Pivot duality of universal interpolation and extrapolation spaces

Christian Bargetz, Sven Ake Wegner

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Abstract

It is a widely used method, for instance in perturbation theory, to associate with a given C0-semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L2(R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff and already complete if we consider a C0-semigroup acting on a reflexive Banach space. If the space is Hilbert, then the inductive limit can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the latter duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space.

Original languageEnglish
Pages (from-to)321-331
Number of pages11
JournalJournal of Mathematical Analysis and Applications
Volume460
Issue number1
DOIs
Publication statusPublished - 2 Dec 2017

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Pivot
Extrapolation
Interpolation
Duality
Interpolate
Inductive Limit
Banach spaces
Projective Limit
Interpolation Spaces
Universal Space
C0-semigroup
Semigroup
Reflexive Banach Space
Hilbert spaces
Topology
Functional Calculus
Self-adjoint Operator
Perturbation Theory
Hilbert
Completion

Cite this

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Pivot duality of universal interpolation and extrapolation spaces. / Bargetz, Christian; Wegner, Sven Ake.

In: Journal of Mathematical Analysis and Applications, Vol. 460, No. 1, 02.12.2017, p. 321-331.

Research output: Contribution to journalArticle

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