On Means and the Laplacian of Functions on Hilbert Space

In his book Problèmes concrets d'analyse fonctionnelle, Paul Lévy introduced the concept of the mean M(f,\,a,\,\rho) of the function f on Hilbert space over the ball of radius \rho with center at the point a, and investigated the properties of the Laplacian \displaystyle Lf(a)=\frac{M(f,\,a,\,\rho)-f(a)}{\rho^2},but he did not determine which functions have means. Moreover, the mean M(f,\,a,\,\rho) and the Laplacian Lf(a) are not invariant, in general, under rotation about the point a.In the present paper we give a class of functions with invariant means on Hilbert space. An example of such a class is the set of functions f(x) for which f(x)=\gamma(x)I+T(x), where the function \gamma(x) is uniformly continuous and has invariant means, I is the identity operator, and T(x) is a symmetric, completely continuous operator whose eigenvalues, arranged in decreasing order of absolute value \lambda_j(x), have the property that (1/n)\sum_{i=1}^n\lambda_i(x)\to0 uniformly in x (§3). The invariant mean of such a function exists and is given by the formula \displaystyle M(f,\,x,\,r)=f(x)+\int_0^r\rho M(\gamma,\,x,\,\rho)d\rho,and its Laplacian is Lf(a)=\gamma(a)/2. In §4 we consider the Dirichlet problem and the Poisson problem for the ball and give sufficient conditions for the solution to be expressed by the Lévy formulas.Bibliography: 7 entries.