B Commutators of functions of operators

LagrangeEuler
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I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute? Also I have a question is it possible that
##e^A## is matrix where all elements are ##\infty## so that ##e^A \cdot e^B-e^B\cdot e^A## has all elements that are ##\infty##?
 
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LagrangeEuler said:
I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute?
Have you tried to prove that?

LagrangeEuler said:
Also I have a question is it possible that
##e^A## is matrix where all elements are ##\infty## so that ##e^A \cdot e^B-e^B\cdot e^A## has all elements that are ##\infty##?
That makes no sense. Matrices cannot have infinite elements.
 
LagrangeEuler said:
I would like to ask whether if operators ##A## and ##B## commute also operators ##e^A## and ##e^B## commute?
I would try to prove that by showing with mathematical induction that any partial sum MacLaurin expansions ##\displaystyle\sum\limits_{k=0}^{N}\frac{1}{k!}A^k## and ##\displaystyle\sum\limits_{k=0}^{N}\frac{1}{k!}B^k## with a finite ##N## commute when ##[A,B]=0##. It results just from the fact that ##A^m B^n = B^n A^m## for any ##m,n\in\mathbb{N}## when ##[A,B]=0##, so the order of operators doesn't make any difference when forming that polynomial of two operator variables.
 
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