Gauss–Seidel Convergence for Operators on Hilbert Space

Abstract

For invertible linear operator A on Hilbert space $\mathcal{H}$, we consider classes of splittings for A (viz., $A = A_1 + A_2 $, where $A_1 ,A_2 $ and $A_1^{ - 1} $ exist as a bounded linear operator on $\mathcal{H}$) for which $\rho (A_1^{ - 1} A_2 )$, the spectral radius of $A_1^{ - 1} A_2 $, is less than one. For example, we introduce the class of $\alpha $-splittings (Definition 3.2) which generalizes the classical notion of the lower-triangular $A_1 $ and upper-triangular $A_2 $ splitting for an $n \times n$ matrix A. Tests are presented which guarantee Gauss–Seidel convergence (i.e., $\rho (A_1^{ - 1} A_2 ) < 1$) relative to $\alpha $-splittings (cf. Theorem 3.6, Corollary 3.7, Corollary 3.8). Another class of splittings for (self-adjoint) linear operators $A = A_1 + A_2 $ is an operator-parameter generalization of the $\omega $-splitting which occurs in the matrix method of successive overrelaxation. In fact, Theorem 4.2, which establishes conditions on this splitting sufficient for Gauss–Seidel convergence, has, as a consequence, the theorems of Ostrowski (Corollary 4.3) and, hence, of Reich (cf. Example, § 4).