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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