All Questions
10,051 questions
8
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167
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A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters
Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?
(...
- 4,535
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
8
votes
0
answers
260
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 503
8
votes
0
answers
211
views
Superharmonic functions and amenability
Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$.
Assume that there is a set of non-...
- 4,705
8
votes
0
answers
194
views
Hilbert spaces over the semi-field $\mathbb R_+$
Let $\mathbb R_+$ be the semi-field of non-negative real numbers.
Definition (preliminary): A Hilbert space over $\mathbb R_+$ is a pair $(H,P)$, where $H$ is a complex Hilbert space, and $P\subset H$...
- 43.2k
8
votes
0
answers
265
views
$L^2$ norms of Whittaker vectors and zeros of Intertwining operators
For $\mu,\nu\in \mathbb{C}^2$ we denote $I(\mu,\nu)$ to be the principal series of $\mathrm{GL}_2(\mathbb{Q}_p)$ induced from $|.|^\mu\otimes |.|^\nu$. For $s=\mu-\nu$ one defines the standard ...
- 1,692
8
votes
0
answers
686
views
The function space defined by deep neural nets
Given a deep net graph and the activation functions on the hidden vertices do we have a description of the function space spanned by it? (even if for some specific architectures and activation ...
- 2,246
8
votes
0
answers
384
views
What is the name for a Banach space property closed under ultraproducts?
In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (...
- 6,287
8
votes
0
answers
421
views
Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
- 6,206
8
votes
0
answers
208
views
(Un)bounded Geometry and Sobolev Spaces
This post is related to this and this post.
It is known that on a complete Riemannian manifold, the space $C^\infty_c(M)$ is generally not dense in the Sobolev spaces $W^{k, p}(M)$ ($1 \leq p < \...
- 9,853
8
votes
0
answers
278
views
Pseudodifferential operators on compact manifolds with boundary
I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
- 81
8
votes
0
answers
1k
views
On the classification of injective Banach spaces
A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a norm-...
- 4,461
8
votes
0
answers
6k
views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
- 8,512
8
votes
0
answers
512
views
Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$
Question : Is the following $(\star)$ true for $a,b,c\in\mathbb Z$ ?
$$\begin{align}\int_{0}^{\frac{\pi}{2}}(ax^4+b\pi x^3+c{\pi}^{2}x^2)\log(\sin x)dx=0\Rightarrow a=b=c=0\qquad(\star)\end{align}$$
...
- 4,757
8
votes
0
answers
221
views
Density of odd and even eigenstates of an integral operator
Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
- 81
8
votes
0
answers
357
views
Ultrapowers of Banach spaces without the continuum hypothesis
Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
- 4,461
8
votes
0
answers
952
views
About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
8
votes
0
answers
452
views
Preduals of $\ell_1$
The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices.
Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. ...
- 191
8
votes
0
answers
1k
views
Strictly singular operators and their adjoints
This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...
- 1,073
8
votes
0
answers
751
views
The log kernel and Bochner Theorem
I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...
- 2,135
8
votes
0
answers
196
views
Parametrizing derivations from the algebra of smooth functions on a manifold to its dual
$\newcommand{\Der}{\operatorname{Der}}$
$\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
- 25.8k
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
8
votes
1
answer
207
views
Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure
Any information about the following questions would be welcome.
I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...
- 81
8
votes
4
answers
888
views
$\ell^p$ version of singular values
I am embarrassed to pose this question. It is a generalization of a question asked less than 24 hours ago by an unknown (Google), which has been deleted since then, presumably by its author themself.
...
- 52.3k
7
votes
7
answers
3k
views
Gelfand representation and functional calculus applications beyond Functional Analysis
I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way.
I am curious about ...
7
votes
3
answers
2k
views
A question on fractional derivatives
I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.
I just wanted to ask if there is a notion of ...
user168611
7
votes
3
answers
4k
views
Measures on infinite dimensional Banach spaces
Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure ?
7
votes
3
answers
1k
views
Index of an Operator
I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the ...
- 205
7
votes
2
answers
684
views
Hölder continuity for operators
Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/...
user127612
7
votes
2
answers
3k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 21.8k
7
votes
1
answer
968
views
a claim for a proof of the invariant subspace problem [closed]
Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states
Does every bounded operator on a separable Hilbert space have a non-trivial ...
- 87
7
votes
1
answer
1k
views
Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 4,058
7
votes
4
answers
2k
views
Invariant means on the integers
Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density $d^+(A)=\lim\sup_{n\...
- 6,034
7
votes
3
answers
1k
views
Compactness properties of plurisubharmonic functions
I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.
Let $\...
- 60.6k
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
7
votes
3
answers
935
views
Elegant proofs of $\bar{\partial}z^{-1} = 2\pi \delta_0$
For a function $f(x,y)$ on $\mathbb{R}^2,$ defined possibly outside the origin, write
$$\int_\epsilon ' f \,dx\,dy : = \int_{\mathbb{R}^2\setminus D_\epsilon}f \, dx\,dy,$$
(the integral on the ...
- 7,962
7
votes
3
answers
409
views
Are nearby subalgebras of matrix algebras conjugate?
Let $k=\mathbb{R}$ or $\mathbb{C}$ and let $A$ be a finite-dimensional $k$-algebra. If $A$ is simple, then the Skolem-Noether theorem says that any two algebra homomorphisms $f, g: A \to M_n(k)$ are ...
- 9,853
7
votes
3
answers
881
views
Criterion for compactness
Let $T:H\to H$ be a continuous operator on a Hilbert space.
Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero.
Must $T$ be compact?
user130903
7
votes
3
answers
521
views
Count of non-trivial ergodic measures of a topological dynamical system
Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\...
- 467
7
votes
3
answers
676
views
Noncommutative torus as a von Neumann algebra
Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
- 9,340
7
votes
3
answers
934
views
Norms as Points in $C(X)$
$\newcommand\abs[1]{\lvert{#1}\rvert}$Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$.
For each point $x$, there is a ...
user30211
7
votes
1
answer
1k
views
A variational problem - some guidance
This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.
Let
$\...
user125029
7
votes
2
answers
988
views
Missing mass conjecture
Let $n,t$ be positive integers and $p_1,p_2,\ldots,p_n$ positive numbers summing to 1. Conjecture:
$$
\sum_{i=1}^n p_i (1-p_i)^t \le \frac{n(1-1/n)^n}{t}
$$
always holds.
The motivation comes from my ...
- 6,541
7
votes
3
answers
1k
views
Nice orthonormal basis for L^2(Cantor set)
Let X be the Cantor set, which we view as the space $2^\mathbb{N}$ (the set of all infinite binary sequences), equipped with the product topology. We can construct a Borel probability measure $\mu$ on ...
- 81
7
votes
2
answers
648
views
If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?
Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
- 1,054
7
votes
1
answer
556
views
Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC
Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm.
Is it possible to show a non-continuous linear form on $E$ exists ...
- 1,355
7
votes
4
answers
1k
views
Radon-Nikodým property of $\ell^\infty$
I am wondering whether $\ell^\infty(\mathbb N)$ has the Radon-Nikodým property. Of course $\ell^1(\mathbb N)$ does, but I was unable to find out whether (e.g.) duals of spaces with the R-N property ...
- 3,388
7
votes
1
answer
330
views
Is this a characterization of commutative $C^{*}$ algebras?
Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$
Is $A$ necessarily a commutative ...
- 366
7
votes
4
answers
1k
views
Is the space of tempered distribution second countable?
Let $\mathcal S '(\mathbb R^d)$ be the space of Schwartz tempered distributions equipped with the weak-* topology. I need to know if this space is second countable, i.e. if this topology has a ...
- 630