Timeline for Is exp(rA) = (exp(A))^r for real r and A in a Banach space?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 27, 2010 at 0:32 | vote | accept | soulphysics | ||
| Dec 24, 2010 at 22:52 | answer | added | Gerald Edgar | timeline score: 5 | |
| Dec 24, 2010 at 17:38 | comment | added | Pietro Majer | You are asking why $(f\circ g)(A)= f(g(A))$. The answer is, it is just a matter of functional calculus; check any textbook on the topic. | |
| Dec 24, 2010 at 16:59 | comment | added | fedja | The Taylor expansion of the logarithm diverges more often than not. There is no reason to expect it to converge for $e^A$. Of course, if everything is defined as a series and you can meaningfully plug series into series justifying all passages to the limits, the identity holds. On the other hand, if the right hand side just fails to exist or has multiple values like non-integer powers of complex numbers (the first example to look at), the question hardly makes sense. | |
| Dec 24, 2010 at 16:44 | comment | added | soulphysics | @fedja -- One way is the Taylor expansion: $A^r = \Sigma^\infty_{n=0}\frac{r^n \log^n(A)}{n!},$ where the logarithms are again defined by their Taylor expansions. @arsmath -- That's a way to define the operator $e^X$ (where $X$ is in the Banach algebra), but what's needed is $X^r$. | |
| Dec 24, 2010 at 16:09 | comment | added | arsmath | The conventional definition is to use the limit of (1 + x/n)^n as n goes to infinity. (There's a slightly different variants that's more likely to converge, but I can't remember what it is.) | |
| Dec 24, 2010 at 15:59 | comment | added | Denis Serre | follows from the Baker-Hausdorff lemma and the fact that $[A,A]=0$: isn't is pedantic ? Just say that $[A,B]=0$ implies $e^{A+B}=e^Ae^B$. | |
| Dec 24, 2010 at 15:36 | comment | added | Mariano Suárez-Álvarez | Using the binomial series, I guess? | |
| Dec 24, 2010 at 15:35 | comment | added | fedja | How do you define the non-integer powers for arbitrary Banach algebra elements? | |
| Dec 24, 2010 at 15:29 | history | asked | soulphysics | CC BY-SA 2.5 |