Limit operators and their applications in operator theory
The purpose of the present book is two-fold: to develop a method of limit operators and to apply limit operators to the study of Fredholm properties of band and band dominated operators. In the space of two-sided infinite sequences \(\ell^2(\mathbb Z)\), a standard basis \(e^k=\{\delta_{kj}\}_{j=-\infty}^\infty\), \(k=0,\pm1,\dots\), is chosen, where \(\delta_{kj}\) is the Kroneker's symbol. Any linear bounded operator \(A\in{\mathcal L}(\ell^2(\mathbb Z))\) is represented in this standard basis as a two-sided infinite matrix \(A=[a_{jk}]_{\mathbb Z\times\mathbb Z}\), where \(a_{jk}=(Ae^j,e^k)\). The band operators are those which possess only a finite number of non-trivial diagonals in their representation \(a_{jk}=0\) whenever \(| j-k| \geq N\) for some \(N\). The band dominated operators are limits (in the operator norm) of convergent sequences of band operators and belong to the Banach subalgebra of \({\mathcal L}(\ell^2(\mathbb Z))\) generated by the band operators. We encounter such operators rather often among partial differential (more generally, among pseudodifferential) equations; their solution is a dominating problem in numerical methods for differential and pseudodifferential equations. The authors concentrate mainly on the Fredholm properties of band and band dominated operators. These properties play a crucial role not only in solving the equations explicitly (which turns out usually to be a rather theoretical result), but also in elaborating an efficient numerical approximation with strongly motivated convergence and stability. The study is based on the fact that Fredholm properties of an operator are stable under compact perturbations. Therefore, no finite parts of the matrix representation \(A=[a_{jk}]_{\mathbb Z\times\mathbb Z}\) of the operator influences the Fredholm properties, and only the asymptotic behaviour of \(a_{jk}\) as \(j,k\to\pm\infty\) matters. To extract information about the behaviour of the operator at ``infinity'', one can consider \(V_{-n}AV_ne^k\), \(n\to\infty\), \(k=0,\pm1,\ldots\), where \(V_je^m=e^{m+j}\) are the shift operators. If the diagonals of the band operator \(A\) stabilize at infinity (the limits \(\lim_{j\to\pm\infty}a_{j+k,j}\) exist for all \(k\)), then the strong limits \(V_{-n}AV_n\to A_\pm\) as \(n\to\pm\infty\) exist and represent the operator \(A\) at infinity. If the limits do not exist, then the strong limits \(V_{-g(n)}AV_{g(n)}\to A_g\) exist for certain subsequences \(\{g(n)\}_{n\to\pm\infty}\). The variety of these partial limits forms the operator spectrum, which carries exact information about Fredholm properties of the operator \(A\): a band dominated operator is Fredholm if and only if the all operators in the spectrum are invertible and the inverses are uniformly bounded. Moreover, the authors consider band and band dominated operators in the spaces \(\ell^p(\mathbb Z^N,X)\), \(1\leq p\leq\infty\), where \(X\) is a complex Banach space. Such an approach enables them to consider the operators in \(L^p(\mathbb R^n)\), which become band dominated after appropriate discretization. In the first two chapters, the authors develop the theory of limit operators in the spaces \(\ell^p(\mathbb Z^N,X)\), \(1\leq p\leq\infty\), and prove the principal theorem on Fredholm properties exposed above. In Chapter 3, the obtained results are applied to integral convolution type operators on \(\mathbb R^N\), generated by convolutions and functions. In Chapter 4, the authors give an introduction to the theory of pseudodifferential operators, adapted to the purposes of the book. Aside from classical results, operators with slowly oscillating and with almost periodic symbols, as well as Schrödinger operators are considered. Separate sections are devoted to pseudodifferential operators with Mellin symbols and to singular integral operators on Carleson curves with Muckenhoupt weight. In Chapter 5, the authors continue their consideration of pseudodifferential operators. The theory of limit operators, developed in Chapter 2, is applied to study the Fredholm property of operators considered in Chapter 4. In Chapter 6, finite sections of band dominated operators are studied. Aside of the undoubted theoretical interest, approximation by finite sections has important applications in numerical methods. Stability and convergence of finite sections of band dominated operators are studied in general and for convolution type operators in particular. The spectral approximation and the fractality of approximation methods are considered in separate sections. The concluding Chapter 7 is devoted to the axiomatization of the limit operator approach. As an application, convolution operators on homogeneous groups are studied.
scientific article; zbMATH DE number 2117720