Boundary multipliers of a family of M\"obius invariant function spaces

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.