1
$\begingroup$

I have a problem with understanding the definition of strong continuity and uniform continuity for the families of operators, e.g. semigroups.

Let $(X_t)_{t \geq 0}$ be a family of bounded linear operators on a Hilbert space $H$. If somebody says that the map $t \mapsto X_t$ is strongly continuous does it mean the following

  1. $ \forall_{x \in H} \forall_{\varepsilon >0} \exists_{\delta >0} \forall_{t, s \geq 0} |t-s| < \delta \Rightarrow \|(X_t-X_s)x\| < \varepsilon, $

    which is uniform continuity in time with a strong topology considered on $B(H)$.

or 2. $ \forall_{x \in H} \forall_{t \geq 0} \forall_{\varepsilon >0} \exists_{\delta >0} \forall_{ s \geq 0} |t-s| < \delta \Rightarrow \|(X_t-X_s)x\| < \varepsilon, $ which is a continuity in time with a strong topology considered on $B(H)$?

The same for so called norm continuous (or uniformly continuous) semigroups, that is, is it

  1. $ \forall_{\varepsilon >0} \exists_{\delta >0} \forall_{t, s \geq 0} |t-s| < \delta \Rightarrow \|X_t-X_s\| < \varepsilon, $

    which is uniform continuity in time with a uniform topology considered on $B(H)$

or 2. instead of uniform continuity in time it is just a continuity in time for all $t\geq 0$?

Thank you for the help.

$\endgroup$
  • 1
    $\begingroup$ Ok, I got it it is the 2nd option in both cases, but of course the continuity is uniform on every compact intervals of $[0, \infty)$. $\endgroup$ – ResetHead255 Aug 26 '13 at 16:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.