Convolution-Dominated Matrices in Groups of Polynomial Growth

Abstract

In this thesis, we use a variation of a commutator technique to prove that l^p-stability is independent of p, for p greater than or equal to one, and for convolution-dominated matrices indexed by relatively separated sets in groups of polynomial growth. Moreover, from the inverse-closedness of the Schur matrices we deduce a Wiener type Lemma for the matrices in the intersection of the weighted convolution-dominated matrices, over all polynomial weights.
Finally, applications of the convolution-dominated matrices are presented. We prove the inverse-closedness of a non-commutative space generated by a discrete series representation restricted to a lattice in a nilpotent Lie group. In addition, we apply the aforementioned result on l^p-stability to show that if \pi(\Lambda)g is a p-frame for the coorbit space Co(L^p) for some p in [1,\infty], then \pi(\Lambda)g is a q-frame for the coorbit space Co(L^q) for each q in [1,\infty], where (\pi, H) is a discrete series representation of a group G of polynomial growth, \Lambda is a relatively separated set in G, and g is a vector in H such that the matrix coefficient V_g g is in the Amalgam space W_{w_a}(G). Moreover, we prove that the frame operator of the frame \pi(\Lambda)g is invertible on the coorbit spaces Co(L^p) for each p in [1,\infty], under the assumption that g is a vector in H, such that V_g g belongs in W_{w_a}(G) for each polynomial weight w_a.