Let $A$ be a Banach algebra
if $ab=ba$ then prove that $e^{a+b}=e^ae^b$
I've started by $e^a=\sum _{n=0}^{\infty }\frac{a^n}{n!}$, I want to know if this is correct way?
Any help will be greatly appreciated.
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yes, and then multiply it with $e^{b}$. Do you know how to multiply two convergent series? There’s a we’ll known formula – Fakemistake Nov 19 '18 at 17:58
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@Dear Fakemistake, I think I have to youse Cachy formula – user62498 Nov 19 '18 at 18:09
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Correct it‘s Cauchy formula – Fakemistake Nov 19 '18 at 18:10
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Yes. Expand as power series, and you can rearrange the $a$'s and $b$'s because they commute (so $(a+b)^2=a^2+2ab+b^2$ for example), so you can pretend you are doing high-school maths with numbers. You might want to read about the BCH (Baker Campbell Hausdorff) expansion. A slightly harder result holds when $a$ and $b$ both commute with their commutator $[a,b]$, and if that fails then it gets harder ...
Richard Martin
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@DRichard Martinear , please explain a little more about, A slightly harder result holds when $a$ and $b$ both commute with their commutator $[a,b]$, and if that fails then it – user62498 Nov 19 '18 at 18:13