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