The Finite Section Method in the Space of Essentially Bounded Functions: An Approach Using Limit Operators

Abstract

We present an approach to the finite section method for band-dominated operators—the norm-limits of band operators on . We hereby show that the sequence of finite sections is stable if and only if some associated operator is invertible at infinity. By means of the theory in Lindner and Silbermann (Lindner, M., Silbermann, B. (2003). Invertibility at infinity of band-dominated operators in the space of essentially bounded functions, (accepted at) Integral Equations and Operator Theory.) and Lindner (Lindner, M. (2003). Classes of multiplication operators and their limit operators (submitted to) Zeitschrift für Analysis und ihre Anwendungen), we study this invertibility at infinity using limit operators. Having the mentioned criterion at our disposal, we will give some applications in an algebra of convolution and multiplication operators: one for the usual finite section method and one for an approximation method of operators on the space of continuous functions.

Keywords:

• Finite section method
• Stability
• Invertibility at infinity
• Band-dominated operators
• Limit operators

Acknowledgment

I am grateful to my advisors and friends Bernd Silbermann and Steffen Roch for many fruitful conversations, inspirations, and hints.

Notes

^aIn limit operators are defined exactly as in Definiton 2.28.

^bNote that the notation of a -convergent operator sequence is more general in Roch and Silbermann (1989) than here.

^cOccasionally we will replace Eq. (9) by the strong convergence *A _τ  → *A _{τ 0} on the pre-dual of X.