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Every new inner product in a Hilbert space is obtained from the original oneby means of a unique positive operator$.$ The first part of the paper is asurvey on applications of such a technique, including a characterization ofsimilarity to isometries$.$ The second part focuses on Banach limits fordealing with power bounded operators. It is shown that if a power boundedoperator for which the sequence of shifted Ces\`aro means converges (at leastin the weak topology) uniformly in the shift parameter, then it has a Ces\`aroasymptotic limit coinciding with its $\varphi$-asymptotic limit for all Banachlimits $\varphi$.

Asymptotic limits, Banach limits, and Ces\`aro means