ANISOTROPIC HARMONIC ANALYSIS ON TREES 9
We further define
7rn(y,x) = {On,*-- ,xo) £ G n + 1 ; x0 = x, x
n
= y }
oo
7r(y,x;B) = ( J 7rn(y,x;P)
oo
7T(2/»^) = I J 7Tn(2/,^)
(2.2) Definition. Let the matrix R define a bounded linear map on
£1(G).
Let P = (xn, ,xo) be a path in G. Define the evaluation of R along P as
E(P;R) = E(P) n?=o ^( x i+i5 x j)- The evaluation of a 0-path is defined
to be 1. If 7r is any set of paths we also define the evaluation of R along n as
E(ir,R) = E(ir) = ZP^E(P;R).
(2.3)
LEMMA.
Let R be a matrix which defines a bounded linear map on
£1(G),
let 7 be a complex number with \j\ \\R\\, and let R1 = (7
R)~l.
Then
(a) R*(y,x) = E(irn(y,x);R)
(b) Ry(y,x)
=-y-1E(ir(y,x);R/y).
The sums in (a) and (b) are absolutely convergent.
Proof. Observe that E(TCo(y, x); R) = 6xy. This means that (a) is true for
n = 0. We now use induction to prove that
J2 \E(P;R)\ = £ E \E{P',R)\\R(t,x)\ WRir'WRW =
\\R\\n
.
Penn(y,x) teG P€7rn_i(i/,t)
This shows absolute convergence in both (a) and (b). The same induction gives
(a). The formula (7 -
R)-1
= 7"
1 Yl™=o(Rft)n
y
i e l d s (b)- D
If Pi = (xn,'-' ,xo) and P2 = (x
m + n
,--- , xn) are two paths, and if P2
starts at the same vertex where Pi ends, we define the product P2P1 as the
path (x
m + n
, , x
n
, , xo). We also use the notation xP\ = (xxn, , xxo).
Assume now that P is G-invariant. Then, for some \x G £l(G), P / = / / / .
Moreover ||P|| = ||/x||i and
Rn(y,x)
= /^(ar^y). Let £(7r;/x) mean P(7r; P)
where 7r is any set of paths. The identity operator is convolution with 6e G
£X(G),
so just as 7 P means 7 id P, 7 [i means 75e //.
(2.4) Remarks. Let \i G 0{G), 7 G C, |-y| ll^lli; x,y,t G G, P C G; P,
Pi, and P2 be paths inside G. Then
(a)
fj,n(x)
= E(irn(x,e);fjb)
(b) (1-a)-\x) =
1-lE^(x,e)'^h)
(c)
E(P2PI\/JL)
=
E(P2\/JL)E(PI;/JL)
whenever the product P2P1 is defined
(d) E(tP;n) = E(P;ii)
(e) E(n(ty, tx\ tB)\ //) = E(n(y, x\ B)\u).
At this point assume further that /i is supported on a finite set. We will
eventually prove that under this hypothesis (7 /i)_1(x) is an algebraic function.
To illustrate the method we will first consider the case of a nearest neighbor
random walk.
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