Action of an operator with nonclosed range on orthonormal bases in a Hilbert space (Q750841)

A basis \(\{x_ n\}\) in a Banach space X is called unconditional if after the arrangement of its terms in an arbitrary way it still remains a basis for this space. But there are conditional norms in Hilbert space H which can be obtained as a result of applying to an orthonormal basis an operator A with nonclosed range. The author investigates the problem whether there exists an orthonormal basis in H such that the sequence \(\{Ax_ n\}\) will serve (act) as a basis in H having some or other features. He show that there exists such an orthonormal basis \(\{x_ n\}\) in H that \(\{y_ n=Ax_ n\}\) appears to be a unconditional basis in H provided: 1) Ker \(A=\{0\}\); 2) \(R(A)=H\) and 3) R(A)\(\neq H.\) It is also pointed out that in this assertion the ``orthonormal'' basis can not be replaced by ``unconditional'' one.