Abstract
In this paper we study the bicomplex version of weighted Hardy spaces. Further, we describe reproducing kernels for the bicomplex weighted Hardy spaces. In particular, we generalize some results which holds for the classical weighted Hardy spaces. We also introduce the notion of bicomplex C*-algebra and discuss some of its properties.
Similar content being viewed by others
References
Alpay, D., Luna-Elizarraras, M.E., Shapiro, M., Struppa, D.C.: Basics of functional analysis with bicomplex scalars, and bicomplex schur analysis. In: Springer Briefs in Mathematics. Springer, Berlin (2014)
Colombo F., Sabadin I., Struppa D.C., Vajiac A., Vajiac M.B.: Singularities of functions of one and several bicomplex variables. Ark. Mat. 49, 277–294 (2011)
Colombo F., Sabadini I., Struppa D.C.: Bicomplex holomorphic functional calculus. Math. Nachr. 287(13), 1093–1105 (2013)
Conway, J.B.: A course in functional analysis, 2nd edn. Springer, Berlin (1990)
Cowen, C.C., MacCluer, B.D.: Composition operators on spaces of analytic functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)
De Bie H., Struppa D.C., Vajiac A., Vajiac M.B.: The Cauchy–Kowalewski product for bicomplex holomorphic functions. Math. Nachr. 285(10), 1230–1242 (2012)
Gervais Lavoie R., Marchildon L., Rochon D.: Infinite-dimensional bicomplex Hilbert spaces. Ann. Funct. Anal. 1(2), 75–91 (2010)
Gervais Lavoie R., Marchildon L., Rochon D.: Finite-dimensional bicomplex Hilbert spaces. Adv. Appl. Clifford Algebr. 21(3), 561–581 (2011)
Kumar, R., Kumar, R., Rochon, D.: The fundamental theorems in the framework of bicomplex topological modules (2011). arxiv:1109.3424v1
Kumar, R., Singh, K.: Bicomplex linear operators on bicomplex Hilbert spaces and Littlewood’s subordination theorem. Adv. Appl. Clifford Algebr. doi:10.1007/s00006-015-0531-3
Luna-Elizarraras M.E., Perez-Regalado C.O., Shapiro M.: On linear functionals and Hahn-Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebr. 24, 1105–1129 (2014)
Luna-Elizarraras M.E., Shapiro M., Struppa D.C.: On Clifford analysis for holomorphic mappings. Adv. Geom. 14(3), 413–426 (2014)
Luna-Elizarraras M.E., Shapiro M., Struppa D.C., Vajiac A.: Bicomplex numbers and their elementary functions. Cubo 14(2), 61–80 (2012)
Luna-Elizarraras M.E., Shapiro M., Struppa D.C., Vajiac A.: Complex Laplacian and derivatives of bicomplex functions. Complex Anal. Oper. Theory 7, 1675–1711 (2013)
Price, G.B.: An introduction to multicomplex spaces and functions, 3rd edn. Marcel Dekker, New York (1991)
Riley J.D.: Contributions to the theory of functions of a bicomplex variable. Tohoku. Math. J. (2) 5(2), 132–165 (1953)
Rochon D., Shapiro M.: On algebraic properties of bicomplex and hyperbolic numbers. Anal. Univ. Oradea Fasc. Math. 11, 71–110 (2004)
Rochon D., Tremblay S.: Bicomplex quantum mechanics II: the Hilbert space. Adv. Appl. Clifford Algebr. 16(2), 135–157 (2006)
Shapiro J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)
Shields, A.L.: Weighted shift operators and analytic function theory. In: Topics in Operator Theory, Math. Surveys Monographs, vol. 13, pp. 49–128. American Mathematical Society, Providence (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of K. Singh is supported by CSIR-UGC (New-Delhi, India).
Rights and permissions
About this article
Cite this article
Kumar, R., Singh, K., Saini, H. et al. Bicomplex Weighted Hardy Spaces and Bicomplex C*-algebras. Adv. Appl. Clifford Algebras 26, 217–235 (2016). https://doi.org/10.1007/s00006-015-0572-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-015-0572-7