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Bulletin of the London Mathematical Society Advance Access originally published online on January 24, 2009
Bulletin of the London Mathematical Society 2009 41(1):103-108; doi:10.1112/blms/bdn111
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© 2009 London Mathematical Society

On a theorem of K. Schmidt

G. M. Feldman

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
Lenin Ave. 47
Kharkov 61103
Ukraine
http://www.ilt.kharkov.ua/bvi/index_e.html

Received 22 March 2008.

Let Y be a locally compact group, Aut(Y) be the group of topological automorphisms of Y and P(Y) be the set of continuous positive definite functions on Y which have unit value at the identity. A function {Phi} isin P (Y2) is said to be of product type if there are such functions {Phi}j isin P (Y) that {Phi}(u, v) = {Phi} 1(u){Phi}2(v). Define the mapping T: Y2 -> Y2 by the formula T(u, v) = (A1 uA2 v, A3 u A4 v), where Aj isin Aut(Y), and assume that T is a one-to-one transform. K. Schmidt proved: (i) if both {Phi}(u, v) and {Phi}(T(u, v)) are of product type, then the functions {Phi}j are infinitely divisible; (ii) if Y is Abelian, both {Phi}(u, v) and {Phi}(T(u, v)) are of product type, and {Phi}(u, v) != 0, then the functions {Phi}j are Gaussian. We show that statement (i), generally, is not valid, but K. Schmidt's proof holds true if {Phi}(u, v) != 0. We also give another proof of statement (ii). Our proof uses neither the Levy–Khinchin formula for a continuous infinitely divisible positive definite function nor (i) on which K. Schmidt's proof is based.

2000 Mathematics Subject Classification 43A35 (primary), 60B15, 62E10 (secondary).


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