Abstract
In previous works we analysed conditions for linearization of Hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real linear space generated by positive definite kernels. The aim of the present note is to find more concrete expressions of the Schwartz type conditions: in the Hamburger moment problem for Hankel type kernels on the free semigroup, in dilation theory (Stinespring type dilations and Haagerup decomposability), as well as in multi-variable holomorphy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Agler, J.E. McCarthy, Complete Nevanlinna-Pick kernels, J. Funct. Anal., 175 (2000), 111–124.
S. Albeverio, H. Gottschalk, J.-L. Wu, Models of local relativistic quantum fields with indefinite metric (in all dimensions), Commun. Math. Phys., 184 (1997), 509–531.
D. Alpay, Some remarks on reproducing kernel Krein spaces, Rocky Mountain J. Math., 21 (1991), 1189–1205.
W. Arveson, Subalgebras of C*-algebras. III. Multivariable operator theory, Acta Math., 181 (1998), 159–228.
D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Birkhäuser Verlag, Basel, (1997).
T. Constantinescu, Orthogonal polynomials in several variables. I, lanl, math. FA/0205333.
T. Constantinescu, A. Gheondea, Representations of Hermitian kernels by means of Krein spaces, Publ. RIMS. Kyoto Univ., 33 (1997), 917–951.
T. Constantinescu, A. Gheondea, Representations of Hermitian kernels by means of Krein spaces II. Invariant kernels, Commun. Math. Phys., 216 (2001), 409–430.
S. Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, (1999).
E. Effros, Zh.-J. Ruan, Operator Spaces, London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York (2000).
D.E. Evans, J.T. Lewis, Dilations of Irreducible Evolutions in Algebraic Quantum Theory, Dublin Institute for Advanced Studies, Dublin, (1977).
U. Haagerup, Injectivity and decomposition of completely bounded maps, in Operator Algebras and their Connections with Topology and Ergodic Theory (Buşteni, 1983), Lecture Notes in Math., Springer, Berlin, 1132 (1985), 170–222
G. Hofmann, On GNS representations on inner product spaces I. The structure of the representation space, Commun. Math. Phys., 191 (1998), 299–323.
G. Morchio, D. Pierotti, F. Strocchi, Infrared and vacuum structure in two-dimensional local quantum field theory models. The massless scalar field, J. Math. Phys., 31 (1990), 1467–1477.
J. Mujica, Complex analysis in Banach spaces, North-Holland, Amsterdam, (1986).
V.I. Paulsen,Completely bounded maps and dilations, Pitman Research Notes in Math., Longman, Wiley, New York, 146 (1986).
K.R. Parthasaraty, K. Schmidt, Positive-Definite Kernels, Continous Tensor Products and Central Limit Theorems of Probability Theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 272 (1972).
L. Schwartz, Sous espace Hilbertiens d'espaces vectoriel topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math., 13 (1964), 115–256.
F. Strocchi, Selected Topics on the General Properties of Quantum Field Theory, Lecture Notes Phys., World Scientific, 51 (1993).
G. Wittstock, Ein operatorwertiger Hahn-Banach Satz, J. Functional Analysis., 40 (1981), 127–150.
Author information
Authors and Affiliations
Corresponding author
Additional information
Prof. Tiberius Constantinescu died unexpectly on 29th of July 2005.
Rights and permissions
About this article
Cite this article
Constantinescu, T., Gheondea, A. On L. Schwartz's Boundedness Condition for Kernels. Positivity 10, 65–86 (2006). https://doi.org/10.1007/s11117-005-0010-5
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11117-005-0010-5