Limit Operators And Their Applications In Operator Theory (Hardcover)

1 Limit Operators.- 1.1 Generalized compactness, generalized convergence.- 1.1.1 Compactness, strong convergence, Fredholmness.- 1.1.2 P -compactness.- 1.1.3 P -Fredholmness.- 1.1.4 P -strong convergence.- 1.1.5 Invertibility of P -strong limits.- 1.2 Limit operators.- 1.2.1 Limit operators and the operator spectrum.- 1.2.2 Operators with rich operator spectrum.- 1.3 Algebraization.- 1.3.1 Algebraization by restriction.- 1.3.2 Symbol calculus.- 1.4 Comments and references.- 2 Fredholmness of Band-dominated Operators.- 2.1 Band-dominated operators.- 2.1.1 Function spaces on $${\mathbb{Z}^N}$$.- 2.1.2 Matrix representation.- 2.1.3 Operators of multiplication.- 2.1.4 Band and band-dominated operators.- 2.1.5 Limit operators of band-dominated operators.- 2.1.6 Rich band-dominated operators.- 2.2 P-Fredholmness of rich band-dominated operators.- 2.2.1 The main theorem on P-Fredholmness.- 2.2.2 Weakly sufficient families of homomorphisms.- 2.2.3 Symbol calculus for rich band-dominated operators.- 2.2.4 Appendix A: Second version of a symbol calculus.- 2.2.5 Appendix B: Commutative Banach algebras.- 2.3 Local P-Fredholmness: elementary theory.- 2.3.1 Local operator spectra and local invertibility.- 2.3.2 PR-compactness, PR -Fredholmness.- 2.3.3 Local P-Fredholmness of band-dominated operators.- 2.3.4 Allan's local principle.- 2.3.5 Local P-Fredholmness of band-dominated operators in the sense of the local principle.- 2.3.6 Operators with continuous coefficients.- 2.4 Local P-Fredholmness: advanced theory.- 2.4.1 Slowly oscillating functions.- 2.4.2 The maximal ideal space of $$SO\left( {{\mathbb{Z}^N}} ight)$$.- 2.4.3 Preliminaries on nets.- 2.4.4 Limit operators with respect to nets.- 2.4.5 Local invertibility at points in $${M^\infty }\left( {SO\left( {{\mathbb{Z}^N}} ight)} ight)$$.- 2.4.6 Fredholmness of band-dominated operators with slowly oscillating coefficients.- 2.4.7 Nets vs. sequences.- 2.4.8 Appendix A: A second proof of Theorem 2 4 27.- 2.4.9 Appendix B: A third proof of Theorem 2 4 27.- 2.5 Operators in the discrete Wiener algebra.- 2.5.1 The Wiener algebra.- 2.5.2 Fredholmness of operators in the Wiener algebra.- 2.6 Band-dominated operators with special coefficients.- 2.6.1 Band-dominated operators with almost periodic coefficients.- 2.6.2 Operators on half-spaces.- 2.6.3 Operators on polyhedral convex cones.- 2.6.4 Composed band-dominated operators on $${\mathbb{Z}^2}$$.- 2.6.5 Difference operators of second order.- 2.6.6 Discrete Schrodinger operators.- 2.7 Indices of Fredholm band-dominated operators.- 2.7.1 Main results.- 2.7.2 The algebra $$\mathcal{A}\left( \mathbb{Z} ight)$$ as a crossed product.- 2.7.3 The Kl-group of $$\mathcal{A}\left( \mathbb{Z} ight)$$.- 2.7.4 The Kl-group of A±.- 2.7.5 Proof of Theorem 2.7.1.- 2.7.6 Unitary band-dominated operators.- 2.8 Comments and references.- 3 Convolution Type Operators on $${\mathbb{R}^N}$$.- 3.1 Band-dominated operators on $${L^p}\left( {{\mathbb{R}^N}} ight)$$.- 3.1.1 Approximate identities and P-Fredholmness.- 3.1.2 Shifts and limit operators.- 3.1.3 Discretization.- 3.1.4 Band-dominated operators on $${L^p}\left( {{\mathbb{R}^N}} ight)$$.- 3.2 Operators of convolution.- 3.2.1 Compactness of semi-commutators.- 3.2.2 Compactness of commutators.- 3.3 Fredholmness of convolution type operators.- 3.3.1 Operators of convolution type.- 3.3.2 Fredholmness.- 3.4 Compressions of convolution type operators.- 3.4.1 Compressions of operators in $$\mathcal{A}\left( {BUC\left( {{\mathbb{R}^N}} ight),{\mathcal{C}_p}} ight)$$.- 3.4.2 Compressions to a half-space.- 3.4.3 Compressions to curved half-spaces.- 3.4.4 Compressions to curved layers.- 3.4.5 Compressions to curved cylinders.- 3.4.6 Compressions to cones with smooth cross section.- 3.4.7 Compressions to cones with edges.- 3.4.8 Compressions to epigraphs of functions.- 3.5 A Wiener algebra of convolution-type operators.- 3.5.1 Fredholmness of operators in the Wiener algebra.- 3.5.2 The essential spect