The null space theorem

The following results are proved.Theorem 0.1 The Null Space Theorem. Let  X,YX,Ybe vector spaces,  P∈L(X),Q∈L(Y)P∈L(X),Q∈L(Y)be projections and  T∈L(X,Y)T∈L(X,Y)be invertible. (The restriction of QTP to  R(P)R(P)can be viewed as a linear operator from  R(P)R(P)to  R(Q)R(Q). This is called a section of T by P and Q and will be denoted by  TP,QTP,Q.) Then there is a linear bijection between the null space of the section  TP,QTP,Qof T and the null space of its complementary section  TIY−Q,IX−P−1of  T−1T−1.Theorem 0.2. Let X be a Banach space with a Schauder basis  A={a1,a2,…}A={a1,a2,…}. Let T be a bounded (continuous) linear operator on X. Suppose the matrix of T with respect to A is tridiagonal. If T is invertible, then every submatrix of the matrix of  T−1T−1with respect to A that lies on or above the main diagonal (or on or below the main diagonal) is of rank ≤1.