Partial inner product spaces and operators on them

Antoine, Jean-Pierre [UCL] Trapani, Camillo [Università di Palermo]

Many families of function spaces play a central role in analysis, in particular in signal processing (e.g. wavelet or Gabor analysis). Such are L^{p} spaces, Besov spaces, amalgam spaces or modulation spaces. In all such cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases of partial inner product spaces (PIP spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, frames on them should be defined globally, for the whole family, instead of individual spaces. In this paper, we shall give an overview of PIP spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. The interesting fact is that they allow a global definition of operators, and various operator classes on them have been defined.