6
votes
Accepted
On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces
This is a much-studied problem. If you do not require the $V$'s to commute, then a dilation (even a unitary dilation) always exists, this is a theorem of Bozejko. For commuting operators $T$ (and ...
- 2,351
5
votes
Accepted
Generation of strict contraction semigroups
Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see Proposition V.1.7. in
Engel, Klaus-Jochen; Nagel, Rainer, ...
- 4,649
4
votes
On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces
To add to what Mike Jury wrote:
Paulsen's book also contains what he calls the Sz.-Nagy-Foias Theorem, stating that $n$ doubly commuting contractions dilate to $n$ doubly commuting unitaries. We say $...
- 3,894
3
votes
Accepted
Gradient flows: convex potential vs. contractive flow?
Doesn't this follow from dependence on initial data?
Consider the flow mapping $\Phi(t,X)$ which solves
$$ \frac{d}{dt}\Phi(t,X) = - \nabla V(\Phi(t,X)) $$
so taking the derivative in $X$ we have
...
- 33.9k
2
votes
Accepted
Small contraction for Hyperkähler Varieties
Let $f:X\to Y$ be a birational contraction where $X$ is hyperkähler, then $K_X\sim 0$ and $K_Y=f_*K_X\sim 0$, and hence $K_X=f^*K_Y$. In particular, this means that $Y$ has canonical singularities. ...
- 1,054
2
votes
Gradient flows: convex potential vs. contractive flow?
It should be noticed that already on $R^d$ equipped with a non-Euclidean norm $\|.\|$ the answer to your question is no. Ohta-Sturm [1] proved the following: let $\lambda\in R$ and consider the ...
- 608
1
vote
Accepted
Is this a contraction mapping for small $T$?
$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\De}{\Delta}\newcommand\R{\mathbb R}$Edit: This answer is insufficient, even though (almost) all the reasoning appears relevant to the ...
- 98.4k
1
vote
Set operations over iterated function systems
If $F = f_1, \ldots, f_n$ is an IFS, then the attractor of any subset of F is a subset of the attractor of F. This is readily seen through coding points in the attractor (see Barnsley's book for the ...
- 4,900
Only top scored, non community-wiki answers of a minimum length are eligible