We consider Toeplitz operators T_u with symbol u on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols φ for which the sequence \(T_\varphi ^{*k}\) f or \(T_\varphi ^k\) f converges to 0 or ∞ as k → ∞ in norm for every nonzero Bergman function f. Also, we characterize analytic symbols φ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result.

中文翻译：

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Bergman空间上Toeplitz算子的幂序列

我们考虑在单位球的Bergman空间上具有符号u的Toeplitz算子T _u，然后研究Toeplitz算子的幂序列的收敛性和可积性。我们首先对解析符号φ进行特征化，对于每个非零Bergman函数，当范数为k→∞时，序列\（T_ \ varphi ^ {* k} \）f或\（T_ \ varphi ^ k \）f收敛到0或∞。f。同样，我们描述了解析符号φ，其序列的范数可累加或不可累加。我们还研究了伯格曼空间的无限直接和上的相应问题，作为结果的推广。