Vortragsankündigung

Im Rahmen des Kolloquiums spricht
Herr Prof. Dr. Olaf Post (Universität Trier)

Distances between operators acting on different Hilbert spaces

ABSTRACT
In this talk we will define and compare several distances (or metrics) between operators acting
on different (separable) Hilbert spaces. We consider here three main cases of how to measure
the distance between two bounded operators: first by taking the distance between their unitary
orbits, second by isometric embeddings (this generalises a concept of Weidmann) and third by
quasi-unitary equivalence.
Our main result is that the unitary and isometric distances are equal provided the operators are
both self-adjoint and have 0 in their essential spectra. Moreover, the quasi-unitary distance is
equivalent (up to a universal constant) with the isometric distance for any pair of bounded
operators. The unitary distance gives an upper bound on the Hausdorff distance of their
spectrum. If both operators have purely essential spectrum, then the unitary distance equals
the Hausdorff distance of their spectra. Using a finer spectral distance respecting multiplicity of
discrete eigenvalues, this spectral distance equals the unitary distance also for operators with
essential and discrete spectrum. In particular, all operator distances mentioned above are
equal to this spectral distance resp. controlled by it in the quasi-unitary case for self-adjoint
operators with 0 in the essential spectrum. We also show that our results are sharp by
presenting various(counter-)examples (joint work with Sebastian Zimmer).
 

Der Vortrag findet statt am
Mittwoch, den 08.07.2026
um 17.15 Uhr im Raum W01 0-006
Kaffee/Tee um 16.45 Uhr im Raum W1 2-213

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