Kreiss bounded and uniformly Kreiss bounded operators

A. Bonilla, V. Müller

If $T$ is a Kreiss bounded operator on a Banach space, then $\Vert T^n\Vert =O(n)$. Forty years ago Shields conjectured that in Hilbert spaces, $\Vert T^n\Vert =O(\sqrt{n})$. A negative answer t…