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Limit Operators and Their Applications in Operator Theory

'Operator Theory, Advances and Applications'. Auflage 200…
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Titel: Limit Operators and Their Applications in Operator Theory
Autor/en: Vladimir Rabinovich, Steffen Roch, Bernd Silbermann

ISBN: 3764370815
EAN: 9783764370817
'Operator Theory, Advances and Applications'.
Auflage 2004.
Bibliographie.
Book.
Sprache: Englisch.
Springer Basel AG

25. Juni 2004 - gebunden - XV

This is the first monograph devoted to a fairly wide class of operators, namely band and band-dominated operators and their Fredholm theory. The main tool in studying this topic is limit operators. Applications are presented to several important classes of such operators: convolution type operators and pseudo-differential operators on bad domains and with bad coefficients.
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.
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