On convergence of power series of $L_p$ contractions
On convergence of power series of $L_p$ contractions
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关于$L_p$收缩的幂级数的收敛
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发表时间：
2017
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通讯作者：
Michael Lin
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作者：
G. Cohen;C. Cuny;Michael Lin
抽象的。令 T 为（实数或复数）Banach 空间上的幂有界算子。我们研究当 {βk} 是具有发散和的 Kaluza 序列使得 βk → 0 且 Σ∞ k=0 βkz k 在开单位圆盘中收敛时，幂级数 Σ∞ k=0 βkT x 的收敛性。我们证明了弱收敛和强收敛是等价的，并且在自反空间中supn ‖ Σn k=0 βkT x‖ < ∞ 也等价于级数的收敛。当 T 是 L1 的平均遍历收缩时，最后一个断言也得到证明。对于希尔伯特空间上的正规算子，我们获得了 Σ∞ n=0 βnT x 收敛的谱表征，以及以遍历平均范数表示的充分条件，这在某些情况下也是必要的。对于 σ 有限测度空间的 T Dunford-Schwartz 或 Lp 的正收缩，1 < p < ∞，我们证明当 {βk} 也是完全单调（即 Hausdorff 矩序列）且 βk = O(1/k) 时，Σ∞ k=0 βkT f 的范数收敛性意味着 a.e.收敛。对于 T 的 Lp 正收缩，p > 1，f ∈ Lp 且 β ε R，我们证明，如果级数 Σ∞ n=0 (log(n+1)) (n+1)1−1/r T f 对于某些 r ∈ ( p p−1 ,∞] 收敛于 Lp-范数，则它收敛 a.e.
Abstract. Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the power series ∑∞ k=0 βkT x when {βk} is a Kaluza sequence with divergent sum such that βk → 0 and ∑∞ k=0 βkz k converges in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also supn ‖ ∑n k=0 βkT x‖ < ∞ is equivalent to the convergence of the series. The last assertion is proved also when T is a mean ergodic contraction of L1. For normal operators on a Hilbert space we obtain a spectral characterization of the convergence of ∑∞ n=0 βnT x, and a sufficient condition expressed in terms of norms of the ergodic averages, which in some cases is also necessary. For T Dunford–Schwartz of a σ-finite measure space or a positive contraction of Lp, 1 < p <∞, we prove that when {βk} is also completely monotone (i.e. a Hausdorff moment sequence) and βk = O(1/k), the norm convergence of ∑∞ k=0 βkT f implies a.e. convergence. For T a positive contraction of Lp, p > 1, f ∈ Lp and β ∈ R, we show that if the series ∑∞ n=0 (log(n+1)) (n+1)1−1/r T f converges in Lp-norm for some r ∈ ( p p−1 ,∞], then it converges a.e.