Banach Journal of Mathematical Analysis

Truncation Method For Random Bounded Self-Adjoint Operators

K. Kumar

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Abstract

This article addresses the following question; 'how to approximate the spectrum of random bounded self-adjoint operators on separable Hilbert spaces'. This is an attempt to establish a link between the spectral theory of random operators and the rich theory of random matrices; including various notions of convergence. This study tries to develop a random version of the truncation method, which is useful in approximating spectrum of bounded self-adjoint operators. It is proved that the eigenvalue sequences of the truncations converge in distribution to the eigenvalues of the random bounded self-adjoint operator. The convergence of moments are also proved with some examples. In addition, the article discusses some new methods to predict the existence of spectral gaps between the bounds of essential spectrum. Some important open problems are also stated at the end.

Article information

Source
Banach J. Math. Anal., Volume 9, Number 3 (2015), 98-113.

Dates
First available in Project Euclid: 19 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1419001705

Digital Object Identifier
doi:10.15352/bjma/09-3-7

Mathematical Reviews number (MathSciNet)
MR3296127

Zentralblatt MATH identifier
06430446

Subjects
Primary: 47B80: Random operators [See also 47H40, 60H25]
Secondary: 47H40: Random operators [See also 47B80, 60H25] 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)

Keywords
Random operators spectrum truncation

Citation

Kumar, K. Truncation Method For Random Bounded Self-Adjoint Operators. Banach J. Math. Anal. 9 (2015), no. 3, 98--113. doi:10.15352/bjma/09-3-7. https://projecteuclid.org/euclid.bjma/1419001705


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