B-spline interpolation problem in Hilbert C*-modules

Abstract

We introduce the $B$-spline interpolation problem corresponding to a $C^{\ast }$-valued sesquilinear form on a Hilbert $C^{\ast }$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^{\ast }$-module is self-dual. Extending a bounded $C^{\ast }$-valued sesquilinear form on a Hilbert $C^{\ast }$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^{\ast }$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^{\ast }$-valued sesquilinear to the dual ${\mathcal{X}}^{\prime }$ of the Hilbert $W^{\ast }$-module $\mathcal{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^{\ast }$-module $\mathcal{X}$ is orthogonally complemented with respect to another $C^{\ast }$-inner product on $\mathcal{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^{\ast }$-modules over $C^{\ast }$-ideals of $W^{\ast }$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.

25 pages, final version, to appear in J. Operator Theory

References in corpus (1)

• Orthogonality-preserving, C*-conformal and conformal module mappings on Hilbert C*-modules