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 ...

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
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
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
...

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. ...

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 ...

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 ...

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 ...

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