Steffen Roch

Marko Lindner, Steffen Roch

This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations $Ax=b$ in infinitely many variables, where $A$ is a random Jacobi operator. In other words, we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the (large) finite volume case via a particular truncation technique. For most of the paper we consider non-selfadjoint operators $A$ but we also comment on the self-adjoint case when simplifications occur.

Marko Lindner, Steffen Roch

In this article we compare the set of integer points in the homothetic copy $nΠ$ of a lattice polytope $Π\subseteq\R^d$ with the set of all sums $x_1+\cdots+x_n$ with $x_1,...,x_n\in Π\cap\Z^d$ and $n\in\N$. We give conditions on the polytope $Π$ under which these two sets coincide and we discuss two notions of boundary for subsets of $\Z^d$ or, more generally, subsets of a finitely generated discrete group.

Steffen Roch

We consider spatial discretizations by the finite section method of the restricted group algebra of a finitely generated discrete group, which is represented as a concrete operator algebra via its left-regular representation. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role. Finally, for commutative groups and for free non-commutative groups, the algebras of the finite sections sequences are shown to be fractal.

Steffen Roch

These are the lecture notes for a course at the Summer School on "Applied Analysis" at the Technical University Chemnitz in September 2011. We start with the definition of a fractal algebra and show that the fractal property is enormously useful for several spectral approximation problems, e.g. for the convergence of spectra. These results will be illustrated by sequences in the algebra of the finite sections method for Toeplitz operators. Then we discuss some structural consequences of fractality, which are related with the notion of a compact sequence. Discretized Cuntz algebras will show that idea of fractality is also a very helpful guide in order to analyze concrete algebras of approximation sequences, which illustrates the importance of the idea of {\em fractal restriction}. Our final example is the algebra of the finite sections method for band operators. This algebra is not fractal, but has a related property which we call {\em essential fractality} and which is related with the approximation of points in the essential spectrum.