Small Toeplitz Operators

Abstract: Toeplitz operators acting on Hilbert spaces of analytic functions are among the most well studied examples of concrete operators. In our work we are interested in a cut-off property of such operators; namely, if the operator is small enough, does it have to be zero? Or more in general, must its symbol be of a particular form? There have been several such results, and in the Hardy space the answer is classical and well known. More recently Daniel Luecking proved such a result in the Bergman space case, with the cutoff being at the finite rank level. We present a new proof of a more general version of that theorem, which unifies several results that followed the publication of Luecking's paper.