Let G be a locally compact group and K be a compact group of automorphisms of G. We consider the functional equation
where is a continuous function, is a weakly continuous function and (,) is a Hilbert space. This equation is a generalization of Gajda's functional equation of d'Alembert type. If is a solution of this equation, then the functions f and a are K-invariant and f is K-positive definite, i.e. the kernel
is positive definite. This kernel is the reproducing kernel of a Hilbert space of functions on G, and this implies several properties for f. If is of finite dimensional, we show that the general solution of this equation is of the form
where is an operator valued K-spherical function, with ( is the adjoint operator of ) and . As an application Chojnacki's and Stetkær's results on operator-valued spherical functions are used to give explicit solution formulas of this equation, in terms of strongly continuous unitary representations of G.
© 2012 by Walter de Gruyter Berlin Boston