In this paper, we give an attempt to extend some arithmetic properties such as multiplicativity and convolution products to the setting of operator theory and we provide significant examples which are of interest in number theory. We also give a representation of the Euler differential operator by means of the Euler totient arithmetic function and idempotent elements of some associative unital algebra.

中文翻译：

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扩展算术功能

在本文中，我们尝试将一些算术性质（例如乘法性和卷积积）扩展到算符理论的设置中，并提供了一些有意义的实例，这些实例在数论中很有意义。我们还通过欧拉定常算术函数和一些联合as代数的幂等元素来表示欧拉微分算子。