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# Limit Operators and Their Applications in Operator Theory

## byVladimir Rabinovich, Steffen Roch, Bernd Silbermann

### Paperback | October 29, 2012

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This text has two goals. It describes a topic: band and band-dominated operators and their Fredholm theory, and it introduces a method to study this topic: limit operators. Band-dominated operators. Let H = [2(Z) be the Hilbert space of all squared summable functions x : Z -+ Xi provided with the norm 2 2 X IIxl1 :=L I iI . iEZ It is often convenient to think of the elements x of [2(Z) as two-sided infinite sequences (Xi)iEZ. The standard basis of [2(Z) is the family of sequences (ei)iEZ where ei = (. . . ,0,0, 1,0,0, . . . ) with the 1 standing at the ith place. Every bounded linear operator A on H can be described by a two-sided infinite matrix (aij)i,jEZ with respect to this basis, where aij = (Aej, ei)' The band operators on H are just the operators with a matrix representation of finite band-width, i. e. , the operators for which aij = 0 whenever Ii - jl > k for some k. Operators which are in the norm closure ofthe algebra of all band operators are called band-dominated. Needless to say that band and band dominated operators appear in numerous branches of mathematics. Archetypal examples come from discretizations of partial differential operators. It is easy to check that every band operator can be uniquely written as a finite sum L dkVk where the d are multiplication operators (i. e.

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Title:Limit Operators and Their Applications in Operator TheoryFormat:PaperbackDimensions:392 pagesPublished:October 29, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034896190

ISBN - 13:9783034896191

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Table of Contents

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.