Volterra operators and Hankel forms on Bergman spaces of Dirichlet series

Abstract

For a Dirichlet series g, we study the Volterra operator T g f (s) = − +∞ s f (w)g (w) dw, acting on a class of weighted Hilbert spaces H 2 w of Dirichlet series. We obtain sufficient / necessary conditions for T g to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of T g in Schatten classes. Moreover, we show that if T g is bounded, then g is in H p w , the L p-version of H 2 w , for every 0 < p < ∞. We also relate the boundedness of T g to the boundedness of a multiplicative Hankel form of symbol g, and the membership of g in the dual of H 1 w .

Helson operators and coinvariant subspaces

“…Due to this connection, the spaces A p α (D ∞ ) received considerable attention in recent years. In particular, these function spaces and related operator theory have been widely studied; see [2,3,4,6,7,8,9,10,12,13,19,21,22,24,27,31] and the references therein.…”

“…Naturally, there are many interesting problems concerning Helson operators, such as boundedness, compactness, membership of Schatten classes, positivity etc. Much valuable work related to these problems is presented in [4,6,8,9,10,21,22,25,27,28], and a detailed overview can be found in [27]. In particular, Perfekt and Pushnitski [27,Theorem 6.6] obtained a complete description of Helson matrices of finite rank and a complete characterization of bounded Helson forms of finite rank in terms of certain factorizable differential operators.…”