Regularization of ill-posed problems involving unbounded operators in Banach spaces (Q1185411)

Let \(A\) be a closed, densely defined, injective linear operator with dense range, acting on a Banach space. Suppose that the resolvent set of \(A\) contains the negative reals and that there is \(c>0\) with the property: \[ \| (A+\alpha)^{-1}\|\leq c\alpha^{-1}\qquad (\alpha> 0). \] The authors propose a regularization scheme for the equation \(Ax=y\), based on natural estimates of powers of \(A\). A similar technique is used in regularizing the equation \(Tx=y\) for a class of power bounded operators \(T\).