Structure and frame properties of noncommutative shift-invariant spaces

PATERNOSTRO VICTORIA

Barranquilla

Conferencia; V Congreso latinoamericano de Matematicos; 2016

In this talk we will discuss the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We shall give characterizations of Riesz and frame sequences associated to group representations extending previous results for abelian groups and for cyclic subspaces of unitary representations of noncommutative discrete groups. For the cyclic case, we will see that the Gramian associated to an orbit of a single function and the Bracket map associated to the representation, agree. This will allow us to obtain all the characterization results for cyclic Riesz and frame sequences, previously proven in several context with very elaborated proofs, in a simply way. For the general case, that is, systems generated by more that one function, we will see that Riesz and frame properties are related to a notion for frame and Riesz basis in Hilbert modules endowed with inner products taking values in spaces of unbounded operators. The results presented in this talk are a joint work with Eugenio Hern\'andez and Davide Barbieri from Universidad Aut\'onoma de Madrid.