The structure of norm-achieved Toeplitz and Hankel operators. (Q1396058)

A linear operator \(A\) on a Hilbert space \(H\) is norm-achieving if \(\{f\in H: \| Af\| =\| A\| \| f\| \}\neq\{0\}\) and is paranormal if \(\| Af\| ^2\leq \| A^2f\| \| f\| \) for all \(f\in H.\) The author studies the structure of norm-achieving Toeplitz and Hankel operators. In particular, it is shown that a norm-achieving paranormal Hankel operator is a normal operator while a norm-achieving paranormal Toeplitz operator is a scalar multiple of an isometry.