All Questions
10,050 questions
9
votes
1
answer
280
views
Fredholm theory on Fr\'echet spaces
Dear everybody,
In my study of the classial Fredholm theory on Banach spaces, I am interested in the corresponding Fredholm theory on Fr\'echet spaces. But it seems to me that there is
little ...
- 515
9
votes
1
answer
395
views
Is there a coalgebraic characterisation of the hyperfinite II_1 factor?
Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
- 5,094
9
votes
1
answer
670
views
Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"
It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
- 191
9
votes
1
answer
621
views
Uniqueness of solutions of Young differential equations
Consider the following one dimensional Young differential equation:
\begin{align*}
&Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1];\\
&Y_0=0.
\end{align*}
Here the driving process $X$ is a bounded ...
- 931
9
votes
1
answer
916
views
Inverse Fourier transform of an $L^2$ function as limit on balls
$B_m :=\{x \in \Bbb R^n : ||x|| \le m\}$ and $\mathscr{F}f$
denotes the $L^2$ fourier transform of an $f \in L^2(\Bbb R^n)$.
I am trying to show that
If $f \in L^2(\Bbb R^n)$ then $f(y)=\lim\limits_{...
- 201
9
votes
1
answer
385
views
A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator
Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
$$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...
- 1,109
9
votes
1
answer
598
views
Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity
I'm currently trying to get familiar with the Jordan normal form for matrices; and after some example I ask for the possible Jordan-form for the Carleman matrix for the function $f(x) = \sin(x)$ when ...
- 5,342
9
votes
3
answers
654
views
measure with given push-forwards
Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps $p_1,\ldots,p_n:Y\...
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
9
votes
0
answers
163
views
Moore-Penrose partial isometries and hermitian elements
Let $A$ be a unital Banach algebra. An element $a \in A$ is hermitian if $\|\mathrm{exp}(ita)\|=1$ for every $t \in \mathbb{R}$. An element $a \in A$ is Moore-Penrose invertible if there exists $b \in ...
- 3,497
9
votes
0
answers
240
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
- 146
9
votes
0
answers
210
views
Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,424
9
votes
0
answers
1k
views
Weak compactness in $\mathcal{F}(X)$
Let $(X,0)$ be a pointed metric space and let $\mathcal{F}(X)$ be the natural predual of ${\rm Lip}_0(X)$, the space of Lipschitz functions on $X$ that map $0$ to $0$; here $\mathcal{F}(X)$ is really ...
- 11.3k
9
votes
0
answers
540
views
Why is spectral theory developed for $\mathbb C$
Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
- 381
9
votes
0
answers
137
views
A self-isometry of the sphere of a strictly convex Banach space that does not move basic vectors
Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points. ...
- 42k
9
votes
0
answers
176
views
Is the switch automorphism inner for continuous-trace $C^*$-algebras?
If $R$ is a commutative ring, and $A$ is an Azumaya algebra over $R$, then the switch (or flip, or exchange, etc.) automorphism of $A\otimes_R A$, given by $a\otimes b\mapsto b\otimes a$, is inner: it ...
- 272
9
votes
0
answers
230
views
Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
9
votes
0
answers
240
views
Reference request: integral formula for $\sum_{\text{roots }\lambda}e^{-|\lambda|^2}$
Consider a polynomial $f(z)=c\prod_m(z-\lambda_m)\in\mathbb{C}[z]$. I am mostly interested in the case where this actually lies in $\mathbb{R}[z]$, but that is not essential. I wanted to find a nice ...
- 56.9k
9
votes
0
answers
164
views
Comparison of the absolute value of an operator with its positive parts, II
Suppose $A,B\in M_n(\mathbb C)$ are self-adjoint. Does there exist a constant $C>0$ depending only on $n$ such that
$$
|A+iB| \leq C(|A| + |B|)?
$$
One can take $C=1$ if $A$ and $B$ commute.
More ...
- 3,984
9
votes
0
answers
261
views
SVD-type decomposition for the tensor product of three Hilbert spaces?
(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the ...
- 25.8k
9
votes
0
answers
953
views
Topologies on compactly supported functions
Let M be a (non-compact) smooth manifold and consider the set $C^\infty_c(M)$ of smooth real-valued functions with compact support. We can give this function space several topologies. Here are four:
...
- 27.5k
9
votes
0
answers
284
views
Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-algebras, where $ -1 < q < 1 $
This question is related to one that I asked some time ago.
Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the $ ...
- 1,396
9
votes
0
answers
979
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
9
votes
0
answers
351
views
How many ideals are there in $B(H)^{**}$?
It is well-known (and easy to prove) that the only closed ideals of $B(\ell_2)$ are $\{0\}$, $B(\ell_2)$ and $K(\ell_2)$, the ideal of compact operators on $\ell_2$. I am curious whether we know what ...
- 91
9
votes
0
answers
305
views
Convergence in $L^2$ of iterated expectations
Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...
- 1,068
9
votes
0
answers
397
views
Does the algebra of bounded variation functions have a "noncommutative geometric" meaning and generalization?
According to Gelfand-Naimark theory, $C^*$-algebras of continuous functions $\mathcal{C}^0(X,\mathbb{C})$ on a compact Hausdorff topological space completely capture its topology. Furthermore, every ...
- 23.3k
9
votes
0
answers
885
views
Continuous projections in $\ell_1$ with norm $>1$
I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $...
- 1,697
9
votes
1
answer
1k
views
Noncompactness of the Sobolev embedding in the critical exponent case
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$.
It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
- 446
8
votes
4
answers
681
views
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
- 364
8
votes
3
answers
2k
views
The "Spaces of Schwartz distributions are finite dimensional" challenge
The more I study Schwartz distributions and the corresponding spaces, the more the latter look "finite dimensional" to me. Of course they are not finite dimensional in the technical sense but they are ...
8
votes
3
answers
1k
views
Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?
Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...
- 221
8
votes
3
answers
1k
views
Are all positive eigenfunctions principal eigenfunctions?
In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$?
Also, more generally, does this also apply for $...
- 421
8
votes
3
answers
1k
views
is every element in a C* algebra a product of normal elements?
I have the following question and since I am not an expert on C*-algebras, I thought I ask it here:
I know that in general the sum and product of normal elements need not be normal. It is even true ...
- 987
8
votes
3
answers
1k
views
Relating a Polynomial equation to the characteristic equation of a Hermitian matrix
This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
- 1,421
8
votes
2
answers
905
views
Continuous linear functionals and the Axiom of Choice
Can one prove without the Axiom of Choice that for every normed vector space $X$ there exist a nonzero continuous linear functional on $X$?
- 935
8
votes
2
answers
837
views
Strict topology between weak and norm topologies
I want to believe that this has an easy answer, but I’ve never considered it before and can’t seem to answer it now either.
Does every infinite-dimensional Banach space admit a locally convex vector ...
- 1,453
8
votes
3
answers
1k
views
Conceptually, what does unitization do?
Let $(\mathcal A,||\cdot||)$ be a normed algebra (with or without a unit). The unitization of $\mathcal A$ is the space $\mathcal A_+:=\mathcal A\oplus \Bbb C$ where the multiplication operation $\...
- 1,245
8
votes
4
answers
13k
views
Eigenvalues of infinite matrices [closed]
I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can ...
- 431
8
votes
3
answers
3k
views
The mean of points on a unit n-sphere $S^n$
A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$
The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-...
- 147
8
votes
3
answers
1k
views
When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
I'm giving some lectures on the trace formula. Here's something I proved in the last lecture. Let $G$ be a locally compact Hausdorff unimodular topological group (e.g. a reductive Lie group), let $\...
- 41.4k
8
votes
2
answers
3k
views
$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$
It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
- 176
8
votes
4
answers
4k
views
Sum of two self-adjoint unbounded operators
Let $H$ be a Hilbert space, and $T:D(T)\subset H\rightarrow H$ and $S:D(S)\subset H\rightarrow H$ be unbounded self-adjoint operators.
Is $T+S:D(T)\cap D(S)\rightarrow H$ self-adjoint?
- 315
8
votes
3
answers
485
views
Does the metric space of compact metric spaces satisfy the binary intersection property?
A metric space $Y$ has the binary intersection property provided that whenever a collection of closed balls in $Y$ intersects pairwise, then there is a common intersection point.
Does the metric ...
- 15.5k
8
votes
2
answers
760
views
If the diagonal of a positive operator is compact, is the operator itself compact?
Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{e_n\}_n$. For a bounded operator $T$ on $H$, the diagonal of $T$ is the unique operator $D_T$ on $H$ which is diagonal with ...
- 2,263
8
votes
3
answers
884
views
abstract evolution equations
Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^*)$). In examples, they always ...
- 107
8
votes
2
answers
865
views
frechet manifolds book
hi, does anyone know a good book or some lecture notes on the theory of frechet manifolds ?
8
votes
2
answers
675
views
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
It is known that an entire function that is nowhere zero must be the exponential of another entire function.
Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
- 286
8
votes
5
answers
2k
views
Topological vector space textbook with enough applications
(Sorry for my bad English.)
For "applications", I mean applications in math, not real-life.
There are many textbooks about topological vector space, for example, GTM269 by Osborne, Modern Methods in ...
8
votes
2
answers
613
views
Pairs of elementary Fourier transforms in $L^2$
It is customary to teach Fourier transform on the real line by starting with functions from $L^1$, $L^2$ or the Schwartz space. It is not so easy to illustrate the theory by computing explicit pairs ...
- 18.7k
8
votes
3
answers
526
views
Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$
Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\...
- 345