Volterra operators and Hankel forms on Bergman spaces of Dirichlet series
- PDF / 601,403 Bytes
- 43 Pages / 439.37 x 666.142 pts Page_size
- 72 Downloads / 175 Views
Volterra operators and Hankel forms on Bergman spaces of Dirichlet series H. Bommier-Hato1 Received: 15 January 2020 / Accepted: 16 February 2020 © The Author(s) 2020
Abstract +∞ f (w)g (w) For a Dirichlet series g, we study the Volterra operator Tg f (s) = − s 2 dw, acting on a class of weighted Hilbert spaces Hw of Dirichlet series. We obtain sufficient / necessary conditions for Tg to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of Tg p in Schatten classes. Moreover, we show that if Tg is bounded, then g is in Hw , the p 2 L -version of Hw , for every 0 < p < ∞. We also relate the boundedness of Tg to the boundedness of a multiplicative Hankel form of symbol g, and the membership of g 1. in the dual of Hw Keywords Volterra operator · Dirichlet series · Hankel forms Mathematics Subject Classification Primary 31B10 · 32A36; Secondary 30B50 · 30H20
1 Introduction Dirichlet series are functions of the form f (s) =
+∞
an n −s , with s ∈ C.
(1.1)
n=1
Communicated by Adrian Constantin. The author was supported by the FWF project P 30251-N35.
B 1
H. Bommier-Hato [email protected] Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
123
H. Bommier-Hato
For a real number θ , Cθ stands for the half-plane {s, s > θ }, and D for the unit disk. D denotes the class of functions f of the form (1.1) in some half-plane Cθ , and P is the space of Dirichlet polynomials. The increasing sequence of prime numbers will be denoted by ( p j ) j≥1 , and the set of all primes by P. Given a positive integer n, n = p κ will stand for the prime number factorization n = p1κ1 p2κ2 · · · pdκd , which associates uniquely to n the finite multi-index κ(n) = (κ1 , κ2 , . . . , κd ). The number of prime factors in n is denoted by (n) (counting multiplicities), and by ω(n) (without multiplicities). The space of eventually zero complex sequences c00 consists in all sequences which ∞ ∞ have only finitely many non zero elements. We set D∞ fin = D ∩ c00 and N0,fin = ∞ N0 ∩ c00 , where N0 = N ∪ {0} is the set of non-negative integers. ∞ Let F : D∞ fin → C be analytic, i.e. analytic at every point z ∈ Dfin separately with respect to each variable. Then F can be written as a convergent Taylor series
F(z) =
α∈N∞ 0,fin
cα z α , z ∈ D∞ fin .
The truncation Am F of F onto the first m variables is defined by Am F(z) = F(z 1 , . . . , z m , 0, 0, . . .). For z, χ in D∞ , we set z.χ := (z 1 χ1 , z 2 χ2 , . . .), and px := ( p1x , p2x , . . .) for a real number x, . −s is the power series The Bohr lift [11] of the Dirichlet series f (s) = +∞ n=1 an n B f (χ ) =
+∞
an χ κ(n) =
α∈N∞ 0,fin
n=1
a˜ α χ α , where a˜ α = a pα , χ ∈ D∞ fin ,
with the multiindex notation χ α = χ1α1 χ2α2 · · · . Given a sequence of positive numbers w = (wn )n = (w(n))n , one considers the Hilbert space (see [21,23]) 2 Hw
:=
+∞
an n
−s
n=1
:
+∞ |an |2 n=1
wn
< +∞ .
The choice wn = 1 corresponds to the space H2 , introd
Data Loading...
