Unitary subgroups and orbits of compact self-adjoint operators

作者：Bottazzi Tamara^*; Varela Alejandro

来源：Studia Mathematica, 2017, 238(2): 155-176.

DOI：10.4064/sm8690-12-2016

• unitary groups
• Lie subgroups
• unitary orbits
• geodesic curves
• minimal operators in quotient spaces

Let H be a separable Hilbert space, and let D(B(H)(ah)) be the anti-Hermitian bounded diagonal operators in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators
U-k,U-d = { u is an element of U(H) : there exists D is an element of D(B(H)(ah)); u -e(D) is an element of K(H) }
in order to obtain a concrete description of short curves in unitary Fredholm orbits O-b = { e(K) be(-K) : K is an element of K(H)(ah) } of a compact self-adjoint operator b with spectral multiplicity one. We consider the recti fi able distance on O-b defined as the in fi mum of curve lengths measured with the Finsler metric defined by means of the quotient space K (H)(ah) / D (K (H)(ah)). Then for every c is an element of O-b and x is an element of T-c (O-b) there exists a minimal lifting Z(0) is an element of B (H)(ah) (in the quotient norm, not necessarily compact) such that gamma(t) = e(tZ0) ce(-tZ0) is a short curve on O-b in a certain interval.

• 出版日期2017