论文标题
Katznelson-tzafriri定理的最佳衰减速率，用于希尔伯特空间的操作员
Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces
论文作者
论文摘要
Katznelson-Tzafriri定理是离散操作员半群的渐近理论的核心结果。它指出，对于Banach空间上的电源运算符$ t $，我们有$ || T^n（i-t）\ | \ to0 $，并且仅当$σ（t）\ cap \ cap \ mathbb {t} \ subseteq \ {1 \ {1 \} $。本文的主要结果对希尔伯特空间上的操作员发生这种衰减的速度进行了详尽的估计，假设分解规范的增长$ \ | r（e^{iθ}，t），t）\ | $ as as $ | to | fo | fo | fo | fo | fo | fo to $ | | to $ to0 $满足轻度的规律性。这显着扩大了第二作者的较早结果，该结果涵盖了多项式分解增长的重要案例。我们进一步表明，在自然的额外假设下，我们对分解增长的状况不仅足够，而且对于结论我们的主要结果得出的条件也是必要的。通过考虑一类合适的Toeplitz运营商，我们表明我们的理论甚至超出了正常运营商的设置，我们还获得了更一般的结果。
The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator $T$ on a Banach space we have $||T^n(I-T)\|\to0$ if and only if $σ(T)\cap\mathbb{T}\subseteq\{1\}$. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms $\|R(e^{iθ},T)\|$ as $|θ|\to0$ satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case of polynomial resolvent growth. We further show that, under a natural additional assumption, our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold. By considering a suitable class of Toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators, for which we in addition obtain a more general result.
