Factorizations of semi-separable operators along continuous chains of projections (Q2640114)

The authors consider operators of the form \(T=\int^{1}_{0}P(t)K_ 1dP(t)+\int^{1}_{0}dP(t)K_ 2(t)\) acting on a separable Hilbert space \({\mathcal H}\), where \(K_ 1\) and \(K_ 2\) are bounded linear operators of finite rank on \({\mathcal H}\) and P(t) is a continuous chain of orthogonal projections on \({\mathcal H}\) \((P(0)=0\), \(P(1)=I).\) It is shown that \(I+T\) has the special P-factorization \(I+T=(I+Y_ - )(I+Y_+)\) if and only if a certain Riccati differential equation has a solution. Here \(Y_ -\), \(Y_+\) are Volterra operators which satisfy the conditions \(P(t)Y_+P(t)=P(t)=Y_+(t),\) \(P(t)^{\perp}P_ - (t)^{\perp}=Y_ -P(t)^{\perp}\) for all \(t\in [0,1]\). The Riccati equation mentioned above has the form \(R'(t)=I-Q_ m+R(t)Q_ m)M(P(t)\psi,\phi)(Q_ m-R(t)Q_ m),\) \(R(0)=0\), where \(\phi\), \(\psi\) and M(P(t)\(\psi\),\(\phi\)) are a column, a row and an \((m+n)\times (m+n)\) matrix, respectively, which are constructed with the help of the operators \(K_ 1\) and \(K_ 2\) (rank \(K_ 1=m\), rank \(K_ 2=n)\).