Map between Bochner spaces induced by map between Hilbert spaces

Question

Assume I have a map $F:U \to V$ where $U$ and $V$ are Hilbert spaces. Consider now the spaces $L^p(T;U)$ and $L^p(T;V)$, where $T$ is a measure space. Will $F$ induce a mapping between these two Bochner spaces?

It feels reasonable that it in some cases should, for instance if $[a,b]=T$, but I can't really show anything.

I get that it essentially boils down to showing that $$ \int_a^b \|F(u(t))\|^2\, \mathrm{d}t <\infty $$ given that $u \in L^2([a,b],U)$ and $F(u(t)) \in L^2([a,b],V)$, but the above does not have to hold just by assuming that $u(t) \in U$ and $F(u(t)) \in V$, since integrating something finite really does not have to be finite. So When does it work? Any assumptions that can be made on the spaces to make it work?

Example: $F$ is a bounded linear map. – user10354138 Jun 21 '21 at 17:51 — user10354138, Jun 21 '21 at 17:51

score 1 · Accepted Answer · answered Jun 21 '21 at 18:01

The following conditions are sufficient:

$F$ is continuous,
and $\exists a \in L^p(T;\mathbb R),b\in \mathbb R$ such that $$ \|F(u)\|_V \le a + b \|u\|_U \quad \forall u\in U. $$

Then $F(u)$ is measurable and belongs to $L^p(V)$ for all $u\in L^p(U)$.

Map between Bochner spaces induced by map between Hilbert spaces

1 Answers1