Annals of Functional Analysis

Infinite-dimensional bicomplex Hilbert spaces

Raphaël Gervais Lavoie, ‎Louis Marchildon, and Dominic Rochon

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Abstract

‎This paper begins the study of infinite-dimensional‎ ‎modules defined on bicomplex numbers‎. ‎It generalizes‎ ‎a number of results obtained with finite-dimensional‎ ‎bicomplex modules‎. ‎The central concept introduced‎ ‎is the one of a bicomplex Hilbert space‎. ‎Properties‎ ‎of such spaces are obtained through properties of‎ ‎several of their subsets which have the structure of‎ ‎genuine Hilbert spaces‎. ‎In particular‎, ‎we derive the Riesz‎ ‎representation theorem for bicomplex continuous linear‎ ‎functionals and a general version of the bicomplex Schwarz‎ ‎inequality‎. ‎Applications to concepts relevant to quantum‎ ‎mechanics‎, ‎specifically the bicomplex analogue of the quantum‎ ‎harmonic oscillator‎, ‎are pointed out‎.

Article information

Source
Ann. Funct. Anal. Volume 1, Number 2 (2010), 75- 91.

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1399900590

Digital Object Identifier
doi:10.15352/afa/1399900590

Mathematical Reviews number (MathSciNet)
MR2772041

Zentralblatt MATH identifier
1216.46023

Subjects
Primary: 16D10: General module theory
Secondary: 30G35‎ ‎46C05‎ ‎46C50

Keywords
Bicomplex numbers ‎bicomplex quantum mechanics ‎Hilbert spaces ‎Banach algebras ‎bicomplex linear algebra

Citation

Gervais Lavoie, Raphaël; Marchildon, ‎Louis; Rochon, Dominic. Infinite-dimensional bicomplex Hilbert spaces. Ann. Funct. Anal. 1 (2010), no. 2, 75-- 91. doi:10.15352/afa/1399900590. https://projecteuclid.org/euclid.afa/1399900590


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