Bull. Sci. math. 129 (2005) 567–590
www.elsevier.com/locate/bulsci
A measure-theoretical approach to the nuclear
and inductive spectral theorems
Manuel Gadella a,∗
, Fernando Gómez b
a Departamento de Física Teórica, Facultad de Ciencias, Prado de la Magdalena s.n. 47005 Valladolid, Spain
b Departamento de Análisis Matemático, Facultad de Ciencias,
Prado de la Magdalena s.n. 47005 Valladolid, Spain
Received 17 February 2005; accepted 28 February 2005
Available online 22 April 2005
Abstract
The usual mathematical implementations for the generalized eigenvectors and eigenfunctions of a
spectral measure (or a normal operator) on a Hilbert space H use direct integral decompositions of the
space H or auxiliary subspaces Φ with their topology τΦ, so that the generalized eigenvectors belong
to the components of the direct integral or to the (anti)dual space Φ×, respectively. In this work the
Gelfand–Vilenkin description of the generalized eigenvectors, in terms of certain Radon–Nikodym
derivatives associated to the spectral measure, permit us to give new proofs of renewed inductive and
nuclear versions of the spectral theorem, casting new insight on the measure-theoretical nature of
these result
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Bull. Sci. math. 129 (2005) 567–590 www.elsevier.com/locate/bulsci A measure-theoretical approach to the nuclear and inductive spectral theorems Manuel Gadella a,∗ , Fernando Gómez b a Departamento de Física Teórica, Facultad de Ciencias, Prado de la Magdalena s.n. 47005 Valladolid, Spain b Departamento de Análisis Matemático, Facultad de Ciencias, Prado de la Magdalena s.n. 47005 Valladolid, Spain Received 17 February 2005; accepted 28 February 2005 Available online 22 April 2005 Abstract The usual mathematical implementations for the generalized eigenvectors and eigenfunctions of a spectral measure (or a normal operator) on a Hilbert space H use direct integral decompositions of the space H or auxiliary subspaces Φ with their topology τΦ, so that the generalized eigenvectors belong to the components of the direct integral or to the (anti)dual space Φ×, respectively. In this work the Gelfand–Vilenkin description of the generalized eigenvectors, in terms of certain Radon–Nikodym derivatives associated to the spectral measure, permit us to give new proofs of renewed inductive and nuclear versions of the spectral theorem, casting new insight on the measure-theoretical nature of these result