Compactness of composition operators on a Hilbert space of Dirichlet series

Abstract

In this paper new L p a -L q b estimates are proved for translation-invariant Radon transforms along curves for apb and poq: For a fixed a and b; if p is sufficiently close to 2 the best possible q is obtained, up to e: The method is related to that of [5].

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

“…As is to be expected when investigating composition operators on H 2 , the symbols with c 0 = 0 are more difficult to handle and require different techniques. In this case, it is proven independently in [2] and [8] that composition operators induced by linear symbols (3) with c 0 = 0 are compact if and only if ϕ has restricted range or d ≥ 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.…”

Espaces de séries de Dirichlet et leurs opérateurs de composition

“…, on a donc C ϕ / ∈ S 2 (résultat déjà dans [21]) mais C ϕ ∈ S p pour p > 2, alors que dans [21] le résultat est seulement prouvé pour p ≥ 4.…”

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

“…In view of (b) in Theorem 9, it is interesting to know when the operator C φ is compact. In [2,3,8] the compactness of such composition operators is studied.…”