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 dν = 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

+ω

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