3. PRELIMINARIES 11
(b) Let f = u + i v, let λ 0 be the spectral radius of Lu : Fθ(ΣC)
+
−→
Fθ(ΣC)
+
and let h = hu Fθ(ΣC)
+
be a corresponding to λ strictly positive
eigenfunction of Lu such that h = 1, where ν = νu. Then:
(i) The spectral radius of Lf : Fθ(ΣC)
+
−→ Fθ(ΣC)
+
is λ, and the
essential spectral radius of Lf is θ λ.
(ii) Suppose Lf has at least one eigenvalue µ with |µ| = λ. Then there
exist α C with |α| = 1 and w Fθ(ΣC)
+
with |w(ξ)| = 1 for all
ξ ΣC
+
such that µ = α λ and
(3.10) Lf = α M Lu
M−1
,
where M : Fθ(ΣC)
+
−→ Fθ(ΣC)
+
is the multiplication operator Mg =
w g. Moreover,
specθ(Lf ) {z : |z| = λ} = {µ1, µ2, . . . , µτ }
where µj = α e2πi j/τ for each j = 1, . . . , τ , and every z specθ(Lf )
with |z| λ satisfies |z|≤ ρ λ.
(iii) Under the assumption in (ii), for each j = 1, . . . , τ there exists an
eigenfunction wj Fθ(ΣC)
+
corresponding to the eigenvalue µj with
|wj| = h and |wj|θ |h|θ + Wf |h|∞, where
(3.11) Wf =
q |h|∞
e2|f |∞
(1 θ)2 min h
|f|θ + 2
h
θ
min h
,
and a projection operator Qj : Fθ(ΣC)
+
−→ C · wj , Qj (g) = qj (g) wj ,
with qj (wj ) = 1 and |qj(g)|
|g|∞
min h
for any g Fθ(ΣC),
+
such that
for every g Fθ(ΣC)
+
and every integer m 0 we have
(3.12) Lf
mg

τ
j=1
µj
m
Qj (g)
θ
Ef
λm ρm
g
θ
,
where ρ =

ρ0 (0, 1), b = bf = max{1, |f|θ} and
(3.13) Ef = (1 + Wf
)2
Du ,
Du being given by (3.7) with f replaced by u. Moreover for any
g Fθ(ΣC)
+
of the form g =
∑τ
k=1
Qk(g) we have
(3.14) max
1≤k≤τ
Qk(g)
θ
Hf (1 + Wf
)2
g
θ
.
It should be stressed that in the above we only assume the matrix C to be
irreducible and not necessarily aperiodic. A proof of the above theorem including
estimates involving the choice of the constants ρ0, Df , Hf , Wf and Ef is given in
[St2].
Next, assume as in Theorem 3.1 that the matric C is irreducible with period
τ IN and let f, ω Fθ(ΣC)
+
be real-valued functions. Given s0 R, let λ be the
spectral radius of L−s0f

on Fθ(ΣC).
+
Using perturbation theory (cf. e.g. [Ka])
and Theorem 3.1 (a) and (b)), it follows that there exist δ 0 and analytic families
hs Fθ(ΣC)
+
and λs C such that L−sf +ωhs = λs hs and |λs| is the spectral radius
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