Integr. equ. oper. theory 47 (2003) 307–314 0378-620X/030307-8, DOI 10.1007/s00020-002-1164-1 c  2003 Birkh¨auser Verlag Basel/Switzerland Integral Equations and Operator Theory Weyl’s Theorem for Algebraically Paranormal Operators Ra´ ul E. Curto and Young Min Han Abstract. Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyl’s theorem holds for f (T ) for every f ∈ H(σ(T )); (ii) a-Browder’s theorem holds for f (S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T . Mathematics Subject Classification (2000). Primary 47A10, 47A53; Secondary 47B20. Keywords. Weyl’s theorem, Browder’s theorem, a-Browder’s theorem, algebraically paranormal operator, single valued extension property. 1. Introduction Throughout this note let B(H) and K(H) denote, respectively, the algebra of bounded linear operators and the ideal of compact operators acting on an infi- nite dimensional separable Hilbert space H. If T ∈B(H) we shall write N (T ) and R(T ) for the null space and range of T , respectively. Also, let α(T ) := dimN (T ), β(T ) := dimN (T * ), and let σ(T ), σ a (T ) and π 0 (T ) denote the spectrum, approxi- mate point spectrum and point spectrum of T , respectively. An operator T ∈B(H) is called Fredholm if it has closed range, finite dimensional null space, and its range has finite co-dimension. The index of a Fredholm operator is given by i(T ) := α(T ) − β(T ). T is called Weyl if it is Fredholm of index zero, and Browder if it is Fredholm “of finite ascent and descent:” equivalently ([Har2, Theorem 7.9.3]) if T is Fredholm and T − λ is invertible for sufficiently small |λ| > 0, λ ∈ C. The essential spectrum σ e (T ), the Weyl spectrum ω(T ) and the Browder spectrum σ b (T ) of T are defined by ([Har1],[Har2]) σ e (T )= {λ ∈ C : T − λ is not Fredholm}, The research of the first author was partially supported by NSF grants DMS-9800931 and DMS- 0099357.