  The spectral Tur\'an theorem states that the $k$-partite Tur\'an graph is the
unique graph attaining the maximum adjacency spectral radius among all graphs
of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This
result is known to be stronger than the classical Tur\'an theorem. In this
paper, we consider hypergraph extensions of spectral Tur\'an theorem. For
$k\geq r\geq 2$, let $H_{k+1}^{(r)}$ be the $r$-uniform hypergraph obtained
from $K_{k+1}$ by enlarging each edge with a new set of $(r-2)$ vertices. Let
$F_{k+1}^{(r)}$ be the $r$-uniform hypergraph with edges: $\{1,2,\ldots,r\} =:
[r]$ and $E_{ij} \cup\{i,j\}$ over all pairs $\{i,j\}\in
\binom{[k+1]}{2}\setminus\binom{[r]}{2}$, where $E_{ij}$ are pairwise disjoint
$(r-2)$-sets disjoint from $[k+1]$. Generalizing the Tur\'an theorem to
hypergraphs, Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] and
Mubayi and Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] respectively
determined the exact Tur\'an number of $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, and
characterized the corresponding extremal hypergraphs.
  Our main results show that $T_r(n,k)$, the complete $k$-partite $r$-uniform
hypergraph on $n$ vertices where no two parts differ by more than one in size,
is the unique hypergraph having the maximum $p$-spectral radius among all
$n$-vertex $H_{k+1}^{(r)}$-free (resp. $F_{k+1}^{(r)}$-free) $r$-uniform
hypergraphs for sufficiently large $n$. These findings are obtained by
establishing $p$-spectral version of the stability theorems. Our results offer
$p$-spectral analogues of the results by Mubayi and Pikhurko, and connect both
hypergraph Tur\'an theorem and hypergraph spectral Tur\'an theorem in a unified
form via the $p$-spectral radius.
