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- W750215164 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 Ω be a homogeneous space of a locally compact group G and let Tσ : L(Ω) → L(Ω) be a convolution operator induced by a measure σ on G, where 1 ≤ p <∞. When σ is symmetric and absolutely continuous, we describe the Lspectrum of Tσ in terms of the Fourier transform of σ. An operator T is said to be hypercyclic if there is a vector x ∈ L(Ω) such that the orbit {x, Tx, . . . , T x, . . .} is dense in L(Ω). Given a positive weight w on Ω, we consider the weighted convolution operator Tσ,w(f) = wTσ(f) on L (Ω) and study hypercyclic properties of Tσ,w. For a unit point mass σ, we show that Tσ,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 Ω of a discrete group G and the Laplacian L on Ω can be viewed as a convolution operator. We can therefore apply the above result on L-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 Schrodinger operator on an invariant homogeneous graph." @default.
- W750215164 created "2016-06-24" @default.
- W750215164 creator A5055201977 @default.
- W750215164 date "2009-01-01" @default.
- W750215164 modified "2023-09-27" @default.
- W750215164 title "Convolution operators and the discrete Laplacian" @default.
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- W750215164 hasPublicationYear "2009" @default.
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