Limit Operators and Their Applications in Operator TheoryThis 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. |
Contents
II | 1 |
III | 17 |
IV | 23 |
V | 29 |
VI | 31 |
VII | 44 |
VIII | 59 |
IX | 72 |
XXVII | 263 |
XXVIII | 265 |
XXX | 271 |
XXXI | 274 |
XXXII | 278 |
XXXIII | 286 |
XXXIV | 287 |
XXXV | 291 |
Other editions - View all
Limit Operators and Their Applications in Operator Theory Vladimir Rabinovich,Steffen Roch,Bernd Silbermann No preview available - 2004 |
Limit Operators and Their Applications in Operator Theory Vladimir Rabinovitch,Steffen Roch,Bernd Silbermann No preview available - 2004 |
Common terms and phrases
approximate identity approximate projection assertion Banach algebra Banach space band operators band-dominated operators belongs C*-algebra choose closed subalgebra compact operators converge strongly convolution operators Corollary coset defined Definition discrete equivalent essential spectrum exists finite section method follows fractal Fredholm operator Further g of h Gelfand Hence Hilbert space homomorphism implies inverse closed inverses are uniformly IP(ZN L²(RN Lemma let h lim inf lim sup limit operator Ah limit operators locally invertible mapping multiplication operator norm operator of multiplication operator Op(a operator spectrum operators with slowly P-compact P-Fredholm P-strong P-strongly proof of Theorem Proposition pseudodifference operators pseudodifferential operators respect S(ZN satisfied Schrödinger operators sequence h singular integral operators slowly oscillating functions SO(ZN strong convergence strong limit operator subsequence g subset tends to infinity Toeplitz operators topology uniformly bounded uniformly invertible whence Wiener algebra
Popular passages
Page 373 - Mouche, and B. Simon, On the measure of the spectrum for the almost Mathieu operator, Comm. Math. Phys.