User stephan fackler - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:24:05Z http://mathoverflow.net/feeds/user/21704 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129613/sz-nagy-dilation-for-uniformly-convex-banach-spaces/129622#129622 Answer by Stephan Fackler for Sz.-Nagy dilation for uniformly convex Banach spaces Stephan Fackler 2013-05-04T08:42:18Z 2013-05-04T08:42:18Z <p>I would give this partial answer as a comment but it seems that I do not have earned enough credit points yet.</p> <p>I do not know whether such a general dilation theorem holds. But on $L^p$-spaces there is an analogue for <em>positive, contractive</em> operators: the Akcoglu-Sucheston dilation theorem. I personally like the presentation in <a href="http://arxiv.org/abs/1202.5425" rel="nofollow">On Dilations and Transference for Continuous One-Parameter Semigroups of Positive Contractions on $\mathcal{L}^p$-spaces</a> by G. Fendler and the <a href="http://cms.math.ca/10.4153/CMB-1982-054-8" rel="nofollow">lattice-theoretic approach of R. Nagel &amp; G. Palm</a>. </p> http://mathoverflow.net/questions/90112/inequalities-for-uniformly-convex-normed-spaces/90165#90165 Answer by Stephan Fackler for Inequalities for uniformly convex normed spaces Stephan Fackler 2012-03-04T00:17:57Z 2012-03-04T00:17:57Z <p>In addition to S. Ivanov's proof, I give you a reference to the literature. A proof can be found in Classical Banach Spaces II by J. Lindenstrauss &amp; L. Tzafriri in Section 1.e directly after the definition of uniform convexity on p. 60.</p> http://mathoverflow.net/questions/89441/the-part-of-an-operator-as-an-analytic-generator/89499#89499 Answer by Stephan Fackler for The part of an operator as an analytic generator Stephan Fackler 2012-02-25T16:48:47Z 2012-02-25T16:48:47Z <p>This is wrong. Let us assume that $Y$ is a closed subspace of $X$ to clearify the problem. As Matthew Daws already said, you have to assume that the semigroup $(T(t))$ generated by $A$ leaves $Y$ invariant: suppose that $A_Y$ indeed generates a (strongly continuous) semigroup $(S(t))$ on $Y$. Then for example the Yosida-approximation shows that $T_Y(t) = S(t)$ for all $t$.</p> <p>I have the impression that you have a concrete application in mind. So maybe you should reformulate the question in this concrete setting?</p> http://mathoverflow.net/questions/89441/the-part-of-an-operator-as-an-analytic-generator/89499#89499 Comment by Stephan Fackler Stephan Fackler 2012-02-25T19:45:51Z 2012-02-25T19:45:51Z $A_Y$ can be a generator, but is not in general. If $Y$ is a closed subspace as in your original question, the argument above shows that the invariance of $(T(t))$ is necessary. It is also sufficient: $(T(z))$ can be restricted to an analytic semigroup on $Y$ with generator $A_Y$. So the problem is not interesting in this case. However, if $Y$ is only continuously embedded into $Y$, your question is interesting and has a positive answer in natural situations: for example if $X$ is a Hilbert space and $A$ is induced by a form, see &quot;Analysis of Heat Equations&quot; by Ouhabaz if you're interested.