Decomposition of quotients of bounded operators with respect to closability and Lebesgue-type decomposition of positive operators (Q582550)

Let A and B be bounded linear operators on an infinite dimensional Hilbert space H with ker \(A\subset \ker B\). A quotient [B/A] is defined as the linear operator: Ax\(\mapsto Bx\), \(x\in H\). The J-decomposition of [B/A] by Q is \([B/A]=[QB/A]+[Q^{\perp}B/A]\), if Q is an orthogonal projection such that [QB/A] is closable and \([Q^{\perp}B/A]\) is singular. The L-decomposition is defined: if S is a positive operator then every positive operator T is decomposable into the sum \(T=U+V\) of two positive operators U and V such that U is S-absolutely continuous and V is S-singular. This paper gives some equivalent conditions for uniqueness of the J-decomposition. Then it shows that every J- decomposition of a quotient [B/A] induces an L-decomposition of \(B^*B\) w.r.t. \(A^*A\), and conversely that every L-decomposition of T w.r.t. S (ker \(S\subset \ker T)\) is induced from a J-decomposition of [B/A] such that \(A^*A=S\) and \(B^*B=T\).