Abstract
The aim of this paper is to present a unified framework in the setting of Hilbert \(C^*\)-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and \(C^*\)-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the \(C^*\)-algebra of adjointable operators acting on a Hilbert \(C^*\)-module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert \(C^*\)-modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and present an interpolation theorem. Moreover, we study some basic properties of the so-called relative reproducing kernel Hilbert \(C^*\)-modules and give a characterization of dual modules. Among other things, we prove that every conditionally negative definite kernel gives us a reproducing kernel Hilbert \(C^*\)-module and a certain map. Several examples illustrate our investigation.
Similar content being viewed by others
References
Alpay, D., Jorgensen, P., Volok, D.: Relative reproducing kernel Hilbert spaces. Proc. Amer. Math. Soc. 142(11), 3889–3895 (2014)
Amyari, M., Chakoshi, M., Moslehian, M.S.: Quasi-representations of Finsler modules over \(C^*\)-algebras. J. Operator Theory 70(1), 181–190 (2013)
Arambašić, L., Bakić, D., Moslehian, M.S.: A treatment of the Cauchy-Schwarz inequality in \(C^*\)-modules. J. Math. Anal. Appl 381(2), 546–556 (2011)
Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)
Arveson, W.: An invitation to C\(^*\)-algebras, Graduate Texts in Mathematics, No. 39. Springer, New York-Heidelberg, (1976)
Ball, J.A., Marx, G., Vinnikov, V.: Noncommutative reproducing kernel Hilbert spaces. J. Funct. Anal. 271(7), 1844–1920 (2016)
Barreto, S.D., Bhat, B.V.R., Liebscher, V., Skeide, M.: Type I product systems of Hilbert modules. J. Funct. Anal. 212(1), 121–181 (2004)
Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, Graduate Texts in Mathematics. Springer, New York (1984)
Carmeli, C., De Vito, E., Toigo, A.: Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. Anal. Appl. (Singap.) 4(4), 377–408 (2006)
Fang, X., Moslehian, M.S., Xu, Q.: On majorization and range inclusion of operators on Hilbert \(C^*\)-modules. Linear Multilinear Algebra 66(12), 2493–2500 (2018)
Frank, M.: Self-duality and \(C^*\)-reflexivity of Hilbert \(C^*\)-moduli. Z. Anal. Anwendungen 9(2), 165–176 (1990)
Ghaemi, M., Manuilov, V., Moslehian, M. S.: Left multipliers of reproducing kernel Hilbert\(C^*\)-modules and the Papadakis theorem, J. Math. Anal. Appl. 505 (2022), no. 1, Paper No. 125471, 14 pp
Jorgensen, P.E.T.: Unbounded Hermitian operators and relative reproducing kernel Hilbert space. Cent. Eur. J. Math. 8(3), 569–596 (2010)
Kumari, R., Sarkar, J., Sarkar, S., Timotin, D.: Factorizations of kernels and reproducing kernel Hilbert spaces. Integral Eq. Oper. Theory 87(2), 225–244 (2017)
Lance, E.CHilbert C\(^*\)-Modules, London Math. Soc. Lecture Note Series, vol. 210, Cambridge Univ. Press, (1995)
Magajna, B.: Hilbert \(C^*\)-modules in which all closed submodules are complemented. Proc. Amer. Math. Soc. 125(3), 849–852 (1997)
Manuilov, V., Moslehian, M.S., Xu, Q.: Douglas factorization theorem revisited. Proc. Amer. Math. Soc. 148(3), 1139–1151 (2020)
Manuilov, V.M., Troitsky, E.V.: Hilbert C\(^*\)-modules, Translated from the 2001 Russian original by the authors. Trans. Math. Monog. 226. AMS, Providence, RI,( 2005)
Moslehian, M.S.: Conditionally positive definite kernels in Hilbert \(C^*\)-modules. Positivity 21(3), 1161–1172 (2017)
Moslehian, M.S., Bakherad, M.: Chebyshev type inequalities for Hilbert space operators. J. Math. Anal. Appl. 420(1), 737–749 (2014)
Mousavi, Z., Eskandari, R., Moslehian, M.S., Mirzapour, F.: Operator equations \(AX+YB=C\) and \(AXA^*+BYB^*=C\) in Hilbert \(C^*\)-modules. Linear Algebra Appl. 517, 85–98 (2017)
Murphy, G.J.: Positive definite kernels and Hilbert \(C^*\)-modules. Proc. Edinburgh Math. Soc. 40(2), 367–374 (1997)
Paschke, W.L.: Inner product modules over \(B^*\)-algebras. Trans. Amer. Math. Soc. 182, 443–468 (1972)
Paulsen, V.I., Raghupathi, M.: An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)
Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44(3), 522–536 (1938)
Szafraniec, F.H.: Multipliers in the reproducing kernel Hilbert space, subnormality and noncommutative complex analysis, Reproducing kernel spaces and applications, 313–331, Oper. Theory Adv. Appl., 143, Birkhäuser, Basel, (2003)
Szafraniec, F.H.: Murphy’s Positive definite kernels and Hilbert \(C^*\)-modules reorganized, Noncommutative harmonic analysis with applications to probability II, 275–295, Banach Center Publ., 89, Polish Acad. Sci. Inst. Math., Warsaw, (2010)
Takesaki, M.: Theory of Operator Algebras. I, Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences. 124. Operator Algebras and Non-commutative Geometry, 5. Springer, Berlin, (2002)
Acknowledgements
This research was supported by a grant from Ferdowsi University of Mashhad (No. 2/54679).
Author information
Authors and Affiliations
Ethics declarations
Conflict of Interest
The author states that there is no conflict of interest.
Data Availability Statement.
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Additional information
Communicated by Daniel Alpay.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the topical collection “Reproducing kernel spaces and applications” edited by Daniel Alpay.
Rights and permissions
About this article
Cite this article
Moslehian, M.S. Vector-Valued Reproducing Kernel Hilbert \(C^*\)-Modules. Complex Anal. Oper. Theory 16, 2 (2022). https://doi.org/10.1007/s11785-021-01179-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-021-01179-3
Keywords
- Conditionally negative definite kernel
- Reproducing kernel Hilbert module
- Hilbert \(C^*\)-module
- Kolmogorov decomposition