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Minimal norm Hankel operators

Let $φ$ be a function in the Hardy space $H^2(\mathbb{T}^d)$. The associated (small) Hankel operator $\mathbf{H}_φ$ is said to have minimal norm if the general lower norm bound $\|\mathbf{H}_φ\| \geq \|φ\|_{H^2(\mathbb{T}^d)}$ is attained. Minimal norm Hankel operators are natural extremal candidates for the Nehari problem. If $d=1$, then $\mathbf{H}_φ$ has minimal norm if and only if $φ$ is a constant multiple of an inner function. Constant multiples of inner functions generate minimal norm Hankel operators also when $d\geq2$, 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.