The book explores the integration of vector valued functions with vector valued measures, focusing on spectral measures and their applications in operator theory, scattering theory, and semiclassical approximation in quantum physics. It introduces innovative techniques for bilinear integration, addressing scenarios where traditional methods are ineffective.
Central to this work is the exploration of integration theory in vector spaces, closely linked to the representation of evolutions through functional integrals. While the book reflects the philosophical influence of Professor Kluvanek, it diverges from what his direct contributions might have shaped. The Feynman-Kac formula serves as a foundational concept, extending beyond its traditional use to suggest new mathematical structures, inspired by the role of path integrals in quantum physics. The focus remains on theoretical insights rather than practical applications.
Forming functions of operators is a basic task of many areas of linear analysis and quantum physics. Weyl’s functional calculus, initially applied to the position and momentum operators of quantum mechanics, also makes sense for finite systems of selfadjoint operators. By using the Cauchy integral formula available from Clifford analysis, the book examines how functions of a finite collection of operators can be formed when the Weyl calculus is not defined. The technique is applied to the determination of the support of the fundamental solution of a symmetric hyperbolic system of partial differential equations and to proving the boundedness of the Cauchy integral operator on a Lipschitz surface.