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Journal of Operator Theory

Volume 77, Issue 2,  Spring  2017  pp. 481-501.

Essential spectrum and Fredholm properties for operators on locally compact groups

Authors:  Marius Laurentiu Mantoiu
Author institution: Departamento de Matematicas, Facultad de Ciencias, Universitad de Chile, Santiago, 7800003, Chile

Summary:  We study the essential spectrum and Fredholm properties of certain integral and pseudo-differential operators associated to non-commutative locally compact groups $\G$. The techniques involve crossed product $C^*$-alge\-bras. We extend previous results on the structure of the essential spectrum to self-adjoint operators belonging (or affiliated) to the Schr\"odinger representation of certain crossed products. When the group $\G$ is unimodular and type I, we cover a new class of global pseudo-differential differential operators with operator-valued symbols involving the unitary dual of $\G$. We use recent results of Nistor, Prudhon and Roch on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.

Keywords:  locally compact group, pseudo-differential operator, $C^*$-algebra, dynamical system, essential spectrum, Fredholm operator

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