Linear Operators

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June 6th 2011, 08:34 AM
liquidpaper

Linear Operators

I need help with this:

Let E a Banach Space and B(E) the Banach space of linear and bounded operators from E to E with the usual norm. Let S $\in$ B(E).

If the sum $\sum\limits_{n = 1}^\infty {\frac{{{S^k}}}{{k!}}}$ convergs to the operator $e^S$ , show that $Se^S$ = $e^S S$ . And if T,S $\in$ B(E) conmute then $e^{S+T}$ = $e^S$ $e^T$ . In particular prove that $e^S$ is invertible.

Please somebody give me a hand.
June 6th 2011, 08:47 AM
girdav

For the first question, put $S_n :=\sum_{k=0}^n\frac{S^k}{k!}$ . Show that for all $n\in\mathbb N$ we have $SS_n=S_nS$ . Then use the fact that the maps $T\mapsto ST$ and $T\mapsto TS$ are continuous for the usual norm on $\mathcal B(E)$ .
June 6th 2011, 11:30 AM
orbit

Quote:

Originally Posted by girdav

For the first question, put $S_n :=\sum_{k=0}^n\frac{S^k}{k!}$ . Show that for all $n\in\mathbb N$ we have $SS_n=S_nS$ . Then use the fact that the maps $T\mapsto ST$ and $T\mapsto TS$ are continuous for the usual norm on $\mathcal B(E)$ .

Sorry to bother again, but I can´t show $S{S_n} = {S_n}S$ ...
June 6th 2011, 12:02 PM
girdav

Prove it by induction.
June 6th 2011, 01:11 PM
orbit

Quote:

Originally Posted by girdav

Prove it by induction.

Done!

Could you explain to me how to use the continuity of T and S?

Thanks"!
June 6th 2011, 01:17 PM
girdav

Show that $SS_n$ converges to $Se^S$ in $\mathbb{B}(E)$ and similarly that $S_nS$ converges to $e^SS$ in $\mathbb{B}(E)$
June 6th 2011, 01:46 PM
orbit

Quote:

Originally Posted by girdav

Show that $SS_n$ converges to $Se^S$ in $\mathbb{B}(E)$ and similarly that $S_nS$ converges to $e^SS$ in $\mathbb{B}(E)$

Done! Thank you again.

And, could you explain a bit how to do the other part?

Thanks!
June 6th 2011, 01:47 PM
orbit

Thank you..

Sorry double post.
June 7th 2011, 01:05 AM
girdav

Since $S$ and $T$ commute, you can try a proof which is similar to the case $S,T\in\mathbb{C}$ . The main step consists of showing that
$e^Se^T = \sum_{n=0}^{+\infty}\sum_{k=0}^n\dfrac{S^k}{k!} \cdot\dfrac{T^{n-k}}{(n-k)!}$ .

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