On Extensions of Positive Definite Operator-Valued Functions on Ordered Groups
by
Mihaly Bakonyi
Georgia State University

Let G be an abelian group and let \Lambda be a subset of G. A function k:S=\Lambda-\Lambda --> L(H) (the algebra of bounded linear operators on a Hilbert space H) is called positive (semi)definite with respect to \Lambda if, for every finite subset \lambda₁, \lambda₂, ... , \lambda_n in \Lambda , the operator matrix {k(\lambda_i-\lambda_j)}_{i, j=1}ⁿ is positive (semi)definite. M.G.  Krein [Kre] proved that every positive definite continuous scalar function scalar function on a real interval (-a, a) admits a continuous positive definite extension to R. A.P.  Artjomenko [Art] provided a new proof for Krein's Extension Theorem without the continuity requirement. J. Friedrich and L. Klotz [FK] proved that, given 0 < a < \infty and a topological group G, then any strongly continuous positive definite function k:(-a, a)×G --> L(H) admits a positive definite extension to R×G. We generalize the latter result by omitting the continuity requirement and by substituting R with an ordered abelian group. The proof is based upon Artjomenko's approach. Several extension results for positive definite functions are presented as corollaries of our main result.

REFERENCES

[Art] A.P. Artjomenko, Hermitian positive functions and positive functionals, Dissertation, Odessa State University, 1941. Published in Teor. Funkcii, Funk. Anal. i Prilozen, Vol. 41(1983), 1-16; Vol. 42(1984), 1-21 (Russian).

[FK] J. Friedrich and L. Klotz, On extensions of positive definite operator-valued functions, Rep. Math. Phys., Vol. 26, No. 1(1988), 45-65.

[Kre] M.G. Krein, Sur le problème de prolongement des functions hermitiniennes positives et continues, Dokl. Akad. Nauk. SSSR, Vol. 26(1940), 17-22.

Date received: April 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-19.