Analyzing Dunford Property for Operators Satisfying Sr2 TqSr2 = S2r

Abstract

In this study, we introduce a new class of operators defined by the properties S^(r/2) T^q S^(r/2) = S^(2r) and T^(r/2) S^q T^(r/2) = T^(2r) for integers r > q ≥ 0. The main objective is to investigate the Dunford property, commonly referred to as property (C), for the operators S^(r/2) T^q and T^q S^(r/2) under the condition that S^(2r) ∈ B(X). This research expands the framework of operator theory by introducing new operator classes through operator identities and extending existing ones. The motivation stems from the central role of operator equations in functional analysis and operator theory, where many fundamental problems in mathematics and physics can be reformulated as operator equations, yet certain classes remain insufficiently explored. The methodology involves an iterative analysis of local spectral subspaces and their interactions under the given operator identities. The results demonstrate that the introduced classes of operators satisfy the single-valued extension property (SVEP) and possess property (Q). Moreover, it is established that if S^(3r/2) has property (C), then both S^(r/2) T^q and T^q S^(r/2) inherit this property. These findings enrich the theory with broader generalizations and open avenues for further exploration of spectral properties and applications in mathematical and scientific contexts.