  We consider the restriction to $SL_2({\mathbb Q}_p)$ of an irreducible
$p$-adic unitary Banach space representation $\Pi$ of $GL_2({\mathbb Q}_p)$. If
$\Pi$ is associated, via the $p$-adic local Langlands correspondence, to an
absolutely irreducible 2-dimensional Galois representation $\psi$, then the
restriction of $\Pi$ decomposes as a direct sum of $r \le 2$ irreducible
representations. The main result of this paper is that $r$ is equal to the
cardinality $s$ of the centralizer in $PGL_2$ of the projective Galois
representation $\overline{\psi}$ associated to $\psi$, and the restriction is
multiplicity-free, except if $\psi$ is triply-imprimitive, in which case the
restriction of $\Pi$ is a direct sum of two equivalent representations. From
this result we derive a classification of absolutely irreducible $p$-adic
unitary Banach space representations of $SL_2({\mathbb Q}_p)$.
