A Volterra type operator on spaces of analytic functions

Abstract

The main results are conditions on g such that the Volterra type operator Jg(f)(z) = z 0 f (ζ)g (ζ) dζ, is bounded or compact on BMOA. We also point out certain information when Jg is considered as an operator on a general space X of analytic functions on the disc.

Semigroups of composition operators and integral operators in spaces of analytic functions

“…where its symbol g is an analytic function on D. This operator (also called in the literature the Volterra operator or the generalized Cesàro operator) was first considered by Pommerenke [17] and has been widely studied in several recent papers [2,1,24,27]. We will use properties of this operator for a particular choice of the symbol g in order to better describe information for the maximal space of strong continuity in various cases.…”

“…For the space V M OA, this result is contained in [24,Corollary 3.3]. For the space B 0 , the proof is almost contained in [27, Theorem 2.1].…”

Linear differential equations with slowly growing solutions

“…By [28,Lemma 5.3] 5) where S a = {re iθ : |a| < r < 1, |θ − arg(a)| ≤ (1 − |a|)/2} denotes the Carleson square with respect to a ∈ D \ {0} and S 0 = D. See also [35,Lemma 3.4]. Solutions in VMOA, the closure of polynomials in BMOA, are discussed in Section 6 in which Theorem 3 is proved.…”

“…As expected, LMOA consists of those functions in H(D) which can be represented as the second derivative of a function in LMOA. For more details on LMOA, see [4,35]. Finally, part (iv) of Lemma 6 gives a sufficient condition for a lacunary series to be in LMOA .…”

Products of integral-type and composition operators from generally weighted Bloch space to F(p,q,s) space

“…For some information on the operators I φ and J φ and their n-dimensional extensions, see, for example [6,7,8,9,13,14,16,18,19,20,21,22,24,25,26] as well as the related references therein. Let g ∈ H(D) and ϕ be a holomorphic self-map of D. Products of integral and composition operators on H(D) were introduced by S. Li and S. Stević (see [6], [11], [12], [15], as well as closely related operators in [10] and [23]) as follows…”