What is $\left\| f \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$.

I'm reading a paper Chern-Yamabe flow which said that

**Now for $k$ big enough $(k>n)$, the first eigenvalue of the operator $-\Delta$ on the space $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \mathrm{vol}_{g}=0\right\}$ is strictly negative hence $\left\|e^{-t \Delta}\right\|_{H_{0}^{k}, H_{0}^{k}} \leq$ $C_{0} e^{-c t}$.**

I have no idea what is $\left\| - \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm.