Abstract. Let $T$ be a bounded linear operator on a complex Hilbert space $H$. In this paper, we prove: (i) $T$ has property ($\beta $) if and only if $\widetilde{T}_{p,r}=|T|^pU|T|^r$ $(p+r=1)$ has property ($\beta $). (ii) If $T$ belongs to Class $wF(p,r,q)$ operators, and $\lambda $ is an isolated point of the spectrum of $T$, $E$ the Riesz idempotent, with respect to $\lambda $, of $T$, then $\mathop{\rm Ker} (T -\lambda )=EH$ if $\lambda\not=0$. (iii) Weyl's theorem and a-Browder's theorem hold for Class $wF(p,r,q)$ operators. (iv) The spectral mapping theorem holds for the Weyl spectrum of $T$ and for the essential approximate point spectrum of $T$.

AMS Subject Classification (1991): 47A10, 47B20

Keyword(s): wF(p, class, r, operators, q), single valued extension property, \beta, Bishop's property(), Weyl's theorem, a-Browder's theorem, Browder's theorem

Received October 19, 2006, and in revised form June 12, 2007. (Registered under 6017/2009.)