On~means and the Laplacian of functions on Hilbert space

I. Ya. Dorfman

Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 181-196

Résumé

In his book Problemes concrets d'analyse fonctionnelle, Paul Levy 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 $$ Lf(a)=\lim_{\rho\to0}\frac{M(f,a,\rho)-f(a)}{\rho^2}, $$ 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 $\frac1n\sum_{i=1}^n\lambda_i(x)\to0$ uniformly in $x$ (§ 3). The invariant mean of such a function exists and is given by the formula $$ M(f,x,r)=f(x)+\int_0^r\rho M(\gamma,x,\rho)\,d\rho, $$ and its Laplacian is $Lf(a)=\frac{\gamma(a)}2$. 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 Levy formulas. Bibliography: 7 titles.