Weyl's theorem for f(T{\hskip1}) when T is a dominant operator

Ko, Eungil; Lee, Hong Youl

doi:10.1017/s0017089501030154

Cited by 3 publications

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“…The class of M -hyponormal operators has been studied by many authors (cf. [1][2][3], [5][6][7][8][9][10][11][12]). However examples of M -hyponormal non-hyponormal operators seem to be scarce from the literature.…”

Section: Introductionmentioning

confidence: 99%

On M-hyponormal weighted shifts

Ham

Lee

2003

Journal of Mathematical Analysis and Applications

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Section: Introductionmentioning

confidence: 99%

On M-hyponormal weighted shifts

Ham

Lee

2003

Journal of Mathematical Analysis and Applications

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Dunford's property (C) and Weyl's theorems

Duggal

Djordjević

2002

Integr equ oper theory

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This paper studies Weyl's theorems, and some related results for operators with Dunford's property (C). Weyl's theorem in some classes of operators (e.g. Mhyponormal, p-hyponormal and totally paranormal operators) is considered. NotationLet X and Y be infinite dimensional complex Banach spaces and let L(X, Y)(L(X)) denote the algebra of bounded linear transformations (operators) on X into Y (respectively, X into itself). An operator T E L(X) satisfies (Dunford's) condition (C) if XT(F) = {x E X : there exists an analytic X-valued f : C\F --~ Z such that(T-A)f(A) = x} is closed for every subset F C C. T is said to have the single valued extension property (shortened to, SVEP) at Ao C C if, for a neighborhood U of Ao, f = 0 is the only analytic function f : U -* X satisfying (T -A)f(A) = 0; we say that T has the SVEP if T has this property at every A E C. It is known that (C) ==:v SVEP (see [13, Proposition 1.2]).If T E B(X) write Ker (T) and R(T) for the null space and the range of T, respectively. If R (T) is closed, let a(T) = dim Ker (T) and f~(T) = dim (X/R (T)). We say that the operator T E B(X) is semi-Fredholm if R(T) is closed and either Ker (T) is finite dimensional or its range has finite co-dimension. The index of a semi-Fredholm operator T E B(X) is defined by ind (T) = a(T) -f~(T).Let crp(T), 7r00(T) and 7r0(T) denote, respectively, the point spectrum of T, the set of isolated A E ap(T) with finite multiplicity and the set of Riesz points of T (i.e., the set of A E a(T) for which the corresponding spectral projection has finite dimensional range).Let a~(T), a~(T) and a~F(T) denote the respectively Weyl, Browder and semi-Fredholm spectrum of T (see [6]). Recall that T obeys Weyl's theorem if c%(T) = cr(T) \ 7r00(T), and that T obeys Browder's theorem if cry(T) = ab(T) (see [7,17]). Duggal, Djordjevi6 291 Operators with property (C)An operator S E B(Y, X) is said to be a quasi-ajfinity if S is injective and has dense range. The operators A E B(X) and B E B(Y) are said to be quasisimilar, denoted A ,-~ B, if there exist quasiaffinities T E B(X, Y) and S E B(Y, X) such that AS = SB and BT = TA. LEMMA 2.1. Let AS = SB, where A E B(X) satisfies condition (C) and S E B(Y, X) is a quasi-a]finity. Then B E B(Y) has the SVEP, B satisfies Browder's theorem (i.e., a(S) \ ~rb(S) = ~ro(B)) and a(A) C a(S).Proof. Let v be a fixed point in X , and let ul(.) and u2(.) be two analytic extensions of (B -A)-lv with common domain E. Then, since A satisfies (C), A has SVEP and (A -A)Yu,()~) = Y(B -~)u~(~) = Yv for all ~ E E and i --1, 2. This implies that Yu~()~) is an analytic extension of (A -~)-lYv for all A E E and i = 1,2. Hence, since YuI(A) = Yu2(A) for all A E E, it follows that B has SVEP. Now, let A E c~(B)\ ~(B). Then B-A is a Fredholm operator with index zero, but B -), is not invertible. Also, A is not an interior point of a(B). For suppose to the contrary that A is an interior point of a(B). Then there exists a neighborhood U~ of A such that dimA/(B -#) > 0 for every # E U~ . This however contradicts the fac...

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M-Hyponormal Judgment of a Class of Unilateral Weighted Shift Operators

代¹

2022

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Weyl's theorem for f(T{\hskip1}) when T is a dominant operator

Abstract: Abstract. Let T be a dominant operator that is a quasi-affine transform of an M-hyponormal operator. In this paper we show that if f is a function analytic on a neighborhood of the spectrum of T, then Weyl's theorem holds for fðT Þ.

Cited by 3 publications

References 6 publications

On M-hyponormal weighted shifts

On M-hyponormal weighted shifts

Dunford's property (C) and Weyl's theorems

M-Hyponormal Judgment of a Class of Unilateral Weighted Shift Operators

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