Measurability of Generalized Inverses of Random Linear Operators

Abstract

Let $( {\Omega ,\mathcal{B}} )$ be a measurable space, ${\rm X},Y$ be separable Hilbert spaces. Let T be a random linear operator from $\Omega \times {\rm X}$ into Y. Let $T^\dag ( \omega )$ denote the generalized inverse of $T( \omega )$, for $\omega \in \Omega $. Questions of measurability of $T^\dag ( \omega )$ are investigated in this paper, and in particular the following results are established: (i) If T is bounded, then $T^\dag $ is a random operator. (ii) If T is a closed operator with dense domain and if $T^\dag $ is bounded, then $T^\dag $ is a random operator under some mild restriction on the domains of $T( \omega )$ and $T^ * ( \omega ),\omega \in \Omega $. The results are applied to the measurability of best approximate solutions of random linear operator equations.