2017
Том 69
№ 7

All Issues

Weyl's theorem for algebrascally $wF(p, r, q)$ operators with $p, q > 0$ and $q \geq 1$

Rashid M. H. M.

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Abstract

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 $f \in \text{Hol}(\sigma(T))$, where $ \text{Hol}(\sigma(T))$ denotes the set of all analytic functions in an open neighborhood of $\sigma(T)$. 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 mapping theorems for the Weyl spectrum and essential approximate point spectrum of T for every $f \in \text{Hol}(\sigma(T))$, respectively. Finally, we examine the stability of the Weyl theorem and $a$-Weyl theorem under commutative perturbation by finite-rank operators.

English version (Springer): Ukrainian Mathematical Journal 63 (2011), no. 8, pp 1256-1267.

Citation Example: Rashid M. H. M. Weyl's theorem for algebrascally $wF(p, r, q)$ operators with $p, q > 0$ and $q \geq 1$ // Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1092-1102.

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