Let X be a complex in®nite dimensional Banach space. We use ' a T and ' ea T , respectively, to denote the approximate point spectrum and the essential approximate point spectrum of a bounded operator T on X. Also, % a T denotes the set iso ' a T n ' ea T . An operator T on X obeys the a-Browder's theorem provided that ' ea T ' a T n % a T . We investigate connections between the Browder's theorems, the spectral mapping theorem and spectral continuity. 1991 Mathematics Subject Classi®cation.

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... 10. Introduction. Let X be a complex in®nite-dimensional Banach space and let BX and KX denote respectively the algebra of bounded operators and the ideal of compact operators on X. If T P BX, then 'T denotes the spectrum of T and &T denotes the resolvent set of T. It is well known that the following sets form semigroups of semi±Fredholm operators on X: È X fT P BX X T is losed nd dim x T 'Ig and È À X fT P BX X T is losed nd dim XaT 'IgX