Abstract. For a bounded linear operator $T$ on a Banach space, the $\beta $-spectrum $\sigma_{\beta }(T)$ is defined and then is used to determine Nagy's spectral residuum. Duality theory for the $\beta $-spectrum is established and is used to prove the spectral mapping theorem for $\sigma_{\beta }(T)$. The relation between property ($\beta $) and the local spectral approximation property is also discussed. In particular, it is shown that, among other things, each operator has the local spectral approximation property in the complement of its spectral residuum. A few examples to illustrate various spectral properties of operators are also given.

AMS Subject Classification (1991): 47A10, 47B40

Received November 30, 1995 and in revised form August 22, 1996. (Registered under 6131/2009.)