Let X be a real or complex Banach space, let \({\mathcal {A}}\) be a standard operator algebra on X, let n be a positive integer, and let \(D:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\) be a linear mapping such that the equality \(D(A^{2n})=D(A^n)A^n+A^nD(A^n)\) holds for every \(A\in {\mathcal {A}}\). We prove that D can be written in a unique way as \(D=D_1+D_0\) where \(D_1:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\) is of the form \(A\rightarrow AB-BA\) for some \(B\in {\mathcal {L}} (X)\), and \(D_0:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\) is a linear mapping such that \(D_0(A^n)=0\) for every \(A\in {\mathcal {A}}\). The case \(n=1\) of this result refines a theorem of Chernoff.

令X为实数或复数 Banach 空间，令\({\mathcal {A}}\)为X上的标准算子代数，令n为正整数，令\(D:{\mathcal {A}} \rightarrow {\mathcal {L}} (X)\)是一个线性映射，使得等式\(D(A^{2n})=D(A^n)A^n+A^nD(A^n )\)对每个\(A\in {\mathcal {A}}\) 成立。我们证明D可以以独特的方式写成\(D=D_1+D_0\)其中\(D_1:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\)是形成\(A\rightarrow AB-BA\)对于某些\(B\in {\mathcal {L}} (X)\)，和\(D_0:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\)是一个线性映射，使得\(D_0(A^n)=0\)对于每个\(A\in {\mathcal {A}}\)。这个结果的情况\(n=1\)细化了切尔诺夫定理。