Inequality related to spectral radius

For some reason, you switched the indices from the paper. The "$n=1$" from the paper is your "$d=1$".

So what the paper is saying is that, when $d=1$, $$ r_a(A)=\lim_n\|A_1^{n*}A_1^n\|^{1/2n}=\lim_n\|A_1^n\|^{1/n}=r(A_1). $$ Also $$ \|A_1^n\|^{1/n}\leq\|A_1\|, $$ so $$ r_a(A)\leq r(A_1)\leq \|A_1\|=\sup_{\|x\|=1}\|A_1x\|. $$

Tags:

Functional Analysis

Operator Algebras

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