Weighted Composition Groups on the Little Bloch Space

Dedicated to Prof. Len Miller (PhD advisor to J. O. Bonyo) and Prof. Vivien Miller of Mississippi State University on their retirement

Logarithmic Bloch space and its predual

We consider the space B1log?, of analytic functions on the unit disk D, defined by the requirement ?D|f?(z)|?(|z|) dA(z) < ?, where ?(r) = log?(1/(1?r)) and show that it is a predual of the ?log?-Bloch? space and the dual of the corresponding little Bloch space. We prove that a function f(z)=??n=0 an zn with an ? 0 is in B1 log? iff ??n=0 log?(n+2)/(n+1) < ? and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in B1 log?. Some properties of the Cesaro and the Libera operator are considered as well.

SUMS OF WEIGHTED COMPOSITION OPERATORS ON COP

Let COP =0∩H∞, where0is the little Bloch space on the open unit disk, andA() be the disk algebra on. For non-zero functionsu1,u2,. . .,uN∈A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕNsatisfying ϕj∈A() and ∥ϕj∥∞=1 for everyj, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕjmaps COP intoA().

On an integral-type operator from the Bloch space into the QK(p,q) space

Let n be a positive integer, 1 ? H(D) and ? be an analytic self-map of D. The boundedness and compactness of the integral operator (Cn ?,1 f )(z) = ?z 0 f (n)(?(?))1(?)d? from the Bloch and little Bloch space into the spaces QK(p, q) and QK,0(p, q) are characterized.

Linear Combinations of Composition Operators on the Bloch Spaces

We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space.

Distances from Bloch functions to some Möbius invariant function spaces in the unit ball of ℂn

Distance formulae from Bloch functions to some Möbius invariant function spaces in the unit ball of ℂnsuch asQsspaces, little Bloch spaceℬ0and Besov spacesBpare given.