Multiplication operators on weighted Banach spaces which are an isomorphism into
by
K. Bogalska
A. Mickiewicz University, Poznan

Let v:ID --> IR⁺ be a radial continuous strictly positive function and M_j be a pointwise multiplication operator, M_j(f)(z):=j(z) f(z), where j:D --> C denotes a bounded non-constant analytic function on the unit disc D. We consider these operators on the weighted Banach space,

A\inftyv:={ f:D --> C;    f   analytic,     ||f||v := sup z in D   v(z) |f(z) | < \infty}.

We characterize when M_j: A^\infty_v --> A^\infty_v is an isomorphism into for the class of weights tending rapidly to zero at the boundary. In particular, our result holds for exponential weights like v(z)=exp(-[ 1/((1 - |z|²)^\delta)]) or v(z)=exp(-[ 1/((1 - |z|)^\delta)]) for any \delta > 0.

The problem when M_j, acting between various Bergman spaces, is an isomorphism into was studied for example in papers [2] and [1]. Luecking [2] solved this problem for M_j: A^p_v --> A^p_v, 1 <= p < \infty,

Apv:={f:D --> C;    f   analytic,     ||f||v := æ è ó õ D   |f(z) |pv(z)dA(z) ö ø 1/p   < \infty},

where dA(z) is the Lebesgue area measure and v(z): = (1 - |z|²)^\alpha , \alpha > 0, or v(z)=1. In the paper [1] the above result was extended to the weighted Bergman spaces A^\infty_v, (i.e., for p=\infty) even for weights of polynomial decay.

References

[1] J. Bonet, P. Doma\'nski, M. Lindström, Pointwise multiplication operators on weighted Banach spaces of analytic function, Studia Math. 137 (1999) 177-194.

[2] D. Luecking, Inequalities on Bergman spaces, Illinois J. Math. 25 (1981), 1-11.

(T)

Date received: November 18, 1999