Wolff's Ideal Problem in the Multiplier Algebra on weighted Dirichlet Space

Analysis Seminar
Wednesday, March 27, 2013 - 14:00
1 hour (actually 50 minutes)
Skiles 005
University of Alabama
In 1980, T. M. Wolff has given the following version of the ideal membership for finitely generated ideals in $H^{\infty}(\mathbb{D})$: \[\ensuremath{\mbox{If \,\,}\left\{ f_{j}\right\} _{j=1}^{n}}\subset H^{\infty}(\mathbb{D}),\, h\in H^{\infty}(\mathbb{D})\,\,\mbox{and }\]\[\vert h(z)\vert\leq\left(\underset{j=1}{\overset{n}{\sum}}\vert f_{j}(z)\vert^{2}\right)^{\frac{1}{2}}\,\mbox{for all \ensuremath{z\in\mathbb{D},}}\]then \[h^{3}\in\mathcal{I}\left(\left\{ f_{j}\right\} _{j=1}^{n}\right),\,\,\mbox{the ideal generated by \ensuremath{\left\{ f_{j}\right\} _{j=1}^{n}}in \ensuremath{H^{\infty}}\ensuremath{(\mathbb{D})}. }\]In this talk, we will give an analogue of the Wolff's ideal problem in the multiplier algebra on weighted Dirichlet space. Also, we will give a characterization for radical ideal membership.