The Henstock-Kurzweil Functional Calculus on Self-Adjoint Operators
Marin Matei-Luca
TL;DR
The dissertation introduces a novel Henstock-Kurzweil functional calculus for self-adjoint operators, using regulated functions as a bridge between the continuous and Borel calculi. It develops both bounded and unbounded theories, proves a general spectral mapping theorem, and provides a spectral-dominant representation for solutions to abstract differential equations via HK integration. Key contributions include a Lipschitz, operator-norm-robust calculus that extends the continuous calculus, an explicit spectrum description for regulated functions, and applications to A.I.C.P. that connect spectral theory with semigroup methods. The work advances both the theory of functional calculi and their practical deployment in operator-valued differential equations, with clear avenues for extending to normal operators and contour-based HK integration.
Abstract
This dissertation focuses on developing a new construction of a functional calculus using Henstock-Kurzweil integration methods. The assignment of a functional calculus will be applied to self-adjoint operators. We will address both the bounded and unbounded cases, examine the advantage of the underlying function space compared to larger spaces, prove the spectral mapping theorem, and explore one application of this functional calculus in abstract differential equations.