Volterra operators and Hankel forms on Bergman spaces of Dirichlet series

Abstract

AbstractFor a Dirichlet series g, we study the Volterra operator $$T_g f(s)=-\int ^{+\infty }_{s} f(w)g'(w)dw,$$ T g f ( s ) = - ∫ s + ∞ f ( w ) g ′ ( w ) d w , acting on a class of weighted Hilbert spaces $${{\mathcal {H}}^{2}_{w}}$$ H w 2 of Dirichlet series. We obtain sufficient / necessary conditions for $$T_g$$ T g to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of $$T_g$$ T g in Schatten classes. Moreover, we show that if $$T_g$$ T g is bounded, then g is in $${{\mathcal {H}}}^p_w$$ H w p , the $$L^p$$ L p -version of $${{\mathcal {H}}^{2}_{w}}$$ H w 2 , for every $$0<p<\infty $$ 0 < p < ∞ . We also relate the boundedness of $$T_g$$ T g to the boundedness of a multiplicative Hankel form of symbol g, and the membership of g in the dual of $${{\mathcal {H}}}^1_w$$ H w 1 .