Chapter 1 Jacobi operators This chapter introduces to the theory of second order difference equations and Jacobi operators. All the basic notation and properties are presented here. In addition, it provides several easy but extremely helpful gadgets. We investigate the case of constant coefficients and, as an application, discuss the infinite harmonic crystal in one dimension. 1.1. General properties The issue of this section is mainly to fix notation and to establish all for us relevant properties of symmetric three-term recurrence relations in a self-contained manner. We start with some preliminary notation. For / C Z and M a set we denote by t(I,M) the set of M-valued sequences (f(n))nej. Following common usage we will frequently identify the sequence / = /(.) = (f(n))nej with f(n) whenever it is clear that n is the index (I being understood). We will only deal with the cases M = M, R 2 , C, and C 2 . Since most of the time we will have M = C, we omit M in this case, that is, £{I) = £(I,C). For Ni,N2 G Z we abbreviate £(NUN2) = e({n G Z|iVi n 7V2}), £{Nuoo) = £({n G Z\NX n}), and £( oc, 7V2) = £({n G Z|n N2}) (sometimes we will also write £{N2, oo) instead of £( oc,N2) for notational convenience). If M is a Banach space with norm |.|, we define F ( J , M) = {/ G £(I, M)\ J2 \f(n)\p oo}, 1 p oo, nei (1.1) e*(I,M) = { / G ^ ( / , M ) | s u p | / ( n ) | o o } . nei Introducing the following norms (1.2) | | / | | P = ( £ l / M r ) , l p o o , | | / | | o c = s u p | / ( n ) | , makes (P(I, M), 1 p oo, a Banach space as well. 3 http://dx.doi.org/10.1090/surv/072/01
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