I have a question about $C_{0}$-semigroup theory.
Let $(H,(\, , \,))$ be a real Hilber space and $(H_{\mathbb{C}}, (\, , \,))$ the its complexification. Any linear operator $(L,D(L))$ on $H$ can be extended to an operator $(L^{\mathbb{C}},D(L^{\mathbb{C}}))$ on $H_{\mathbb{C}}$ by
$L^{\mathbb{C}}([u,v])=[Lu,Lv]$ for $[u,v] \in D(L^{\mathbb{C}}):=\{[u,v] \in H^{\mathbb{C}}: u,v \in D(L) \}$.
Define for $K \in (0,\infty)$, $S(K)=\left\{ z \in \mathbb{C}: |\text{Im} z| \le K \text{Re}z\right\}$.
Definition
Let $K \in (0, \infty)$. A family $(T_{z})_{z \in S(K)}$ of bounded linear operators on $H_{\mathbb{C}}$ is called a holomorphic contraction semigroup on $S(K)$ if
- $T_{0}=id_{H_{\mathbb{C}}}$.
- $T_{z_{1}+z_{2}}=T_{z_{1}}T_{z_{2}}$ for all $z_{1},z_{2} \in S(K)$.
- $T_{z}f \to f$ in $H_{\mathbb{C}}$ as $z \to 0$ in $S(\tilde{K})$ for all $f \in H_{\mathbb{C}}$ and all $\tilde{K}<K$.
- $\|T_{z}\| \leq 1$ for all $z \in S(K)$.
- $z \to (T_{z}f,g)$ is analytic on the interior of $S(K)$ for all $f,g$ in $H_{\mathbb{C}}$.
My question
Let $(X,\mu)$ be a $\sigma$-finite measure space and $(T_{t})$ be a strongly continuous contraction semigroup on $L^{2}(X;\mu)$. Suppose that $(e^{-t}T_{t}^{\mathbb{C}})_{t>0}$ is the restriction of a holomorphic contraction semigroup on the sector $S(K^{-1})$ for some $K \in (0,\infty)$. Then can we find $q_{0}>2$ such that $(T_{t})$ has a strongly continous extention on $L^{p}(X;\mu)$ for any $p \in [2,q_{0}]$?
My researh
If $(T_{t})_{t>0}$ satisfies the Markovian property, we can find such extension on $L^{p}(X;\mu)$ for any $p \in [2,\infty)$...
If you know, please let me know!
Thank you in advance.