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 to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss bounded operators. More precisely, for every \(\varepsilon >0\) there exists a uniformly Kreiss bounded operator T on a Hilbert space such that \(\Vert T^n\Vert \sim (n+1)^{1-\varepsilon }\) for all \(n\in \mathbb {N}\). On the other hand, any Kreiss bounded operator on Hilbert spaces satisfies \(\Vert T^n\Vert =O(\frac{n}{\sqrt{\log n}})\). We also prove that the residual spectrum of a Kreiss bounded operator on a reflexive Banach space is contained in the open unit disc, extending known results for power bounded operators. As a consequence we obtain examples of mean ergodic Hilbert space operators which are not Kreiss bounded.

如果T是Banach空间上的Kreiss有界算子，则\（\ Vert T ^ n \ Vert = O（n）\）。四十年前，希尔兹（Shields）猜想在希尔伯特空间中，\（\ Vert T ^ n \ Vert = O（\ sqrt {n}）\）。Spijker，Tracogna和Welfert在2003年对这个猜想给出了否定的答案。我们改进了他们的结果，并表明，即使对于统一的Kreiss有界算子，这种猜想也不成立。更精确地说，对于每个\（\ varepsilon> 0 \），在希尔伯特空间上存在一个统一的Kreiss有界算子T，使得\（\ Vert T ^ n \ Vert \ sim（n + 1）^ {1- \ varepsilon} \）所有\（n \ in \ mathbb {N} \）中的\）。另一方面，希尔伯特空间上的任何Kreiss有界算子都满足\（\ Vert T ^ n \ Vert = O（\ frac {n} {\ sqrt {\ log n}}）\）。我们还证明了自反Banach空间上Kreiss有界算子的剩余谱包含在开放单位圆盘中，从而扩展了有功算子的已知结果。结果，我们得到了不受Kreiss约束的平均遍历Hilbert空间算子的例子。