MULTIPLICATION OPERATORS ON THE ENERGY SPACE

We consider the multiplication operators on H-epsilon (the space of functions of finite energy supported on an infinite network), characterize them in terms of positive semidefinite functions. We show why they are typically not self-adjoint, and compute their adjoints in terms of a reproducing kernel. We also consider the bounded elements of H-epsilon and use the (possibly unbounded) multiplication operators corresponding to them to construct a boundary theory for the network. In the case when the only harmonic functions of finite energy are constant, we show that the corresponding Gel'fand space is the 1-point compactification of the underlying network.