Representation Theorems for Operators on Free Banach Spaces of Countable Type

  • J. AguayoEmail author
  • M. Nova
  • J. Ojeda
Research Articles


This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

Key words

C-algebras representation theorems compact operators self-adjoint operators spectral measure and integration 


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  1. 1.
    J. Aguayo, M. Nova and K. Shamseddine, “Inner product on B*-algebras of operators on a free Banach space over the Levi-Civita field,” Indag. Math. (N.S.) 26 (1), 191–205 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Aguayo and M. Nova, “Compact and self-adjoint operators on free Banach spaces of countable type,” Advances in non-Archimedean Analysis, Contemp. Math. 665, 1–17 (Amer. Math. Soc., Providence, RI, 2016).MathSciNetzbMATHGoogle Scholar
  3. 3.
    V. Berkovich, Spectral Theory and Analytic Geometry over non-Archimedean Fields, Mathematical Surveys and Monograph 33 (AMS, 1990).Google Scholar
  4. 4.
    B. Diarra, “Bounded linear operators on ultrametric Hilbert spaces,” Afr. Diaspora J. Math. 8 (2), 173–181 (2009).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. K. Katsaras, “On p-adic vector measure space,” J. Math. Anal. Appl. 365 (1), 342–357 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Narici and E. Beckenstein, “A non-Archimedean inner product,” Contemp. Math. 384, 187–202 (Amer. Math. Soc., 2005).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. van Rooij, Non-Archimedean Functional Analysis (Marcel Dekker, New York, 1978).zbMATHGoogle Scholar
  8. 8.
    M. Vishik, “Non-Archimedean spectral theory,” J. Soviet Math. 30, 2513–2554 (1985).CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Departamento de Matemática, Facultad de Ciencias Físicas y MatemáticasUniversidad de ConcepciónConcepciónChile
  2. 2.Departamento de Matemática y Física Aplicadas, Facultad de IngenieríaUniversidad Católica de la Santísima ConcepciónConcepciónChile

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