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Some Classes of Operators on Partial Inner Product Spaces

  • Jean-Pierre AntoineEmail author
  • Camillo Trapani
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)

Abstract

Many families of function spaces play a central role in analysis, such as \(L^p\) 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.Ac tually 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 paper, we shall give an overview of pip-spaces and operators on them, defined globally.W e will discuss a number of operator classes, such as morphisms, projections or certain integral operators.W e also explain how a pip-space can be generated from a *-algebra of operators on a Hilbert space and we prove that, under natural conditions, every lattice of Hilbert spaces is obtained in this way.

Keywords

Partial inner product spaces function spaces, operators homomorphisms 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institut de Recherche en Mathématique et Physique (IRMP)Université catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Dipartimento di Matematica e InformaticaUniversità di PalermoPalermoItalia

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