Spectral shorted operators

Abstract

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

19 pages