Local spectral analysis of class wF(p, r, q) operators via Dunford property theory

Abstract

This paper investigates the local spectral properties of class (w mathcal{F}(p,r,q)\) operators. By establishing connections with a related operator-equation framework, we show that these operators satisfy important spectral properties, including the single-valued extension property (SVEP), property ((mathcal{Q})), and Dunford's property ((mathcal{C})). A key contribution is the explicit construction of the remainder operator (mathcal{R}) arising from the deficit in the defining inequalities, with bounds established under appropriate conditions. We address the parameter-domain mismatch between integer-power operator equations and real-parameter operator classes through the continuous functional calculus and the Riesz--Dunford holomorphic functional calculus. The main results show that, under the stated exact-transfer and class-theoretic conditions, the local spectral behavior of generalized Aluthge transformations is linked to that of the original operators. We also examine spectral inclusion relations, connections with established operator classes, and stability under powers. The analysis connects operator inequalities with local spectral theory, extending the existing framework and providing further insight into the structural properties of these operator classes.