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We investigate the numerical radius of weighted shifts on $\ell^{2}$ as anorm on $\ell^{\infty}$. Along the way, we prove that the largest value aBanach limit can attain on the sequence of absolute values of a boundedsequences is larger or equal to the spectral radius and smaller or equal to thenumerical radius of the corresponding weighted shift. Further, we compare theoperator norms on $\ell^{\infty}$ with respect to the uniform norm and thenumerical radius of weighted shifts as a norm. We show that they generallydiffer, but for multiplication operators and weighted shifts on$\ell^{\infty}$, they are both equal to the uniform norm of the correspondingweight. Moreover, we prove that all complex-valued Banach limits satisfy thenorm inequality with respect to the numerical radius of weighted shifts andprovide an application of our results to the theory of Banach limits.

On the numerical radius of weighted shifts on $\ell^{2}$ as a norm on $\ell^{\infty}$ and connections to Banach limits