From graph to Riesz continuity

TL;DR

This paper demonstrates that graph continuous families of unbounded operators can be transformed into Riesz continuous families using unitary operators, simplifying the definition of the index and establishing homotopy equivalences for certain operator spaces.

Contribution

It introduces a method to convert graph continuous families into Riesz continuous families via unitary multiplication, and extends this to operators on Hilbert bundles and self-adjoint operators with compact resolvents.

Findings

Graph continuous families become Riesz continuous after unitary transformation.

The index for such families aligns with Ivanov's index, providing a simpler definition.

Homotopy equivalence between spaces of operators with different topologies is established.

Abstract

We show that every graph continuous family of unbounded operators in a Hilbert space becomes Riesz continuous after one-sided multiplication by an appropriate family of unitary operators. This result provides a simple definition of the index for graph continuous families of Fredholm operators, and we show that for such families this index coincides with the index defined by N. Ivanov in arXiv:2111.15081. This result also has two corollaries for operators with compact resolvents: (1) the identity map between the space of such operators with the Riesz topology and the space of such operators with the graph topology is a homotopy equivalence; (2) every graph continuous family of such operators acting between fibers of Hilbert bundles becomes Riesz continuous in appropriate trivializations of the bundles. For self-adjoint operators, multiplication by unitary operators should be replaced by conjugation. In general, a graph continuous family of self-adjoint operators with compact resolvents cannot be made Riesz continuous by an appropriate conjugation. We obtain a partial analogue of the "trivialization" result above for self-adjoint operators and describe obstructions to existence of such a trivialization in the general case. This motivates the notion of a polarization of a Hilbert bundle, and we prove a similar result for polarizations. These results are closely related to the recent work arXiv:2111.15081 of Ivanov and provide alternative proofs for some of his results. We then show that, under a minor assumption on the space of parameters and for operators which are neither essentially positive nor essentially negative, there is always a trivialization making the family Riesz continuous.