Alex Grossmann, from Nested Hilbert Spaces to Partial Inner Product Spaces and Wavelets

Antoine, Jean-Pierre [UCL]

This chapter is devoted to the long collaboration between Alex Grossmann and I, the focal point being the notion of partial Inner Spaces, introduced around 1970. Next I will review the basic facts of the theory. Then I will explore all the consequences and applications of that notion, to operator theory, spectral analysis of self-adjoint operators, metric operators that generate lattices of Hilbert spaces, signal processing, etc. As a matter of fact, new developments are still arising today. In addition, we'll describe the interaction with Alex that lead to 2D continuous wavelets.