  We study the invertibility of Banach algebras elements in their extensions,
and invertible extensions of Banach and Hilbert space operators with prescribed
growth conditions for the norm of inverses. As applications, the solutions of
two open problems are obtained. In the first one we give a characterization of
E(T)-subscalar operators in terms of growth conditions. In the second one we
show that operators satisfying a Beurling-type growth condition possess
Bishop's property beta. Other applications are also given.
