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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A pointwise spectrum and representation of operators
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by N. Bertoglio, Servet Martínez and Jaime San Martín PDF
Proc. Amer. Math. Soc. 126 (1998), 375-382 Request permission

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
  • MR Author ID: 120575
  • Email: smartine@dim.uchile.cl
  • Jaime San Martín
  • MR Author ID: 265399
  • Email: jsanmart@dim.uchile.cl
  • Received by editor(s): July 20, 1995
  • Received by editor(s) in revised form: April 30, 1996
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 375-382
  • MSC (1991): Primary 47A11, 47D15
  • DOI: https://doi.org/10.1090/S0002-9939-98-04428-1
  • MathSciNet review: 1459108