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.