  In this paper, we give a necessary and sufficient condition for which a
finitely generated group has a property like Kazhdan's Property $(T)$
restricted to one isometric representation on a strictly convex Banach space
without non-zero invariant vectors. Similarly, we give a necessary and
sufficient condition for which a finitely generated group has a property like
Property $(FH)$ restricted to the set of the affine isometric actions whose
linear part are one isometric representation on a strictly convex Banach space
without non-zero invariant vectors. If the Banach space is the $\ell^p$ space
($1<p<\infty$) on a finitely generated group, these conditions are regarded as
an estimation of the spectrum of the $p$-Laplace operator on the $\ell^p$ space
and on the $p$-Dirichlet finite space respectively.
