Abstract:
The consistent invertibility of operators on a Hilbert space is discussed. By means of two spaces introduced by M. Mbekhta, a sufficient and necessary condition for an operator to be consistent invertible is given. Also, the property of consistent invertibility of an upper triangular operator matrix is considered, it is proved that if d(A)=n(B) and R(B) is closed, then there exists C∈B(K, H) such that MC is consistent in invertible if and only if one of the following cases occurs: (1) both A and B are invertible; (2) d(A)≠0 and n(B)≠0;(3) d(B)≠0 and n(A)≠0, where n(A) and d(A) denote the nullity and the defect of A, respectively. Also, a new spectrum corresponding to the consistent invertibility is defined and the spectral mapping theorem for this new spectrum is studied. It is proved that if A is a bounded linear operator on a Hilbert space, then the spectral mapping theorem holds for σ1(A) if and only if σ1(A)=.
