All Questions
966 questions
9
votes
1
answer
359
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
9
votes
2
answers
848
views
$\zeta$-function regularized determinants
In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in ...
- 21.8k
9
votes
4
answers
1k
views
Boundedness of nonlinear continuous functionals
Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...
- 4,060
9
votes
1
answer
758
views
Convergence of Schwartz kernels implies convergence of operators
Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
- 2,998
9
votes
2
answers
553
views
Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
9
votes
4
answers
2k
views
How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
9
votes
1
answer
3k
views
Is every finite Borel measure on a locally compact Hausdorff, $\sigma$-compact and separable space automatically regular?
The conditions stated in the question seem mouthful and a bit arbitrary, so let me provide some backgrounds.
Definition
Let $\mu$ be a Borel measure on a topological space. We say:
$\...
- 83
9
votes
3
answers
1k
views
Does there exist a notion of discrete riemannian metric on graph?
I would like to know if there is any notion of a discrete Riemannian metric on graphs. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on.
To be more ...
- 91
8
votes
0
answers
246
views
A question related to the separable quotient problem
I have the following question related to the previous posts Hereditarily indecomposable Banach spaces and Separable Quotient problem and Weak star separable and separable quotient problem
Question....
- 986
8
votes
0
answers
182
views
Distribution domination for sums of independent random variables in Banach spaces
Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying
$$
\sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A),
...
- 131
8
votes
1
answer
268
views
Two questions about basic sequences
Suppose $(x_n)$ and $(y_n)$ are two basic sequences in a separable Banach space $X$ such that $\overline{span}\{(x_n), (y_n)\}=X$. Can we always pass to subsequences $(x_{n_k})$ and $(y_{n_k})$ such ...
- 355
8
votes
1
answer
426
views
Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor?
This is a reference request for something that is likely to be well-known to operator algebraists. I will not, therefore, include the technical definition of free product of finite von Neumann ...
- 7,067
8
votes
0
answers
269
views
Shift on trivalent directed tree, operator and von Neumann algebra
Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...
8
votes
3
answers
429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,145
8
votes
2
answers
488
views
If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?
- 4,058
8
votes
1
answer
734
views
Almost Arzela Ascoli
Definitions:
We say a sequence of continuous functions $f_n: [0, 1] \to \mathbb R$ is equicontinuous on average if for every $x \in [0, 1]$ and $\varepsilon > 0$ there exists some $\delta > 0$ ...
- 6,215
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
2
answers
644
views
Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
8
votes
1
answer
522
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
8
votes
1
answer
390
views
Order bounded version of monotone complete $C^*$-algebras
Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
- 12.6k
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
1
answer
386
views
Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$
Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$.
Question. Can we prove this?
$$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
- 519
8
votes
1
answer
485
views
An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...
- 563
8
votes
2
answers
385
views
Metrizability of a topological vector space where every sequence can be made to converge to zero
This is a follow-up to this answer.
If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a ...
8
votes
3
answers
691
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
- 12.6k
8
votes
0
answers
345
views
Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?
Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.
There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < ...
- 7,800
8
votes
1
answer
5k
views
integration by parts for the fractional Laplacian
Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like
$$
s\in(0,1),\qquad\int\limits_{\mathbb{R}^N}f[(-\Delta)^sg] =\int\limits_{\mathbb{R}^...
- 5,380
8
votes
1
answer
340
views
characterization of commutative Banach algebras
Let $A$ be a Banach algebra with the following property:
For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges.
...
- 356
8
votes
3
answers
1k
views
Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
- 505
8
votes
2
answers
785
views
Is taking the product of signed measures weakly continuous?
For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
- 29.8k
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
2
answers
1k
views
What does the unique mean on weakly almost periodic functions look like?
There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
- 4,432
8
votes
3
answers
1k
views
Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional
Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$.
When $X$ and $Y$ are both finite dimensional, it follows from the formula
$$\|u\|_{B(X,Y)} = \sup_{\|x\|_X <...
- 9,486
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
811
views
Schwartz space of functions with values in a Frechet space
While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...
- 601
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
3
answers
2k
views
Measures on general topological groups
I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...
- 1,152
8
votes
1
answer
2k
views
Fastest decay of Fourier Transform for Generalized Functions of compact support
What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ...
- 83
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
- 1,407
8
votes
4
answers
2k
views
Manifold-Valued Sobolev Spaces
I have the following basic question about Sobolev-spaces which take their values in a
Riemannian manifold $(M,g)$, i.e.
functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that ...
- 233
8
votes
2
answers
583
views
Does every operator from a Hilbert space to $L^0$ factor through a canonical one?
Let $A:H\to L^0(S, \mu)$ be a continuous operator from a Hilbert space to the space of (equivalence classes of) measurable functions on a probability measure space $S$ with convergence in measure. Let'...
- 4,010
8
votes
2
answers
2k
views
when a pseudo-differential operators to be compact?
In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...
- 1,644
8
votes
1
answer
1k
views
Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
- 81
8
votes
1
answer
656
views
When is the adjoint of a hypoelliptic operator also hypoelliptic?
Suppose that $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.
Recall that $L$ is a hypoelliptic differential operator if for ...
- 329
8
votes
1
answer
434
views
Self-dual finite-dimensional complex normed spaces
Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space?
Remarks: There are easy counterexamples in the real case, and in ...
- 11.4k
8
votes
1
answer
355
views
Proving a certain $ C^{*} $-algebraic inequality
Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} \|...
- 1,396
8
votes
1
answer
455
views
Closure of $L(\ell^2,\ell^2)$ in $L(\ell^2,\ell^\infty)$
Let $\ell^2$, $\ell^\infty$ denote the usual sequence spaces and let $L(\ell^2,\ell^2)$ the Banach space of bounded linear operators from $\ell^2$ to $\ell^2$ as well as $L(\ell^2,\ell^\infty)$ the ...
- 261
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
2
answers
1k
views
What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?
Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...
- 7,197