Essential Norms of Volterra Type Operators between Zygmund Type Spaces

Abstract

In this paper, we investigate the boundedness of some Volterra-type operators between Zygmund type spaces. Then, we give the essential norms of such operators in terms of g, ϕ, their derivatives and the n-th power ϕ n of ϕ.and that Z α 0 is a closed subspace of Z α . When α = 1, we get the classical Zygmund space Z 1 = Z and the little Zygmund space Z 1 0 = Z 0 . It is clear that f ∈ Z if and only if f ′ ∈ B 1 . We consider the weighted Banach spaces of analytic functionsWe notice the standard weightsIt is straightforward to show that v log = v log . Let u be a fixed analytic function on D and an analytic self-map ϕ : D → D. Define a linear operator uC ϕ on the space of analytic functions on D, called a weighted composition operator, by uC ϕ f = u·(f •ϕ), where f is an analytic function on D. It is interesting to provide a function theoretic characterization when ϕ and u induces a bounded or compact composition operator on various function spaces. Some results on the boundedness and compactness of concrete operators between some spaces of analytic functions one of which is of Zygmund-type can be found, e.g., in ([1, 4, 5, 6, 11, 12, 13, 14, 18, 19, 21]).Suppose that g : D → C is a analytic map. Let U g and V g denote the Volterra-type operators with the analytic symbol g on D respectively:andIf g(z) = z, then U g is an integral operator. While g(z) = ln 1 1−z , then U g is Cesàro operator. Pommerenke introduced the V olterra type operator U g and characterized the boundedness of U g between H 2 spaces in [10] . More recently, boundedness and compactness of V olterra type operators between several spaces of analytic functions have been studied by many authors, one may see in [17,20].In this paper, we consider the following integral-type operators, which were introduced by Li and Stevic (see e. g. [5,7]), they can be defined by:We will characterize the boundedness of those integral-type operators between Zygmund type spaces, and also estimate their essential norms. The boundedness and compactness of these operators on the logarithmic Bloch space have been characterized in [20].

Riemann-Stieltjes Operators Between Zygmund-Type Spaces

“…to the classical Zygmund space. For some results on B α , Z α and a variety of operators on them, see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Suppose that ( ) g H ∈  .…”