Convolution operators and the discrete Laplacian

Abstract

In this thesis, we obtain new results for convolution operators on homogeneous spaces and give applications to the Laplacian on a homogeneous graph. Some of these results have been published in joint papers [13, 14] with my supervisor. Let be a homogeneous space of a locally compact group G and let T : Lp( ) ! Lp( ) be a convolution operator induced by a measure on G, where 1 p < 1. When is symmetric and absolutely continuous, we describe the L2- spectrum of T in terms of the Fourier transform of . An operator T is said to be hypercyclic if there is a vector x 2 Lp( ) such that the orbit fx; Tx; : : : ; Tnx; : : :g is dense in Lp( ). Given a positive weight w on , we consider the weighted convolution operator T ;w(f) = wT (f) on Lp( ) and study hypercyclic properties of T ;w. For a unit point mass , we show that T ;w is hypercyclic under some condition on the weight w. This condition is also necessary in the discrete case, and is equivalent to hereditary hypercyclicity of the operator. The condition can be strengthened to characterise topologically mixing weighted translation operators on discrete spaces. A weighted homogeneous graph is a homogeneous space of a discrete group G and the Laplacian L on can be viewed as a convolution operator. We can therefore apply the above result on L2-spectrum to describe the spectrum of L in terms of irreducible representations of G. We compare the eigenvalues of L with eigenvalues of the Laplacian on a regular tree, and obtain a Dirichlet eigenvalue comparison theorem. We also prove a version of the Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.