The strong reproducing kernel thesis for Hankel operators and the Garsia norm

Abstract

Given a reproducing kernel Hilbert space $\mathcal{H}$ and a bounded operator $T$ on $\mathcal{H}$, we say that $T$ obeys the strong reproducing kernel thesis (SRKT) if $\Vert T\Vert ={\sup }_k\Vert Tk\Vert$, where $k$ ranges over the normalized reproducing kernels in $\mathcal{H}$. Two problems are posed. One of these is to determine which Hankel operators on the Hardy space $H^2$ obey the SRKT. The other concerns the geometry of the unit ball of $BMO$ with respect to the Garsia norm.

arXiv comment7 pages