Generalized weighted composition operators on the Bergman space

Abstract

. In this paper, we characterize boundedness and compactness of D n ϕ,ψ on the Bergman space A 2 . We also compute the upper and lower bounds of essential norm of this operator on the Bergman space. IntroductionThroughout this paper, we denote by H(D), the space of holomorphic functions on the open unit disk D = {z ∈ C : |z| < 1} in the complex plane C. For z, w ∈ D, let β z (w) = (z − w)/(1 − zw) be the Möbius transformation of D which interchanges 0 and z. The nth derivative of β z is βπ r dr dθ be the normalized area measure on D. For 0 < p < ∞, let L p be the Lebesgue space which contains measurable functions f on D such thatthe weighted Bergman space with the norm defined asRecall that the Bergman space A 2 (D) = A 2 is a functional Hilbert space of analytic functions on D and K z (w) = 1/(1 − zw) 2 is the Bergman kernel. Moreover, See [6] and [23] for more details on Bergman spaces. Let ϕ and ψ be holomorphic maps on the open unit disk D such that ϕ(D) ⊂ D. For a non-negative integer n, we define a linear operator D n ϕ,ψ as follows:We call it generalized weighted composition operator, since it include many known operators. For example, if n = 0 and ψ ≡ 1, then we obtain the composition operator C ϕ induced by ϕ, defined asIf ψ = 1 and ϕ(z) = z, then D n ϕ,ψ = D n , the differentiation operator defined as D n f = f (n) . If n = 0, then we get the weighted composition operator ψC ϕ defined as, the product of multiplication and differentiation operator. We provide a unified way of treating these operators on the Bergman space A 2 . Composition and weighted composition operators have gained increasing attention during the last three decades, mainly due to the fact that they provide, just as, for example, Hankel and Toeplitz operators, ways and means to link classical function theory to functional analysis and operator theory.In fact, some of the well known conjectures can be linked to composition operators. Nordgren, Rosenthal and Wintrobe [11] have shown that the invariant subspace problem can be solved by classifying certain minimal invariant subspaces of certain composition operators on H 2 , whereas, Louis de Branges used composition operators to prove the Bieberbach conjecture. For general background on composition operators, we refer [2], [15] and references therein. Weighted composition operators also appear naturally in different contexts. For example, W. Smith in [20] showed that the Brennan's conjecture [1], which says that if G is a simply connected planar domain and g is a conformal map of G onto D, then Ì G |g ′ | p dA < ∞ holds for 4/3 < p < 4, is equivalent to the existence of self-maps of D that make certain weighted composition operators compact. The surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see

Essential norm of linear operators from general weighted‐type spaces to Banach spaces and some applications

“…For some other integral-type operators, see earlier works [7,[19][20][21] and the related references therein. For many other product-type operators, see previous research [3,4,9,11,12,[22][23][24][25][26][27][28] and the cited references therein.…”

Metrical Boundedness and Compactness of a New Operator between Some Spaces of Analytic Functions

“…Soon after the first investigations of the operators DC ϕ and C ϕ D, the following operator D m ϕ,u := M u C ϕ D m containing also the operators C ϕ , M u and D, was introduced and considerably studied (see, e.g., [15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the related references therein).…”

Sums of Weighted Differentiation Composition Operators

“…The operator D n u,ϕ and some of its special cases such as DC ϕ were studied in [9,[14][15][16][18][19][20]22,25,26]. In particular, see [1,9,15,18,[20][21][22]26], which are focused on these types of operators on weighted Bergman spaces. Some good references for weighted Bergman spaces are [6,7].…”

“…Thus generalized weighted composition operators provide a unified way of treating these operators. The operator D n u,ϕ and some of its special cases such as DC ϕ were studied in [9,[14][15][16][18][19][20]22,25,26]. In particular, see [1,9,15,18,[20][21][22]26], which are focused on these types of operators on weighted Bergman spaces.…”