# Finite Section Method and Stability

• Albrecht Böttcher
• Bernd Silbermann
Chapter
Part of the Universitext book series (UTX)

## Abstract

Let $$A = \left( {{a_{jk}}} \right)_{j,k = 1}^\infty$$ be an infinite matrix and suppose A generates a bounded operator on l2. In order to solve the equation Ax = y, i.e., the infinite linear system
$$\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{11}}}{{a_{12}}}{{a_{13}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{21}}}{{a_{22}}}{{a_{23}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{31}}}{{a_{32}}}{{a_{33}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} \ldots \ldots \ldots \ldots \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \\ \vdots \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ {{y_3}} \\ \vdots \end{array}} \right)$$
(2.1)
we consider the truncated systems
$$\left( {\begin{array}{*{20}c} {a_{11} } & \ldots & {a_{1n} } \\ \vdots & {} & \vdots \\ {a_{n1} } & \cdots & {a_{nn} } \\ \end{array}} \right)\left( \begin{array}{l} x_1^{(n)} \\ \vdots \\ x_n^{(n)} \\ \end{array} \right) = \left( \begin{array}{l} y_1 \\ \vdots \\ y_n \\ \end{array} \right).$$
(2.2)