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Let $A$ and $B$ be commuting $n\times n$ positive definite complex matrices and let $C$,$D$ be other complex matrices. I wish to think of $C,D$ as small and $A,B$ as any size.

Suppose we are looking for a function $f(C,D)$ such that $$ e^{A+C} e^{B+D} = e^{A+B+f(C,D)}$$

I would have thought that one can find a neighbourhood $U$ around the zero matrix such that one can define $f(C,D)$ for $C,D\in U$, and even probably arrange so that $f$ is analytic on $U\times U$ and $f(0,0)=0$. Is this true?

Moreover, is it the case that one can find a neighbourhood $U$ that works no matter how large $A$ and $B$ are?

Any help/related ideas/references much appreciated.

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    $\begingroup$ en.wikipedia.org/wiki/… — The first terms are going to be $A+B+C+D+\frac{1}{2}[A+C,B+D]+\cdots$ so you can't expect $f(C,D)$ not to depend on $A$ and $B$. $\endgroup$
    – Gro-Tsen
    Commented Jan 6, 2017 at 15:16
  • $\begingroup$ I guess this is true if A, B are small. Then the BCH formula will converge if C,D are also small, and how small depends on A,B. But really I was thinking of the case that A,B are large in which case the BCH does not tell me anything at all about a possible f(C,D). Does it? $\endgroup$
    – JRoss
    Commented Jan 6, 2017 at 15:45
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    $\begingroup$ What I'm saying is, if already for small (and commuting) $A,B$ the corrective term given by the BCH formula is not independent of $A$ and $B$, I don't see how you can hope for this to be true for all values (including large ones). But maybe I misunderstood your question. $\endgroup$
    – Gro-Tsen
    Commented Jan 6, 2017 at 16:23

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