  In the present paper, the Karhunen-Lo{\`e}ve eigenvalues for a sub-fractional
Brownian motion are considered in the case of $H>\frac12$. Rigorous large $n$
asymptotics for those eigenvalues are shown, based on functional analysis
method. By virtue of these asymptotics, along with some standard large
deviations results, asymptotically estimates for the closely related problem of
small $L^2$-ball probabilities for a sub-fractional Brownian motion are
derived. By the way, asymptotic analysis on the Karhunen-Lo{\`e}ve eigenvalues
for the corresponding "derivative" process is also established.
