ€ฆ,๐‘˜. We have ๐‘‡๐œ†๎€ท๐‘“1๐‘๎€ธ=๎€ท๐‘‡โŠ•โ‹ฏโŠ•011๎€ธ๐œ†๐‘“1๐‘(๐œ†)โŠ•โ‹ฏโŠ•0โŸถ0,(3.52) and for ๐‘—=2,3,โ€ฆ,๐‘˜, ๐‘‡๐œ†๎€ท0โŠ•โ‹ฏโŠ•๐‘“๐‘—๐‘๎€ธ=๎€ทโŠ•โ‹ฏโŠ•0โˆ’๐‘‡1๐‘—๐‘“๐‘—๐‘๎€ธ๎€ท(๐œ†)โŠ•โ‹ฏโŠ•โˆ’๐‘‡๐‘—โˆ’1๐‘—๐‘“๐‘—๐‘๎€ธโŠ•๎€ท๐‘‡(๐œ†)๐‘—๐‘—๎€ธ๐œ†๎€ท๐‘“๐‘—๐‘๎€ธ(๐œ†)โŠ•0โŠ•โ‹ฏโŠ•0โŸถ0.(3.53) Since ๐‘‡ has Bishopโ€™s property (๐›ฝ),๐‘“๐‘—๐‘(๐œ†)โ†’0 uniformly on ๐พ for ๐‘—=1,2,โ€ฆ,๐‘˜. Thus, ๐‘‡๐‘—๐‘— has Bishopโ€™s property (๐›ฝ).

Proposition 3.33. Let ๐‘‡ be as in Proposition 3.31. Then if ๐‘‡ has Bishopโ€™s property (๐›ฝ) and ๐‘‡11๐‘‡๐‘–๐‘—=๐‘‡๐‘–๐‘—๐‘‡๐‘—๐‘— for ๐‘—=1,2,โ€ฆ,๐‘˜ and ๐‘—=1,2,โ€ฆ,๐‘˜, then ๐‘‡๐‘—๐‘— has Bishop's property (๐›ฝ) for ๐‘—=1,2,โ€ฆ,๐‘˜.

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

4. Berberian Extension

Denote by โˆž(๐ป) the space of all sequences (๐‘ฅ๐‘›)๐‘›, with ๐‘ฅ๐‘›โˆˆ๐ป,๐‘›=1,2,3,โ€ฆ such that โ€–๐‘ฅ๐‘›โ€– is bounded. Let ๐‘0(๐ป) denote the subspace of all null sequences of ๐ป (those such that โ€–๐‘ฅ๐‘›โ€–โ†’0). If we set โ€–(๐‘ฅ๐‘›)โ€–=supโ€–๐‘ฅ๐‘›โ€– for every sequence (๐‘ฅ๐‘›), this defines a seminorm on ๐‘™โˆž(๐ป), which is zero exactly on the elements of ๐‘0(๐ป). By means of the space โˆž(๐ป) and the Banach limits, Berberian [20] constructed an extension ๐ปโˆ˜ of ๐ป and obtained a homomorphism form operators ๐‘‡โˆˆโ„’(๐ป) to operators ๐‘‡โˆ˜โˆˆโ„’(๐ปโˆ˜) such that ๐‘‡โˆ˜ is an extension of ๐‘‡.

Theorem 4.1 (Berberian extention [20]). Let ๐ป be a complex Hilbert space. Then there exists a Hilbert space ๐ปโˆ˜โŠƒ๐ป and a map ฮฆโˆถโ„’(๐ป)โŸถโ„’(๐ปโˆ˜)โˆถ๐‘‡โŸผ๐‘‡โˆ˜,(4.1) satisfying: ฮฆ which is an *-isometric isomorphism preserving the order such that(1)(๐‘‡โˆ—)โˆ˜=(๐‘‡โˆ˜)โˆ—, (2)(๐œ†๐‘‡+๐œ‡๐‘†)โˆ˜=๐œ†๐‘‡โˆ˜+๐œ‡๐‘†0, (3)(๐ผ๐ป)โˆ˜=๐ผ๐ปโˆ˜, (4)(๐‘‡๐‘†)โˆ˜=๐‘‡โˆ˜๐‘†โˆ˜, (5)โ€–๐‘‡โˆ˜โ€–=โ€–๐‘‡โ€–, (6)๐‘‡โˆ˜โ‰ค๐‘†โˆ˜๐‘–๐‘“๐‘‡โ‰ค๐‘†, (7)๐œŽ(๐‘‡โˆ˜)=๐œŽ(๐‘‡),๐œŽ๐‘Ž๐‘(๐‘‡)=๐œŽ๐‘Ž๐‘(๐‘‡โˆ˜)=๐œŽ๐‘(๐‘‡โˆ˜), (8)if ๐‘‡ is a positive operator, then (๐‘‡๐›ผ)โˆ˜=|๐‘‡โˆ˜|๐›ผforall๐›ผ>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 ๐‘‡ is an irreducible operator, then ๐‘‡โˆ˜ is an irreducible operator.

Lemma 4.3. Let ๐ท be a subset of โ„‚, ๐‘ง0,๐‘…>0, such that ๐ต(๐‘ง0,๐‘…)={๐‘งโˆˆโ„‚;|๐‘งโˆ’๐‘ง0|โ‰ค๐‘…}โŠ‚๐ท, let ๐‘”๐‘›โˆถ๐ทโ†’๐ป be a sequence of analytic functions, and let the Taylor expansion of ๐‘”๐‘› be ๐‘”๐‘›(๐‘ง)=โˆž๎“๐‘˜=0๐‘Ž๐‘›๐‘˜๎€ท๐‘งโˆ’๐‘ง0๎€ธ๐‘˜,||๐‘งโˆ’๐‘ง0||<๐‘….(4.2) If ๐‘”๐‘› is uniformly bounded on ๐ต(๐‘ง0,๐‘…)(i.e.,๐‘€=sup๐‘›โ‰ฅ1โ€–๐‘”๐‘›(๐‘ง)โ€–๐ต(๐‘ง0,๐‘…)<โˆž), then โ€–โ€–๐‘”๐‘›(๐‘ง)โˆ’๐‘”๐‘›๎€ท๐‘ง0๎€ธโ€–โ€–โ‰ค๐‘€๐‘Ÿ๐‘…โˆ’๐‘Ÿ,๐‘งโˆˆ๐ต(๐‘ง0,๐‘Ÿ),0<๐‘Ÿ<๐‘….(4.3)

Proof. For all ๐‘› and ๐‘งโˆˆ๐ต(๐‘ง0,๐‘Ÿ) with 0<๐‘Ÿ<๐‘…, by Cauchyโ€™s integral formula, we get the following inequality: โ€–โ€–๐‘”๐‘›(๐‘ง)โˆ’๐‘”๐‘›๎€ท๐‘ง0๎€ธโ€–โ€–=โ€–โ€–โ€–1๎€œ2๐‘–๐œ‹|๐‘ขโˆ’๐‘ง0|=๐‘…๐‘”๐‘›(๐‘ข)1๐‘ขโˆ’๐‘ง๐‘‘๐‘ขโˆ’๎€œ2๐‘–๐œ‹|๐‘ขโˆ’๐‘ง0|=๐‘…๐‘”๐‘›(๐‘ข)๐‘ขโˆ’๐‘ง0โ€–โ€–โ€–โ‰ค1๐‘‘๐‘ข๎€œ2๐œ‹|๐‘ขโˆ’๐‘ง0|=๐‘…||๐‘งโˆ’๐‘ง0||โ€–โ€–๐‘”๐‘›โ€–โ€–(๐‘ข)||๐‘ขโˆ’๐‘งโ€–๐‘ขโˆ’๐‘ง0||||||โ‰ค๐‘‘๐‘ข๐‘€๐‘Ÿ.๐‘…โˆ’๐‘Ÿ(4.4)

Remark 4.4. Let ๐ท be an open subset of โ„‚. A sequence of analytic functions ๐‘”nโˆถ๐ทโ†’๐ป converges uniformly to 0 on every compact subset ๐พ of ๐ท if and only if for any ๐œ–>0 and any ๐‘ง0โˆˆ๐ท there exists ๐‘Ÿ>0 and ๐‘›0โˆˆN such that ๐ต(๐‘ง0;๐‘Ÿ)โŠ‚๐ท and โ€–๐‘”๐‘›โ€–๐ต(๐‘ง0,๐‘Ÿ)<๐œ– for all ๐‘›>๐‘›0.

5. Single-valued Extension Property for ๐‘š-Partial Isometries

In this section we examine the properties of SVEP and Bishopโ€™s property (๐›ฝ) for some ๐‘š-partial isometries operators by using an approach which is different from that used in Section 3. We recall the definition of an ๐‘š-partial isometry given by (1.4) and the operator ๐ต๐‘š(๐‘‡).

Definition 5.1 (see [8]). An operator ๐‘‡โˆˆโ„’(๐ป) is called an ๐‘š-isometry if ๐ต๐‘š(๐‘‡)=๐‘š๎“๐‘˜=0(โˆ’1)๐‘˜โŽ›โŽœโŽœโŽ๐‘š๐‘˜โŽžโŽŸโŽŸโŽ ๐‘‡โˆ—๐‘šโˆ’๐‘˜๐‘‡๐‘šโˆ’๐‘˜=0.(5.1)

Remark 5.2. It is easy to see that ๐‘‡โˆˆโ„’(๐ป) is an ๐‘š-partial isometry if and only if ๐ต๐‘š(๐‘‡)๐‘ฅ=๐‘š๎“๐‘˜=0(โˆ’1)๐‘˜โŽ›โŽœโŽœโŽ๐‘š๐‘˜โŽžโŽŸโŽŸโŽ ๐‘‡โˆ—๐‘šโˆ’๐‘˜๐‘‡๐‘šโˆ’๐‘˜(๐‘ฅ)=0,โˆ€๐‘ฅโˆˆ๐‘(๐‘‡)โŸ‚,(5.2) which shows that the class of ๐‘š-partial isometries generalizes those of ๐‘š-isometries and partial isometries.

Theorem 5.3 (see [10]). If ๐‘‡โˆˆโ„’(๐ป) is reducible, that is, if it has a nontrivial reducing subspace ๐‘(๐‘‡), then the following properties are equivalent.(1)๐‘‡ is an ๐‘š-partial isometry.(2)๐‘‡|๐‘(๐‘‡)โŸ‚ is an ๐‘š-isometry.

Proposition 5.4 (see [10]). Let ๐‘‡ be a reducible ๐‘š-partial isometry. Then(1)๐œ†โˆˆ๐œŽap(๐‘‡)โงต{0} implies ๐œ†โˆˆ๐œŽap(๐‘‡โˆ—), that is, if ๐‘‡๐œ†๐‘ฅ๐‘›โ†’0 for some sequence of bounded vectors {๐‘ฅ๐‘›}โŠ‚๐ป, then (๐‘‡๐œ†)โˆ—๐‘ฅ๐‘›โ†’0,(2)๐œ†โˆˆ๐œŽ๐‘(๐‘‡)โงต{0} implies ๐œ†โˆˆ๐œŽ๐‘(๐‘‡โˆ—),(3)eigenvectors of ๐‘‡ corresponding to distinct eigenvalues are orthogonal, that is, ๐‘(๐‘‡๐œ†)โŸ‚๐‘(๐‘‡๐œ‡) if ๐œ†,๐œ‡โˆˆ๐œŽ๐‘(๐‘‡),๐œ†โ‰ ๐œ‡.

Lemma 5.5. Let ๐‘‡ be a reducible ๐‘š-partial isometry and let ๐œ†,๐œ‡โˆˆ๐œŽ๐‘Ž๐‘(๐‘‡) and (๐‘ฅ๐‘›), (๐‘ฆ๐‘›) be sequences of bounded vectors in ๐ป such that ๐œ†โ‰ ๐œ‡ and โ€–โ€–๐‘‡๐œ†๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘‡โŸถ0,๐œ‡๐‘ฆ๐‘›โ€–โ€–โŸถ0(๐‘Ž๐‘ ๐‘›โŸถโˆž).(5.3) Then we have โŸจ๐‘ฅ๐‘›โˆฃ๐‘ฆ๐‘›โŸฉโŸถ0(as๐‘›โŸถโˆž).(5.4)

Proof. We may assume that ๐œ‡โ‰ 0. Then from Proposition 5.4(1) we have โ€–(๐‘‡๐œ‡)โˆ—๐‘ฆ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž. Hence, (๐œ†โˆ’๐œ‡)โŸจ๐‘ฅ๐‘›โˆฃ๐‘ฆ๐‘›โŸฉ=โˆ’โŸจ๐‘‡๐œ†๐‘ฅ๐‘›โˆฃ๐‘ฆ๐‘›โŸฉ+โŸจ๐‘ฅ๐‘›โˆฃ๎€ท๐‘‡๐œ‡๎€ธโˆ—๐‘ฆ๐‘›โŸฉโŸถ0,๐‘›โŸถโˆž,(5.5) which implies (5.4) in view of ๐œ†โ‰ ๐œ‡ and the proof is complete.

Theorem 5.6. Any reducible ๐‘š-partial isometry has SVEP.

Proof. Let ๐‘ˆ be a bounded subset of โ„‚ and let ๐‘“โˆถ๐‘ˆโ†’๐ป be an analytic function such that ๐‘‡๐œ†๐‘“(๐œ†)=0for๐œ†โˆˆ๐‘ˆ.(5.6) Since ๐‘(๐‘‡๐œ†)โŸ‚๐‘(๐‘‡๐œ‡),๐œ†โ‰ ๐œ‡ (Proposition 5.4(3)), we have โ€–๐‘“(๐œ†)โ€–2=lim๐œ‡โ†’๐œ†โŸจ๐‘“(๐œ†)โˆฃ๐‘“(๐œ‡)โŸฉ=0.(5.7) This shows that ๐‘“(๐œ†)=0.

Lemma 5.7. If ๐‘‡ is an ๐‘š-partial isometry, then ๐‘‡๐‘œ is also an ๐‘š-partial isometry.

Proof. It is a consequence of the properties of ๐‘‡๐‘œ (see Theorem 4.1).

Theorem 5.8. If ๐‘‡ is an ๐‘š-partial isometry with a nontrivial reducing space ๐‘(๐‘‡), then ๐‘‡๐‘œ has the single-valued extension property (SVEP).

Proof. To prove that ๐‘‡โˆ˜ has SVEP, let ๐œ†โˆˆ๐œŽ๐‘Ž๐‘(๐‘‡โˆ˜)โˆ’{0}. Since ๐œŽ๐‘Ž๐‘(๐‘‡โˆ˜)=๐œŽ๐‘Ž๐‘(๐‘‡) by Theorem 4.1, ๐œ†โˆˆ๐œŽ๐‘Ž๐‘(๐‘‡โˆ—)=๐œŽ๐‘Ž๐‘((๐‘‡โˆ˜)โˆ—). In particular if ๐œ†โˆˆ๐œŽ๐‘(๐‘‡โˆ˜), then ๐œ†โˆˆ๐œŽ๐‘((๐‘‡โˆ˜)โˆ—). Hence, ๐‘((๐‘‡๐œ†)โˆ˜)โŸ‚๐‘((๐‘‡๐œ‡)โˆ˜) for ๐œ† and ๐œ‡โˆˆ๐œŽ๐‘(๐‘‡โˆ˜) with ๐œ†โ‰ ๐œ‡. In a similar way as in the proof of Theorem 5.6, we can see that ๐‘‡โˆ˜ has SVEP.

Acknowledgment

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

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