Spectral radius (Banach algebra)

Hi everyone,

I would like to ask for some help about the following problem:

"__A__ is a Banach algebra on Hilbert space with unitary element. Prove that the following property stand for the spectral radius (r):

r(x y) =r (x y) where x and y are elements of the __A __Banach algebra."

Thanks in advance

Johan

Re: Spectral radius (Banach algebra)

Do you mean $\displaystyle r(xy)=r(yx)$? Use the formula $\displaystyle r(a)=\lim_{n\to\infty}\lVert a^n\rVert^{1/n}$ and (yx)^n=y(xy)^{n-1}x$.

Re: Spectral radius (Banach algebra)

Thanks girdav for quick reply