Some properties of Jordan sets of vectors of linear operators in Banach spaces (Q447658)

Let \(E_1,E_2\) be reflexive Banach spaces, \(A,B\:E_1\to E_2\) be linear and closed operators with \(D(A)=D(B)=E_1\), \(\dim\ker A\neq0\) and/or \(\dim\ker B\neq0\). The authors prove: (1)~If there exist \(r\geq 0\) finite Jordan chains of vectors for \(B\) with respect to \(A\), then there exist \(r\) finite Jordan chains of functionals for the adjoint operator \(B^*\) with respect to~\(A^*\), having the same length; (2)~if there exist \(r\geq 0\) cyclic Jordan chains of functionals for \(B^*\) with respect to \(A^*\), then there exist \(r\) auxiliary chains of vectors for \(B\) with respect to~\(A\).