Isometry invariant Finsler metrics on Hilbert spaces

Abstract

In this paper we study isometry-invariant Finsler metrics on inner product spaces over R or C, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results for more specific classes of metrics, including the class of Riemannian metrics. Our main result states that for an isometry-invariant Finsler metric the only possible linear maps under which the metric is invariant are scalar multiples of isometries. Furthermore, we characterise the metrics invariant with respect to all linear maps of this type.

Invariant Hermitean Metric Generated By Reproducing Kernel Hilbert Spaces

“…The following theorem incorporates several results from [8]: Proposition 2.8, Corollary 3.4, Theorem 3.7 and Remark 3.8. We characterise unitary-invariant and congruencyinvariant Hermitean metric on G, i.e.…”

“…We supply the theorem with several of remarks and an example also taken from [8]. In the following section they will be adapted to the case when H * is a RKHS, as well as the theorem above.…”

“…Section 2 consists of preliminary material, including some notations concerning abstract functions of two variables, necessary basics of RKHS theory, geometry of complex domains and rigidity of sesqui-holomorphic functions. It also contains several results on unitary-invariant Hermitean metrics on Hilbert spaces taken from [8], which we will adapt to the case when the Hilbert space is a dual of a RKHS in the next section.…”