Numerical range of composition operators on a Hilbert space of Dirichlet series

Abstract

We study the numerical range of composition operators on a Hilbert space of Dirichlet series with square-summable coefficients. We first describe the numerical range of "nice" composition operators (as invertible, normal and isometric ones). We also focus on the zeroinclusion question for more general symbols.

Compact composition operators with nonlinear symbols on the H2 space of Dirichlet series

“…We will now choose another sequence s ′ k = σ ′ k + i t k k≥1 where Re(s ′ k ) → 0 in order to obtain a contradiction with (7). More precisely, let σ ′ k k≥1 be any sequence of positive real numbers tending to 0 such that, for any n = 2, .…”

“…[3, Lin. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] We begin by looking at the coefficients of uv 2 in the real part of Φ(u, v ). Using on the one hand (13) and on the other hand (14) we conclude that γ 0,2 = −6a 3 1 Im(b 2 ) − 6a 3 2 Im(b 1 ) + a 1 a 2 (a 1 + a 2 )Im(c) 8a 1 a 2 .…”

“…Definition. The Gordon-Hedenmalm class, denoted G , is the set of functions ϕ : C 1/2 → C 1/2 of the form (1) ϕ(s) = c 0 s + [1,2,7,8,11]). In the present work, we are interested in the study of the compactness of C ϕ when ϕ is a polynomial symbol, say Correspondingly, if ϕ(C 0 ) is strictly contained in any smaller half-plane, we say that C ϕ has restricted range.…”

The Fréchet Schwartz Algebra of Uniformly Convergent Dirichlet Series

“…The next Lemma is a particular case of [18,Theorem 4.2] (see also [4,Lemma 12] and [17,Lemma 2]). (2) if ϕ(s) is not constant, then it can be extended to an analytic function ϕ : C 0 → C 0 , such that for each θ > 0 there is η > 0 such that ϕ(C θ ) ⊂ C η ;…”

“…If ϕ(s) is constant, the conclusion is easy. If ϕ(s) is not constant, [17,Lemma 2] implies Re c 1 > 0, the series n a n n −s converges uniformly in C Re c1/2 , and the series formally obtained expanding n a n n −φ(s) converges in C η for η > 0 sufficiently large. A similar argument is implicit in Bayart's result [4, after Corollary 2, p. 217] that if φ satisfies the assumptions in (2), then C φ (g) is a convergent Dirichlet function for each f ∈ H ∞ .…”