Abstract. We consider various Weyl's theorems in connection with the continuity of the reduced minimum modulus, Weyl spectrum, Browder spectrum, essential approximate point spectrum and Browder essential approximate point spectrum. If $H$ is a Hilbert space, and $T\in B(H)$ is a quasihyponormal operator, we prove the spectral mapping theorem for the essential approximate point spectrum and for arbitrary analytic function, defined on some neighbourhood of $\sigma(T)$. Also, if $T^*$ is quasihyponormal, we prove that the $a$-Weyl's theorem holds for $T$.

AMS Subject Classification (1991): 47A53, 47B20

Keyword(s): Semi--Fredholm operators, essential spectra, Weyl's theorems, spectral continuity, quasihyponormal operators

Received August 15, 1996 and in revised form October 29, 1997. (Registered under 2661/2009.)