All Questions
10,240 questions
6
votes
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How to choose phase to give a desired Fourier transform
Cross posted from MSE.
I have a mathematical problem arising from a physics application, which I feel must have been solved before, but I don't know the terminology associated with it. I am looking ...
- 956
6
votes
1
answer
393
views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
$\newcommand{\R}{\mathbb R}$
$\newcommand{\N}{\mathbb N}$
$\newcommand{\de}{\delta}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\Average}[1]{\left\langle#1\right\rangle} $
$\newcommand{\IP}[2]{\Average{...
- 6,741
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1
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567
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Is restriction a closed map?
Originally asked on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ ...
- 5,529
6
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1
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180
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Symmetric C* property
This is a cross-post from math.stackexchange.com
What can be said about elements $a$ of a C*-algebra which fulfil the 'symmetric C* property'
$$\| a^\ast a+ aa^\ast\|=2\| a\|^2$$
I'd guess that this ...
- 623
6
votes
1
answer
898
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Injectivity of a Fredholm operator
While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.
Question : Suppose $K$ is ...
- 406
6
votes
1
answer
2k
views
About weak convergence of probability measure
Suppose $\mu_j$ is a sequence of measures on $\mathbb{R}$. By the definition of weak convergence of measures, $\mu_j$ weak converges to $\mu$ means that for any bounded continuous function $f$, there ...
- 71
6
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4
answers
1k
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Resource recommendation: Spectral theory and $C^*$ algebras
I have formally studied functional analysis, both as university courses, and by myself, but this is one area of mathematics I find so huge and complicated, I have a hard time properly getting into it.
...
- 3,307
6
votes
2
answers
351
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$id:A\to A^{op}$ is completely positive iff $A$ is abelian
Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive.
The claim is: $id$ is completely positive iff $A$ is abelian.
I need ...
user62639
6
votes
2
answers
672
views
Holomorphy of a function with values in a Hilbert space
Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
- 395
6
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1
answer
243
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Existence of a measurable map between metric spaces
Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...
- 21.8k
6
votes
1
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713
views
Equivalence of Gaussian measures
Let $H$ be a separable Hilbert space and $N(0, C)$ and $N(0, D)$ be Gaussian measures on it. Further, for each $v \in H$, define $R_v = \frac{\left\langle v,Cv \right\rangle}{\left\langle v,Dv \right\...
- 157
6
votes
2
answers
622
views
If Gaussian measures on a Hilbert space converge weakly to 0, how do their covariance operators converge?
Suppose we have a sequence of Gaussian measures $N(0, S(n))$ supported on a Hilbert space $H$ and we know that the sequence converges weakly to the delta measure at $0$, what are the necessary and ...
- 93
6
votes
1
answer
278
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Is the ideal of compact operators strongly Borel?
Let $H$ be a separable infinite dimensional Hilbert space. Denote by $\mathcal{B}(H)$ the space of bounded operators on $H$, and $\mathcal{K}(H)$ the ideal of compact operators. When endowed with the ...
- 3,115
6
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1
answer
644
views
Inductive limit of C*-algebras
It is known that an intertwining (or even approximately intertwining) diagram implies the isomorphism of the limit algebras. Under what conditions the converse holds?
- 169
6
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1
answer
526
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Strong convergence of projections in $B(H)$
(I asked this question at math stackexchange 4 months ago, but received no answers)
Let $\{e_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q_k$ by
$$
q_k=\...
- 2,793
6
votes
1
answer
1k
views
Symmetric basis of harmonic homogeneous polynomials
Recently, a question about the beautiful theory of harmonic polynomials made me aware there is something
I've wanted to know for a long time.
As is well known, for any number of variables $n$ and any ...
- 60.6k
6
votes
1
answer
228
views
Question about Bochner measurability
When I study parabolic pde's I often came across the following type of Bochner spaces $L^p([a,b];L^{q}(\Omega),\ W^{1,p}([a,b];L^{q}(\Omega))$ and $L^{q}([a,b];W^{1,p}(\Omega))$ where $p,q\geq 1$ and $...
- 1,759
6
votes
1
answer
252
views
Poisson kernel for the orthogonal groups
For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
- 1,549
6
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1
answer
319
views
How are coordinate charts constructed in noncommutative geometry?
In noncommutative geometry, one is given a triple $(A,D,H)$, where $A$ is a commutative C* algebra, $H$ is a Hilbert space, and $D$ is an operator. There is a somewhat long list of conditions that ...
- 593
6
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1
answer
237
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Radon-Nikodym property in Diestel & Uhl: a definition clarification
I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly ...
- 201
6
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1
answer
268
views
Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors
For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$.
It is well-know that a series $\sum_{n\in\omega}r_n$ of real numbers converges ...
- 42k
6
votes
1
answer
499
views
Can I find a bump function $\psi$ such that $\nabla\log\psi$ vanishes too?
Consider a bump function supported in the ball of radius $1$, that is $\psi:\mathbb R^n\to\mathbb R$ such that
$\ \psi(x)>0$ for $|x|<1$
$\ \psi(x)=0$ for $|x|\geq 1$
$\ \psi\in C^\infty$.
...
- 311
6
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1
answer
276
views
What is the Gelfand dual of an open surjection?
Does anyone have a handy characterisation of open continuous surjections $X \to Y$ in terms of the corresponding injective $*$-homomorphism $C(Y) \to C(X)$? (I'm only interested in the case where $X$ ...
- 508
6
votes
2
answers
509
views
A question on Grothendieck space
A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck ...
- 3,343
6
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1
answer
377
views
quick question about renorming quasi-Banach spaces into p-Banach spaces
I have a quick question which is probably supposed to be obvious, but for some reason I just don't see it: How does one re-norm a quasi-Banach space to produce a $p$-Banach space ($0<p\leq 1$) ...
- 1,591
6
votes
2
answers
906
views
Gaussian measure on function spaces
I'm reading this classic work and I'd like to get deeper inside some of its techniques. In particular, the authors state: "We construct a Gaussian measure $d\mu_{0}(\phi)$ on a measure space of ...
- 3,535
6
votes
1
answer
252
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Arens regularity of Banach algebras
I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity ...
- 243
6
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1
answer
354
views
Equivalence of $\sigma$-weak topology to another topology
Let $\mathcal H$ be a Hilbert space. Define a topology $\tau_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of ...
- 1,879
6
votes
1
answer
1k
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Lipschitz function of independent subgaussian random variables
This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory).
If $X\in\mathbb{...
- 6,541
6
votes
1
answer
200
views
Coarse index of Dirac operator on $\mathbb{R}$
Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(...
- 1,913
6
votes
2
answers
355
views
Complete dual of bornological space
A bornologigal topological vector space is such that any bounded linear function on it is continuous. It is a standard result [Jarchow, Locally convex spaces, 1981] that if the dual $E'$ of a Mackey ...
- 103
6
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1
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1k
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Prove that the flow of a divergence-free vector field is measure preserving
On page 3 of this preprint, after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi_t(\cdot)$ generated by $a(t, \cdot)$ ...
- 839
6
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1
answer
150
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Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces
Let $A$ (resp. $B$) be a unital $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) ...
- 1,586
6
votes
1
answer
1k
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Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
- 969
6
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1
answer
452
views
Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...
- 42k
6
votes
1
answer
238
views
Extending a weak*-converging sequence onto a superspace
Let $X$ be a real Banach space and $Y\subset X$ be a (closed) subspace of $X$. Assume that a sequence $y_n^*\in S_{Y^*}$ weak*-converges to some $y^*\in S_{Y*}$. (Here $S_{Y^*}$ stands for the dual ...
6
votes
1
answer
1k
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Schmidt decomposition on infinite-dimensional Hilbert spaces
The Schmidt decomposition theorem says:
If $H_1,H_2$ are Hilbert-spaces (for simplicity: of same dimension) and
$x\in H_1\otimes H_2$, then there exist orthonormal bases
$\alpha_i,\beta_i$ of $...
- 896
6
votes
2
answers
530
views
Schrödinger eigenfunctions are bounded
Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
- 129
6
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1
answer
780
views
Resolvents of Schrodinger operators
In the free case one can compute the resolvents of the Laplacian $-\Delta$ in many cases explicitly, in the sense that they are given by an integral operator. Often, one uses the Hille-Yosida theorem ...
- 305
6
votes
2
answers
318
views
Natural map $C^*(G) \to M(A\rtimes G)$
Let $A$ be a C*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota_G\colon C^...
- 717
6
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1
answer
946
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Are Sobolev spaces on non-compact manifolds separable?
If $M$ is a Riemannian manifold that is not compact, is it true that the Sobolev spaces on $M$, $W^{k,p}(M)$, still be separable (for $p < \infty$)?
- 83
6
votes
1
answer
916
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Rellich-Kondrachov compacteness Theorem for the Euclidean space with Gaussian measure
Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by
$$
\gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}.
$$
Let $d\gamma_n$ denote the following measure ...
6
votes
3
answers
481
views
Quantum Mechanics and bilinear optimal control theory
I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t)...
- 259
6
votes
1
answer
225
views
Is the space of vectorial functions that are Dunford integrable complete?
Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...
6
votes
1
answer
2k
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Kullback Leibler "variance": does that divergence have a name?
If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...
6
votes
2
answers
1k
views
Characterization of ideals in the bounded operators
Let $\mathcal{B}(H)$ denote the C*-algebra of all bounded operators on a separable infinite dimensional complex Hilbert space $H$. It is a well-known fact that $\mathcal{B}_0(H)$, the ideal of compact ...
- 3,115
6
votes
2
answers
519
views
Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)
This question is concerning the paper, particularly the proof of Lemma 2.1 in Section 2.1:
Matas, A., Merker, J. Existence of weak solutions to doubly degenerate diffusion equations, Appl Math 57 (...
- 266
6
votes
2
answers
405
views
$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$
Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le \...
- 23k
6
votes
2
answers
963
views
Tightness of Measures, Riesz Representation for locally compact spaces
Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, ...
- 71
6
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1
answer
364
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A question about uniformly bounded semigroups
Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there ...
