The Reproducing Kernel Thesis for Toeplitz Operators on the Paley-Wiener Space

Abstract.

It is known that for particular classes of operators on certain reproducing kernel Hilbert spaces, key properties of the operators (such as boundedness or compactness) may be determined by the behaviour of the operators on the reproducing kernels. We prove such results for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over $\mathbb{C}$. Namely, we show that the norm of such an operator is equivalent to the supremum of the norms of the images of the normalised reproducing kernels of the space. In particular, therefore, the operator is bounded exactly when this supremum is finite. In addition, a counterexample is provided which shows that the operator norm is not equivalent to the supremum of the norms of the images of the real normalised reproducing kernels. We also give a necessary and sufficient condition for compactness of the operators, in terms of their limiting behaviour on the reproducing kernels.