We prove that if $f:Z^d\to R$ is harmonic and there exists a polynomial $W:Z^d\to R$ such that f + W is nonnegative, then f is a polynomial.

The Martin compactification of a bounded Lipschitz domain $D\subset {\mathbf{R}}^n$ is shown to be $\bar{D}$ for a large class of uniformly elliptic second order partial differential operators on $D$.Let $X$ be an open Riemannian manifold and let $M\subset X$ be open relatively compact, connected, with Lipschitz boundary. Then $\bar{M}$ is the Martin compactification of $M$ associated with the restriction to $M$ of the Laplace-Beltrami operator on $X$. Consequently an open Riemannian manifold $X$ has at most one compactification which is a compact Riemannian...

An operator-valued multi-variable Poisson type integral is studied. In Section 2 we obtain a new equivalent condition for the existence of a so-called regular unitary dilation of an n-tuple T=(T₁,...,Tₙ) of commuting contractions. Our development in Section 2 also contains a new proof of the classical dilation result of S. Brehmer, B. Sz.-Nagy and I. Halperin. In Section 3 we turn to the boundary behavior of this operator-valued Poisson integral. The results obtained in this section improve upon...