Borel type of the convergence set of a sequence of bounded linear operators in a Banach space (Q913215)

For a sequence \(\{A_ n\}\) of bounded linear operators between two Banach spaces, the set of convergence \(Q(\{A_ n\})\) is \(F_{\sigma \delta}\). Several classical and recent researches studied cases in which this set has some supplementary properties. The present author is interested in the case where Q is of type \(G_{\delta \sigma}\). Using technical results from Theorem 1 and Proposition 1, he proves (Theorem 2) that in this case the limit operator is closed and (Theorem 3) that for each linear submanifold \(Q_ 1\) of Q either \(\{A_ n| Q_ 1\}\) is bounded or there exists a linear submanifold \(Q_ 2\) of the closure of \(Q_ 1\) which is \(F_{\sigma}\), on which the limit operator \(A_ 0\) exists and is bounded, \(A_ 0(Q_ 2)\) being a closed infinite dimensional subspace with bounded complete basis.