Limit Operators and Their Applications in Operator Theory

1 Limit Operators.- 1.1 Generalized compactness, generalized convergence.- 1.2 Limit operators.- 1.3 Algebraization.- 1.4 Comments and references.- 2 Fredholmness of Band-dominated Operators.- 2.1 Band-dominated operators.- 2.2 P-Fredholmness of rich band-dominated operators.- 2.3 Local P-Fredholmness: elementary theory.- 2.4 Local P-Fredholmness: advanced theory.- 2.5 Operators in the discrete Wiener algebra.- 2.6 Band-dominated operators with special coefficients.- 2.7 Indices of Fredholm band-dominated operators.- 2.8 Comments and references.- 3 Convolution Type Operators on $${\mathbb{R}^N}$$.- 3.1 Band-dominated operators on $${L^p}\left( {{\mathbb{R}^N}} \right)$$.- 3.2 Operators of convolution.- 3.3 Fredholmness of convolution type operators.- 3.4 Compressions of convolution type operators.- 3.5 A Wiener algebra of convolution-type operators.- 3.6 Comments and references.- 4 Pseudodifferential Operators.- 4.1 Generalities and notation.- 4.2 Bi-discretization of operators on $${L^2}\left( {{\mathbb{R}^N}} \right)$$.- 4.3 Fredholmness of pseudodifferential operators.- 4.4 Applications.- 4.5 Mellin pseudodifferential operators.- 4.6 Singular integrals over Carleson curves with Muckenhoupt weights.- 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.- 5.4 Slowly oscillating pseudodifference operators.- 5.5 Almost periodic pseudodifference operators.- 5.6 Periodic pseudodifference operators.- 5.7 Semi-periodic pseudodifference operators.- 5.8 Discrete Schrödinger operators.- 5.9 Comments and references.- 6 Finite Sections of Band-dominated Operators.- 6.1 Stability of the finite section method.- 6.2Finite sections of band-dominated operators on $${\mathbb{Z}^1}$$ and $${\mathbb{Z}^2}$$.- 6.3 Spectral approximation.- 6.4 Fractality of approximation methods.- 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.- 7.3 Fredholm criteria for convolution type operators with shift.- 7.4 Comments and references.