Upper Triangular Operator Matrices, SVEP, and Property ( w )
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Upper Triangular Operator Matrices, SVEP, and Property (w ) Mohammad H. M. Rashid1 Received: 5 January 2018 / Revised: 11 August 2018 / Accepted: 22 August 2018 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019
Abstract by MC an operator acting on When A ∈ L (X) and B ∈ L (Y) are given, we denote A C the Banach space X ⊕ Y of the form MC = . In this paper, first we prove that 0 B ∗ ∗ (A)∪S (B). Also, we σw (M0 ) = σw (MC )∪{S(A )∩S(B)} and σaw (MC ) ⊆ σaw (M0 )∪S+ + give the necessary and sufficient condition for MC to be obeys property (w). Moreover, we explore how property (w) survive for 2 × 2 upper triangular operator matrices MC . In fact, we prove that if A is polaroid on E 0 (MC ) = {λ ∈ iso σ (MC ) : 0 < dim(MC − λ)−1 }, M0 satisfies property (w), and A and B satisfy either the hypotheses (i) A has SVEP at points λ ∈ σaw (M0 ) \ σSF+ (A) and A∗ has SVEP at points μ ∈ σw (M0 ) \ σSF+ (A), or (ii) A∗ has SVEP at points λ ∈ σw (M0 ) \ σSF+ (A) and B ∗ has SVEP at points μ ∈ σw (M0 ) \ σSF+ (B), then MC satisfies property (w). Here, the hypothesis that points λ ∈ E 0 (MC ) are poles of A is essential. We prove also that if S(A∗ ) ∪ S(B ∗ ), points λ ∈ Ea0 (MC ) are poles of A and points μ ∈ Ea0 (B) are poles of B, then MC satisfies property (w). Also, we give an example to illustrate our results. Keywords Weyl’s theorem · Weyl spectrum · Polaroid operators · Property (w) · Upper triangular operator matrices · SVEP Mathematics Subject Classification (2010) Primary 47A55 · 47A53 · 47B20; Secondary 47A10 · 47A11
1 Introduction Throughout this paper, X and Y are Banach spaces and L (X, Y) denotes the space of all bounded linear operators from X to Y. For X = Y we write L (X, Y) = L (X). For T ∈ L (X), let T ∗ , ker(T ), (T ), σ (T ) and σa (T ) denote respectively the adjoint, the null space, the range, the spectrum and the approximate point spectrum of T . Let α(T ) and β(T ) be the Mohammad H. M. Rashid
malik [email protected] 1
Department of Mathematics & Statistics, Faculty of Science P. O. Box (7), Mu’tah University, Al-Karak, Jordan
M.H.M. Rashid
nullity and the deficiency of T defined by α(T ) = dim ker(T ) and β(T ) = co dim (T ). Recall that an operator T ∈ L (X) is called upper semi-Fredholm if α(T ) < ∞ and (T ) is closed, while T ∈ L (X) is called lower semi-Fredholm if β(T ) < ∞. Let + (X) and − (X) denote the class of all upper semi-Fredholm operators and the class of all lower semi-Fredholm operators, respectively. If T ∈ L (X) is either an upper or a lower semiFredholm operator, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T ) = α(T ) − β(T ). If both α(T ) and β(T ) are finite, then T is called a Fredholm operator. Let (X) denote the class of all Fredholm operators. Define − + (X) = {T ∈ + (X) : ind(T ) ≤ 0}. The class of Weyl operators is defined by W (X) = {T ∈ (X) : ind(T ) = 0}. The classes of operators defined above generate the following spectra: The Weyl spectrum is
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