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 for measuring the distance between two bounded operators
: first, by taking the distance between their unitary-
-orbits; second, by isometric embeddings (this generalises a concept introduced by 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) to the isometric distance for any pair of bounded
operators. The unitary distance provides an upper bound on the Hausdorff distance of their
spectrum. If both operators have a purely essential spectrum, then the unitary distance equals
the Hausdorff distance of their spectra. Using a finer spectral distance that takes into account the multiplicity of
discrete eigenvalues, this spectral distance also equals the unitary distance for operators with
essential and discrete spectrum. In particular, all the operator distances mentioned above are
equal to this spectral distance, or 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).