Scattering theory for quantum dynamical semigroups. — II
In order to study the absolute summability of an operator T we consider the set ST = {{xn} | ∑||Txn|| < ∞}. It is well known that an operator T in a Hilbert space is nuclear if and only if ST contains an orthonormal basis and it is natural to ask under which conditions two orthonormal basis define the same left ideal of nuclear operators. Using results about ST we solve this problem in the more general context of Banach spaces.
An algebra of bounded linear operators on a Hilbert space is said to be strongly compact if its unit ball is relatively compact in the strong operator topology. A bounded linear operator on a Hilbert space is said to be strongly compact if the algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. First of all, some basic properties of strongly compact algebras...