Spectral properties of weighted composition operators on $\Hol(\D)$ induced by rotations

Abstract

In this article we study the spectrum σ(T ) and Waelbroeck spectrum σ W (T ) of a weighted composition operator T induced by a rotation on Hol(D) and given by Bonet [2] we show that {β n ∶ n ∈ N} ⊂ σ(T ) ≠ T when the weight is m ≡ 1 and β a diophantine number. This shows that the spectrum is not closed in general.

Weighted composition operators: isometries and asymptotic behaviour

“…Following the results of [4], the results depend on the nature of the number β (cf. Definition 3.5).…”

“…2 nd conclusion: If ϕ is periodic, then the operator W m,ϕ is similar to a linear isometry of Hol(D) if and only if the map m N is constant and unimodular, that is if and only if σ p (W m,ϕ ) ∩ T = ∅, by [4,Theorem 3.4].…”

“…2 nd case: If B is a finite Blaschke product, then since B is vanishing on D, we obtain σ p (T ) = ∅ (see [4,Proposition 3.3 and 3.6]). Now, to compute the spectrum of T , consider once again the two subcases depending on the nature of β.…”

“…Finally, in Section 5, we compute, using the results of [4], the spectra of the operators found in the previous sections. The common property of these weighted composition operators is that the symbol is a rotation.…”

“…If β is periodic, then there exists N ∈ N, N ≥ 2 such that β N = 1 and β k = 1 for k < N. It follows that for all f ∈ Hol(D),T N (f )(z) = m N (z)f (z), m N (z) = Hence, by [4, Proposition 3.2], we have σ(T ) = {λ ∈ C : λ N ∈ m N (D)}. r If β is aperiodic, then by [4, Corollary 2.2], {β n B(0) : n ∈ N 0 } ⊂ σ(T ).Moreover, using [4, Proposition 7.2], we have σ(T ) ⊂ D(0, M), with M defined by Re it /r) dt .Note that if m(z) = B(z/r), we can writem(z) = z N m(z), with m(z) = e iθ r N K j=1 z − rα j r − α j z , with α 1 , • • • , α K = 0.Since m is well defined in D, the equation (7.2) of[4] is also valid for M R with R → 1, that is for M 1 = M. Hence, .Therefore, if B has d zeroes (counting the multiplicities), then {β n B(0) : n ∈ N 0 } ⊂ σ(T ) ⊂ D(0, r −d ).…”