Projective regularizability of inverse operators (Q1326050)

A sequence \(\{R_ n\}\) of linear operators in Banach spaces \(R_ n\in {\mathcal L}({\mathcal Y},{\mathcal X})\) is called a linear regularizer of the equation \(Ax= y\), \(A\in {\mathcal L}({\mathcal X},{\mathcal Y})\), where \(A\) has an unbounded inverse \(A^{-1}\), if \(R_ n Ax\to x\), \(n\to\infty\). If there exist projectors \(P_ n: Y\to M_ n\), \(n= 1,2,\dots\) such that \(R_ n= A^{-1} P_ n\), \(n= 1,2,\dots\), then \(\{R_ n\}\) is called a projective regularizer. In four theorems and five corollaries the author describes some necessary and also sufficient conditions for the existence of projective and finite projective (i.e. with finite-dimensional projectors \(P_ n\)) regularizers.