On Random Normal Operators and Their Spectral Measures
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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.
KeywordsStochastic mappings Random operator Normal random operator Self-adjoint random operator Random projection operator-valued measure Integral representation
Mathematics Subject Classification60G60 46F12 42B10
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.
- 2.Gaşpar, P., Popa, L.: Stochastic mappings and random distribution fields. A correlation approach correlation approach. Monatshefte für Mathematik. https://doi.org/10.1007/s00605-018-1234-3
- 7.Hackenbroch, W.: Random Operators (M. Megan, N. Suciu, eds.), The National Conference on Mathematical Analysis and Applications,West University of Timişoara, pp. 135–146 (2000)Google Scholar
- 14.Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert space, Graduate Texts in Mathematics, Vol. 265. Springer, Dodrecht (2012)Google Scholar
- 16.Sz.-Nagy, B., Foiaş, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space. Springer, New York, Revised and Enlarged Edition, (2010)Google Scholar
- 21.Thang, D.H., Ng. Thinh, : Random bounded operators and their extension. Kyushu J. Math. 58(25), 257–276 (2004)Google Scholar