PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 8, Pages 2543–2547 S 0002-9939(02)06808-9 Article electronically published on November 27, 2002 A NOTE ON WEYL’S THEOREM FOR OPERATOR MATRICES SLAVI ˇ SA V. DJORDJEVI ´ C AND YOUNG MIN HAN (Communicated by Joseph A. Ball) Abstract. When A ∈B(X) and B ∈B(Y ) are given we denote by M C an operator acting on the Banach space X ⊕ Y of the form M C =  A C 0 B  , where C ∈B(Y,X). In this note we examine the relation of Weyl’s theorem for A ⊕ B and M C through local spectral theory. 1. Introduction Throughout this note let X and Y be Banach spaces, let B(X, Y ) denote the set of bounded linear operators from X to Y , and abbreviate B(X, X ) to B(X ). If T ∈B(X ) we shall write N (T ) and R(T ) for the null space and range of T . Also, let α(T ) := dim N (T ), β(T ) := dim X/R(T ), and let σ(T ), σ d (T ), σ a (T ) and π 0 (T ) denote the spectrum, defect spectrum, approximate point spectrum and point spectrum of T , respectively. An operator T ∈B(X ) is called upper semi-Fredholm if R(T ) is closed with finite dimensional null space and lower semi-Fredholm if R(T ) is closed with its range of finite co-dimension. If T is both upper semi- and lower semi-Fredholm, we call it Fredholm. The index of a Fredholm operator T is the integer i(T ) := α(T ) − β(T ). An operator T is called Weyl if it is Fredholm of index zero and is called Browder if it is Fredholm of “finite ascent and descent”. The essential spectrum σ e (T ), the Weyl spectrum ω(T ) and the Browder spectrum σ b (T ) of T are defined by ([6, 7]): σ e (T ) := {λ ∈ C : T − λ is not Fredholm}, ω(T ) := {λ ∈ C : T − λ is not Weyl}, σ b (T ) := {λ ∈ C : T − λ is not Browder}, evidently σ e (T ) ⊆ ω(T ) ⊆ σ b (T )= σ e (T ) ∪ acc σ(T ), where we write acc K for the accumulation points of K ⊆ C. If we write iso K := K \ acc K, then we let π 00 (T ) := {λ ∈ iso σ(T ):0 <α(T − λ) < ∞} denote the set of isolated eigenvalues of finite multiplicity. Received by the editors January 21, 2002 and, in revised form, March 27, 2002. 2000 Mathematics Subject Classification. Primary 47A10, 47A55. Key words and phrases. Upper triangular operator matrix, Weyl’s theorem, single valued ex- tension property. c 2002 American Mathematical Society 2543 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use