Best approximation, unitary groups and orbits of compact self-adjoint operators

BOTTAZZI, TAMARA; VARELA, ALEJANDRO

Rio de Janeiro

Congreso; International Congress of Mathematicians (ICM-2018); 2018

IMPA

Let H be a separable Hilbert space, D(B(H)^ah) the set of anti-Hermitian bounded diagonal operators in some fixed orthonormal basis and K(H) the two-sided closed ideal of compact operators. We studied the group of unitary operatorsU_kd=\{u\in U(H): such that there exists D\in D(B(H)^ah), u-e^D \in K(H)\}in order to obtain a concrete description of short curves in unitary Fredholm orbits O_b=\{ e^K b e^{-K}:K\in K(H)\} of a compact self-adjoint operator b with spectral multiplicity one. We considered the rectifiable distance on O_b defined as the infimum of curve lengths measured with the Finsler metric determined by means of the quotient space K(H)^ah/D(K(H)^ah), where the suffix ah means anti-Hermitian and D(K(H)^ah) is the set of anti-Hermitian compact diagonal operators. Then, for every c\in O_b and any vector x in its tangent space T(O_b)_c, there exists a minimal lifting Z_0\in B(H)^ah (in the quotient norm, not necessarily compact) such that \gamma(t)=e^{t Z_0}ce^{-t Z_0} is a short curve on O_b in a certain interval. We show some examples satisfying the above, which motivated us to study the unitary group $\ukd$ mentioned before.