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.

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http://dx.doi.org/10.1090/surv/072/01