Weyl's theorem for algebrascally $wF(p, r, q)$ operators with $p, q > 0$ and $q ≥ 1$ - Rashid, M. H. M. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
If $T$ or $T*$ is an algebraically $wF(p, r, q)$ operator with $p, r > 0$ and $q ≥ 1$ acting on an infinite-dimensional separable Hilbert space, then we prove that the Weyl theorem holds for $f(T)$, for every $f \in \text{Hol}(σ(T))$,where $ \text{Hol}(σ(T))$ denotes the set of all analytic functions in an open neighborhood of $σ(T)$. Moreover, if $T^*$ is a $wF(p, r, q)$ operator with $p, r > 0$ and $q ≥ 1$, then the $a$-Weyl theorem holds for $f(T)$. Also, if $T$ or $T^*$ is an algebraically $wF(p, r, q)$ operators with $p, r > 0$ and $q ≥ 1$, then we establish spectral mappingtheorems for the Weyl spectrum and essential approximate point spectrum of T for every $f \in \text{Hol}(σ(T))$, respectively. Finally, we examine the stability of the Weyl theorem and $a$-Weyl theorem under commutative perturbation by finite-rank operators.