Integral equalities for functions of unbounded spectral operators in Banach spaces

Silvestri, Benedetto 2008. Integral equalities for functions of unbounded spectral operators in Banach spaces. PhD Thesis, Cardiff University.

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Abstract

Let G be a complex Banach space, R an unbounded scalar type spectral operator in G, for example an unbounded self-adjoint operator in a Hilbert space, σ(R) its spectrum and E its resolution of identity. The main results of the thesis are the following ones. (1) Extension procedure leading from local equality (0.0.18) to global equallity (0.0.19) for integration with respect to the σ(B(G), N )−topology (Theorem 3.4.2 if N is an E−appropriate set and Corollary 3.4.3 if N is an E−appropriate set with the duality property). (2) Extension procedure leading from local equality (0.0.18) to global equality (0.0.19) for integration with respect to the sigma-weak topology ( Corollary 3.4.5 and Theorem 3.4.6) and for integration with respect to the weak operator topology (Corollary 3.4.4 and Theorem 3.4.7 or Theorem 2.2.10 and Corollary 2.2.11). (3) Newton-Leibnitz formula (0.0.3) for a suitable analytic map S for integration with respect to the σ(B(G), N )− topology, where N is an E−appropriate set with the duality property (Corollary 3.5.1 and Corollary 3.5.2); for integration with respect to the sigma-weak topology (Corollary 3.5.3) and for integration with respect to the weak operator topology (Corollary 3.5.4 and Theorem 2.3.6). (4) Differentiation formula (0.0.9) for a suitable analytic map S ( Theorem 2.3.2 and Theorem 2.3.4). (5) Formulas (0.0.26), (0.0.27) and (0.0.28) for the Frechet differential of a power ´ series in a Banach algebra in terms of commutants (Theorem 1.0.11). (6) A new proof for the resolvent formula (0.0.2) via formula (0.0.3) (Corollary 2.4.1).

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Additional Information:	Chapter 1 of the thesis has been published at https://doi.org/10.7494/opmath.2010.30.2.155. Chapter 2 and Chapter 3 of the thesis has been published at https://doi.org/10.4064/dm464-0-1. Please see Related URLs.
Funders:	EPSRC
Related URLs:	https://doi.org/10.7494/opmath.2010.30.2... https://doi.org/10.4064/dm464-0-1
Date of First Compliant Deposit:	5 June 2025
Last Modified:	05 Jun 2025 13:42
URI:	https://orca.cardiff.ac.uk/id/eprint/178814

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