Geometric Aspects of Operator Calculus: Non-commutative Chain Rules

To understand the complexity of the operator chain rule, we must first revisit the most fundamental product rule. In a commutative setting, the derivative of a product fg is simply:
　(fg)' = f'g + fg' = f'g + g'f.　Because the order does not matter, we can rearrange the terms freely. However, when the chain rule is exposed to a non-commutative operator situation, the order becomes sacrosanct. For two operators A and B, the derivative of their product is strictly:
(AB)' = A'B + A B',
where A' must be located on the left of B, and B' must be located on the right of A. This rigid preservation of order is essential to all non-commutative calculus, and it is definitely true when differentiating the exponential eA, which is an infinite product of operator A.
In this article, the 'Geometry of Variation in Operator Calculus' is studied for the cases when unbounded operators in a Banach space are included. The standard chain rule, when confronted with a non-commutative operator situation, undergoes a fundamental transformation. This evolution is deeply rooted in the structural relationship between operators. Starting from the generalized Baker–Campbell–Hausdorff (BCH) formula based on the logarithmic representation of operators [1-7], which describes the non-commutative multiplication of exponentials, we can derive the precise dynamics of operator variations. Ultimately, this necessitates the generalization of the classical chain rule into what we know as the non-commutative Duhamel’s identity.

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[6] Y. Iwata, Chaos, Solitons & Fractals: X 13 (2024) 100119.
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