Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors

Abstract: 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.