  For $1<p<\infty$ and $0<s<1$, let $\mathcal{Q}^p_ s (\mathbb{T})$ be the
space of those functions $f$ which belong to
  $ L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset
\mathbb{T}}\frac{1}{|I|^s}\int_I\int_I\frac{|f(\zeta)-f(\eta)|^p}{|\zeta-\eta|^{2-s}}|d\zeta||d\eta|<\infty,
\] where $|I|$ is the length of an arc $I$ of the unit circle $\mathbb{T}$ . In
this paper, we give a complete description of multipliers between
$\mathcal{Q}^p_ s (\mathbb{T})$ spaces. The spectra of multiplication operators
on $\mathcal{Q}^p_ s (\mathbb{T})$ are also obtained.
