Toeplitz Operators with Non-trivial Kernels and Non-dense Ranges on Weak Hardy Spaces

Abstract

The well known Coburn lemma can be stated as follows: a nonzero Toeplitz operator T(a) with symbol $a\in L^\infty (\mathbb {T})$ has a trivial kernel or a dense range on the Hardy space $H^p(\mathbb {T})$ with p ∈ (1, ∞). We show that an analogue of this result does not hold for the Hardy-Marcinkiewicz (weak Hardy) spaces $H^{p,\infty }(\mathbb {T})$ with p ∈ (1, ∞): there exist continuous nonzero functions $a:\mathbb {T}\to \mathbb {C}$ depending on p such that $\operatorname {dim} \left (\operatorname {Ker} T(a)\right ) = \infty$ and $\operatorname {dim} \left ( H^{p,\infty }(\mathbb {T})/ \operatorname {clos}_{H^{p,\infty }(\mathbb {T})} \big (\operatorname {Ran} T(a)\big ) \right ) = \infty$.

To the memory of Harold Widom—one of the pioneers of the theory of Toeplitz operators