Limit Operators and Their Applications in Operator Theory

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 $$SO\left( {{\mathbb{Z}^N}} \right)$$.- 2.4.3 Preliminaries on nets.- 2.4.4 Limit operators with respect to nets.- 2.4.5 Local invertibility at points in $${M^\infty }\left( {SO\left( {{\mathbb{Z}^N}} \right)} \right)$$.- 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 Schrödinger operators.- 2.7 Indices of Fredholm band-dominated operators.- 2.7.1 Main results.- 2.7.2 The algebra $$\mathcal{A}\left( \mathbb{Z} \right)$$ as a crossed product.- 2.7.3 The Kl-group of $$\mathcal{A}\left( \mathbb{Z} \right)$$.- 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 $${L^p}\left( {{\mathbb{R}^N}} \right)$$.- 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 $${L^p}\left( {{\mathbb{R}^N}} \right)$$.- 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 $$\mathcal{A}\left( {BUC\left( {{\mathbb{R}^N}} \right),{\mathcal{C}_p}} \right)$$.- 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 spectrum of Schrödinger operators.- 3.6 Comments and references.- 4 Pseudodifferential Operators.- 4.1 Generalities and notation.- 4.1.1 Function spaces and Fourier transform.- 4.1.2 Oscillatory integrals.- 4.1.3 Pseudodifferential operators.- 4.1.4 Formal symbols.- 4.1.5 Pseudodifferential operators with double symbols.- 4.1.6 Boundedness on $${L^2}\left( {{\mathbb{R}^N}} \right)$$.- 4.1.7 Consequences of the Calderon-Vaillancourt theorem.- 4.2 Bi-discretization of operators on $${L^2}\left( {{\mathbb{R}^N}} \right)$$.- 4.2.1 Bi-discretization.- 4.2.2 Bi-discretization and Fredholmness.- 4.2.3 Bi-discretization and limit operators.- 4.3 Fredholmness of pseudodifferential operators.- 4.3.1 A Wiener algebra on $${L^2}\left( {{\mathbb{R}^N}} \right)$$.- 4.3.2 Fredholmness of operators in $${\mathcal{W}^\$ }\left( {{L^2}\left( {{\mathbb{R}^N}} \right)} \right)$$.- 4.3.3 Fredholm properties of pseudodifferential operators in OPS0,00.- 4.4 Applications.- 4.4.1 Operators with slowly oscillating symbols.- 4.4.2 Operators with almost periodic symbols.- 4.4.3 Operators with semi-almost periodic symbols.- 4.4.4 Pseudodifferential operators of nonzero order.- 4.4.5 Differential operators.- 4.4.6 Schrödinger operators.- 4.4.7 Partial differential-difference operators.- 4.5 Mellin pseudodifferential operators.- 4.5.1 Pseudodifferential operators with analytic symbols.- 4.5.2 Mellin pseudodifferential operators.- 4.5.3 Mellin pseudodifferential operators with analytic symbols.- 4.5.4 Local invertibility of Mellin pseudodifferential operators.- 4.6 Singular integrals over Carleson curves with Muckenhoupt weights.- 4.6.1 Carleson curves and Muckenhoupt weights.- 4.6.2 Logarithmic spirals and power weights.- 4.6.3 Curves and weights with slowly oscillating data.- 4.6.4 Local representatives and local spectra of singular integral operators.- 4.6.5 Singular integral operators on composed curves.- 4.7 Comments and references.- 5 Pseudodifference Operators.- 5.1 Pseudodifference operators.- 5.2 Fredholmness of pseudodifference operators.- 5.3 Fredholm properties of pseudodifference operators on weighted spaces.- 5.3.1 Boundedness on weighted spaces.- 5.3.2 Fredholmness on weighted spaces.- 5.3.3 The Phragmen-Lindelöf principle.- 5.4 Slowly oscillating pseudodifference operators.- 5.4.1 Fredholmness on lP-spaces.- 5.4.2 Fredholmness on weighted spaces, Phragmen-Lindelöf principle.- 5.4.3 Fredholm index for operators in OPSO.- 5.5 Almost periodic pseudodifference operators.- 5.6 Periodic pseudodifference operators.- 5.6.1 The one-dimensional case.- 5.6.2 The multi-dimensional case.- 5.7 Semi-periodic pseudodifference operators.- 5.7.1 Fredholmness on unweighted spaces.- 5.7.2 Fredholmness on weighted spaces.- 5.7.3 Fredholm index.- 5.8 Discrete Schrödinger operators.- 5.8.1 Slowly oscillating potentials.- 5.8.2 Exponential decay of eigenfunctions.- 5.8.3 Semi-periodic Schrödinger operators.- 5.9 Comments and references.- 6 Finite Sections of Band-dominated Operators.- 6.1 Stability of the finite section method.- 6.1.1 Approximation sequences.- 6.1.2 Stability vs. invertibility.- 6.1.3 Stability vs. Fredholmness.- 6.2 Finite sections of band-dominated operators on $${\mathbb{Z}^1}$$ and $${\mathbb{Z}^2}$$.- 6.2.1 Band-dominated operators on $${\mathbb{Z}^1}$$: the general case.- 6.2.2 Band-dominated operators on $${\mathbb{Z}^1}$$: slowly oscillating coefficients.- 6.2.3 Band-dominated operators on $${\mathbb{Z}^2}$$.- 6.2.4 Finite sections of convolution type operators.- 6.3 Spectral approximation.- 6.3.1 Weakly sufficient families and spectra.- 6.3.2 Interlude: Spectra of band-dominated operators on Hilbert spaces.- 6.3.3 Asymptotic behavior of norms.- 6.3.4 Asymptotic behavior of spectra.- 6.4 Fractality of approximation methods.- 6.4.1 Fractal approximation sequences.- 6.4.2 Fractality and norms.- 6.4.3 Fractality and spectra.- 6.4.4 Fractality of the finite section method for a class of band-dominated operators.- 6.5 Comments and references.- 7 Axiomatization of the Limit Operators Approach.- 7.1 An axiomatic approach to the limit operators method.- 7.2 Operators on homogeneous groups.- 7.2.1 Homogeneous groups.- 7.2.2 Multiplication operators.- 7.2.3 Partition of unity.- 7.2.4 Convolution operators.- 7.2.5 Shift operators.- 7.3 Fredholm criteria for convolution type operators with shift.- 7.3.1 Operators on homogeneous groups.- 7.3.2 Operators on discrete subgroups.- 7.4 Comments and references.