2
votes
1answer
61 views

Spaces of class $J_\alpha$

This question is about the spaces of class $J_\alpha$. Given three Banach spaces $Z\subset Y\subset X$ (with continuous embeddings), and given $\alpha\in (0,1)$, we say that $Y$ is of class $J_\alpha$ ...
0
votes
1answer
135 views

how to prove the generator of semigroup is a Banach space

I am not familiar with semigroup theory, so please stand with my dummy question. Say, $A$ is the generator of a semigroup, consider space $X_{n} = D(A^{n})$ with graph norm, $\|f\|_{A^{n}}:=\|f\| + ...
1
vote
3answers
208 views

Is a contraction semigroup infinitesimal operator bounded?

Let $T_t:L\to L$ be a semigroup of linear operators $T_t$ acting on a Banach space $L$. Assume that $$ \|T_t\| := \sup\limits_{f\in L}\frac{\|T_tf\|_L}{\|f\|_L} \leq 1 $$ for all $t\geq 0$. The ...