On Random Normal Operators and Their Spectral Measures

Abstract

The main aim of this paper is to introduce and study the subclass of not necessarily continuous, normal random operators, establishing connections with other subclasses of random operators, as well as with the existing concept of random projection operator-valued measure. Hence, after recalling some basic facts regarding random operators on a complex separable Hilbert space, theorems about transforming the class of not necessarily continuous decomposable random operators into the class of purely contractive random operators are proved. These are applied to obtain integral representations for not necessarily continuous normal or self-adjoint random operators on a Hilbert space with respect to the corresponding random projection operator-valued measures.

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Notes

  1. 1.

    Actually \(\varvec{A}\) is the closure of the restriction to the domain from (1.9) of a sort of random adjoint of \(A^\bullet \) (see Lemma and Theorem 1 from [8]).

  2. 2.

    Another way to define the composition of a pair of (continuous) random operators can be found in [18] Section 4.

  3. 3.

    In [24] (respectively, [25]) this measure was denoted by \(G_x\)

  4. 4.

    For a specific study of pure contractions, see [10]. Moreover, for \(H_1 = H_2 = H\), these form a special subclass of the class of completely non-unitary contractions introduced and studied in [16], for which the unitary dilation is a bilateral shift of multiplicity \(\dim H\) (see [16, Corollary 7.5]).

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Acknowledgements

I express hereby my gratitude to the referees for their careful reading of the manuscript and their comments and suggestions, which led to considerable improvement of the paper. This work was partially supported by the European Union through the European Regional Development Fund (ERDF) under the Competitiveness Operational Program (COP) through a project entitled Novel Bio-inspired Cellular Nano-architectures, Grant POC-A1.1.4-E-2015 nr. 30/01.09.2016.

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Correspondence to Păstorel Gaşpar.

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Gaşpar, P. On Random Normal Operators and Their Spectral Measures. J Theor Probab 32, 2088–2110 (2019). https://doi.org/10.1007/s10959-018-0870-9

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Keywords

  • Stochastic mappings
  • Random operator
  • Normal random operator
  • Self-adjoint random operator
  • Random projection operator-valued measure
  • Integral representation

Mathematics Subject Classification

  • 60G60
  • 46F12
  • 42B10