Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

Streszczenie

Let $X_1$ and $X_2$ be complex Banach spaces, and let $A_1\in {\rm BL}(X_1)$, $A_2\in {\rm BL}(X_2)$, $A_3\in {\rm BL}(X_1,X_2)$ and $A_4\in {\rm BL}(X_2,X_1)$. We propose an iterative procedure which is a modified form of Newton's iterations for obtaining approximations for the solution $R\in {\rm BL}(X_1,X_2)$ of the Riccati equation $A_2R-RA_1 = A_3+RA_4R$, and show that the convergence of the method is quadratic. The advantage of the present procedure is that the conditions imposed on the operators $A_1, A_2, A_3, A_4$ are weaker than the corresponding conditions for Newton's iterations, considered earlier by Demmel (1987), Nair (1989) and Nair (1990) in the context of obtaining error bounds for approximate spectral elements. Also, we discuss an application of the procedure to spectral approximation under perturbations of the operator.