Spectral shorted operators

Abstract

If H is a Hilbert space, S is a closed subspace of H, and A is a positive bounded linear operator on H, the spectral shorted operator ρ(S,A) is defined as the infimum of the sequence ∑(S,An)1/n, where denotes ∑(S,B) the shorted operator of B to S. We characterize the left spectral resolution of ρ(S,A) and show several properties of this operator, particularly in the case that dim S=1. We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.