Normal solvability and the noetherictty of elliptic operators in spaces of functions on Rn, part I

Fredholm theory and finite section method for band-dominated operators

“…Taking into account the separability of l~ and exploiting the diagonalization process once more, we obtain the existence of a subsequenee g = (g,~) of h such that the limit operator with respect to g exists for each function f 9 l~ ~ Hence, the limit lirn~-.oo f(g,~) exists for each f 9 l~ ~ which implies that there is a functional ~ 9 Moo(l~) such that lim f(g~) = rT(f). This verifies (21) and finishes the proof of Theorem 5. ..…”

“…We claim that {A,, r 7 9 Moo(l~)} = e(A) (20) and {A,, r~ 9 Moo(l~~ ---{g~, ~ 9 Moo(l~176 (21) Obviously, (20) and (21) imply (18). Let us start with proving (20).…”

“…Since FARVARD's work, the limit operators have found many successful applications, see e.g. [20], [21] for partial differential operators, [18], [23] for pseudodifferential operators, and the monograph [13] for lots of applications in numerical analysis. The first application of limit operators to band-dominated operators goes back to LANGE and RABINOVICH [17].…”

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

“…Later, Muhamadiev [10] proved that Favard's condition implies the invertibility of Favard's almost periodic differential operator considered as operator from BC 1 (R, R n ) to BC(R, R n ). Extensions of Muhamadiev's result to wider classes of almost periodic operators can be found in [5,11,12,18,19], for example. For operators A with almost periodic coefficients, the connection between A and its limit operators is a lot stronger than in more general settings.…”

“…For infinitely smooth coefficients, Shubin provides a proof of Muhamadiev's result [11] that the Favard condition is equivalent to the invertibility of A on BC ∞ (R N , R). In [12], Muhamadiev showed that, for Hölder continuous coefficients, Favard's condition is equivalent to A being Φ + -semi Fredholm between an appropriate pair of spaces of bounded Hölder continuous functions. Similarly and much more recently, Volpert and Volpert show that, for a general class of scalar elliptic partial differential operators A on an unbounded domain but also for systems of such, the Favard condition is equivalent to the Φ + -semi Fredholmness of A on appropriate Hölder [21,22] or Sobolev [20,22] spaces.…”