PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 3, Pages 715–722 S 0002-9939(01)06110-X Article electronically published on July 31, 2001 a-WEYL’S THEOREM FOR OPERATOR MATRICES YOUNG MIN HAN AND SLAVI ˇ SA V. DJORDJEVI ´ C (Communicated by Joseph A. Ball) Abstract. If M C = ( AC 0 B ) is a 2 × 2 upper triangular matrix on the Hilbert space H ⊕ K, then a-Weyl’s theorem for A and B need not imply a-Weyl’s theorem for M C , even when C = 0. In this note we explore how a-Weyl’s theorem and a-Browder’s theorem survive for 2 × 2 operator matrices on the Hilbert space. 1. Introduction Throughout this note let H and K be Hilbert spaces, let B(H, K) denote the set of bounded linear operators from H to K, and abbreviate B(H, H ) to B(H ) and let K(H ) denote the ideal of compact operators acting on H . If T ∈ B(H ) write N (T ) and R(T ) for the null space and range of T ; α(T ) = dimN (T ). An operator T ∈ B(H ) 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 ∈ B(H ) is the integer i(T )= α(T ) − α(T * ). An operator T ∈ B(H ) is called Weyl if it is Fredholm of index zero and is called Browder if it is Fredholm of “finite ascent and descent”. If T ∈ B(H ) write σ(T ) for the spectrum of T ; σ a (T ) for the approximate point spectrum of T ; π 0 (T ) for the set of eigenvalues of T ; π 00 (T ) for the isolated points of σ(T ) which are eigenvalues of finite multiplicity; π a 00 (T ) for the isolated points of σ a (T ) which are eigenvalues of finite multiplicity. The essential spectrum σ e (T ), the Weyl spectrum ω(T ) and the Browder spectrum σ b (T ) of T ∈ B(H ) are defined by ([10], [11]) σ e (T )= {λ ∈ C : T − λI is not Fredholm}, ω(T )= {λ ∈ C : T − λI is not Weyl}, σ b (T )= {λ ∈ C : T − λI is not Browder}; evidently σ e (T ) ⊆ ω(T ) ⊆ σ b (T )= σ e (T ) ∪ acc σ(T ), Received by the editors February 29, 2000 and, in revised form, August 25, 2000. 2000 Mathematics Subject Classification. Primary 47A50, 47A53. Key words and phrases. Weyl spectrum, essential approximate point spectrum, Browder es- sential approximate point spectrum, a-Weyl’s theorem, Weyl’s theorem, a-Browder’s theorem, Browder’s theorem. This work was supported by the Brain Korea 21 Project (through Seoul National University). c 2001 American Mathematical Society 715 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use