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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A pointwise spectrum
and representation of operators


Authors: N. Bertoglio, Servet Martínez and Jaime San Martín
Journal: Proc. Amer. Math. Soc. 126 (1998), 375-382
MSC (1991): Primary 47A11, 47D15
MathSciNet review: 1459108
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Abstract: For a self-adjoint operator $A:H\to H$ commuting with an increasing family of projections ${\mathcal{P}}=(P_{t})$ we study the multifunction $t\to \Gamma ^{\mathcal{T}}(t)=\bigcap \{\sigma _{I}:I$ an open set of the topology ${\mathcal{T}}$ containing $t\}$, where $\sigma _{I}$ is the spectrum of $A$ on $P_{I}H$. Let $m_{\mathcal{P}}$ be the measure of maximal spectral type. We study the condition that $\Gamma ^{\mathcal{T}}$ is essentially a singleton, $m_{\mathcal{P}}\{t:\Gamma ^{\mathcal{T}}(t)$ is not a singleton$\}=0$. We show that if ${\mathcal{T}}$ is the density topology and if $m_{\mathcal{P}}$ satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that $A$ is a Borel function of ${\mathcal{P}}$. If ${\mathcal{T}}$ is the usual topology then the condition is equivalent to the fact that $A$ is approched in norm by step functions $\sum \limits _{n\in \mathbb{N}}\Gamma ^{\mathcal{T}} (\alpha _{n})\langle P_{I_{n}} f,f\rangle$, where the set of intervals $\{I_{n}:n\in \mathbb{N}\}$ covers the set where $\Gamma ^{\mathcal{T}}$ is a singleton.


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Additional Information

N. Bertoglio
Affiliation: Facultad de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
Email: nbertogl@riemann.mat.puc.cl

Servet Martínez
Affiliation: Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Departamento de Ingeniería Matemática, Casilla 170-3, Correo 3, Santiago, Chile
Email: smartine@dim.uchile.cl

Jaime San Martín
Email: jsanmart@dim.uchile.cl

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04428-1
PII: S 0002-9939(98)04428-1
Received by editor(s): July 20, 1995
Received by editor(s) in revised form: April 30, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society