3
$\begingroup$

Is there any literature corresponding to one or two-parameter semigroups such that e.g. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$?

I have only seen the case where $X(t) \equiv X$ (i.e. there is only one Banach space).

This may be useful for PDE problems where there are moving domains.

Improve this question
$\endgroup$
2
  • $\begingroup$ I don't see how one-parameter semigroups would fit. The semigroup law $T(t)T(s)=T(t+s)$ only makes sense if $T(t)$ and $T(s)$ act on the same Banach space? $\endgroup$
    – gsa
    Commented Oct 5, 2014 at 16:29
  • $\begingroup$ @gsa: This can be remedied if one requires $T(t)\in \bigcap_{a} Hom(X(a),X(a+t))$ or something like that. $\endgroup$
    – Johannes Hahn
    Commented Oct 6, 2014 at 1:11

1 Answer 1

Reset to default
2
$\begingroup$

You will not find many things about such operator families in the literature, but what there is is for the non-autonomous case. See for example

where the evolution family $P_{s,t}$ acts between spaces $L^p(\nu_t)$ and $L^p(\nu_s)$ with a suitable family of measures $\nu_t$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .