Minimal norm Hankel operators

Brevig, Ole

doi:10.1090/proc/15969

Minimal norm Hankel operators
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by Ole Fredrik Brevig PDF

Proc. Amer. Math. Soc. 150 (2022), 4383-4391 Request permission

Abstract:

Let $\varphi$ be a function in the Hardy space $H^2(\mathbb {T}^d)$. The associated (small) Hankel operator $\mathbf {H}_\varphi$ is said to have minimal norm if the general lower norm bound $\|\mathbf {H}_\varphi \| \geq \|\varphi \|_{H^2(\mathbb {T}^d)}$ is attained. Minimal norm Hankel operators are natural extremal candidates for the Nehari problem. If $d=1$, then $\mathbf {H}_\varphi$ has minimal norm if and only if $\varphi$ is a constant multiple of an inner function. Constant multiples of inner functions generate minimal norm Hankel operators also when $d\geq 2$, but in this case there are other possibilities as well. We investigate two different classes of symbols generating minimal norm Hankel operators and obtain two different refinements of a counter-example due to Ortega-Cerdà and Seip.

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