Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy ; Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, 00-950, Warszawa, Poland
Abstract:
Periodicity of bounded solutions for convolution equations on a separable abelian metric group is established, and related Liouville type theorems are obtained. A non-constant Borel and bounded harmonic function is constructed for an arbitrary convolution semigroup on any infinite-dimensional separable Hilbert space, generalizing a classical result by Goodman (1973).