Convolution operators and the discrete Laplacian
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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.
Authors
Chen, Chung-ChuanCollections
- Theses [3834]