Minkowski type inequality in Banach algebras Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
 A: The case $n=1$ holds always.
For $n\geq2$, i don't have any kind of general answer, but I expect such inequality to hold basically for norms that behave like the one-norm, and little else. 
As an example, the inequality fails for any norm in the $2\times2$ matrices. Indeed, if
$$
A=\begin{bmatrix}0&1\\0&0\end{bmatrix},\ \ B=\begin{bmatrix}0&0\\1&0\end{bmatrix},
$$
then
$$
\|(A+B)^n\|>0, \ \mbox{ while } \ A^n=B^n=0.
$$
And of course this idea works for any algebra where you have nilpotent elements with non-nilpotent sum.
A: Having duly noted Martin Argerami's answer, I thought it worth mentioning that for normal $A,B$ in a $C^*$-algebra, we have
$$
\|(A+B)^m\|^{1/m} \leq \|A+B\| \leq \|A\|+\|B\|=\|A^m\|^{1/m}+\|B^m\|^{1/m}.
$$
Observe also that for commuting normal operators, we have the following Cauchy-Schwarz type inequality
$$
\|AB x\|^2=\langle A^*Ax,B^*Bx\rangle\leq \|A^*Ax\|\|B^*Bx\|=\|A^2x\|\|B^2x\|,
$$
which implies
$$
\|(A+B)^2x\|\leq \|A(A+B)x\|+\|B(A+B)x\|\leq (\|A^2x\|^{1/2}+\|B^2x\|^{1/2})\|(A+B)x\|^{1/2},
$$
or
$$
\|(A+B)^2x\|^{1/2}\leq \|A^2x\|^{1/2}+\|B^2x\|^{1/2}.
$$
I'm quite sure more can be done if one employs the functional calculus.
