  This note deals with a problem of the probabilistic Ramsey theory in
functional analysis. Given a linear operator $T$ on a Hilbert space with an
orthogonal basis, we define the isomorphic structure $\Sigma(T)$ as the family
of all subsets of the basis so that $T$ restricted to their span is a nice
isomorphism. Our main result is a dimension-free optimal estimate of the size
of $\Sigma(T)$. It improves and extends in several ways the principle of
restricted invertibility due to Bourgain and Tzafriri. With an appropriate
notion of randomness, we obtain a randomized principle of restricted
invertibility.

Functional Analysis;Probability