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Zbl 1216.46023
Gervais Lavoie, Raphaël; Marchildon, Louis; Rochon, Dominic
Infinite-dimensional bicomplex Hilbert spaces. (English)
[J] Ann. Funct. Anal. AFA 1, No. 2, 75-91, electronic only (2010). ISSN 2008-8752/e

Summary: This paper begins the study of infinite-dimensional modules defined on bicomplex numbers.\par 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.

MSC 2000:: *46C50 Generalizations of inner products
46C05 Geometry and topology of inner product spaces
16D10 General module theory (assoc. rings and algebras)
30G35 Functions of hypercomplex variables and generalized variables

Keywords: bicomplex numbers; bicomplex quantum mechanics; Hilbert spaces; Banach algebras; bicomplex linear algebra

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