All Questions
3,601 questions with no upvoted or accepted answers
5
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341
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Real interpolation of weighted Sobolev spaces with different weights
Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
5
votes
0
answers
196
views
Distributions and functions on the Jacquet module $C_c^\infty(X)_{H,\chi}$
Let $X$ be an $\ell$ space (in the sense of Bernstein-Zelevinski), $H$ be an $\ell$ group which acts on $X$ and $\chi$ be a character of $H$. Denote $C^\infty(X)^{H,\chi}$ the space of locally ...
- 960
5
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0
answers
204
views
quasi-weakly compact operators, co-ideals of operator ideals, and Banach spaces $X$ with $X^{**}/X$ separable
Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator $\overline{T}\in\mathcal{L}(X^{**}/...
- 1,591
5
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0
answers
184
views
Automorphisms of Cuntz algebra
Suppose, $ O_{\infty} $ is the cuntz algebra generated by the orthogonal isometries $ \{S_i\}_{i\in \mathbb{N}} $,i.e. $ S_i^*S_j=\delta_{ij}$ and $ O_{\infty}=C^*(\{S_i\}_{i\in \mathbb{N}}) $.
Then ...
- 817
5
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374
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A question about Carleman linearization
Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...
- 1,590
5
votes
0
answers
120
views
L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension
For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator:
$M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$
(...
- 4,010
5
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168
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Functional equations about Conway's box function
Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence).
The ...
- 213
5
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175
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A Banach space with the BD property and without the weak Gelfand-Phillips property
A subset A of X is called Grothendieck if every operator T from X to $c_0$ maps A to a relatively weakly compact set.
A Banach space has the weak Gelfand-Phillips property (wGP) if every ...
5
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151
views
Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra
While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove:
Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...
- 743
5
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216
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Existence or construction of a sequence of orthogonal matrices with three properties
This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...
- 984
5
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answers
322
views
Differential operators acting on the Schwartz space
I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome!
Let $D$ be a linear differential operator with ...
- 81
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348
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Discrete groups G whose full C*-algebra C*(G) is not MF?
This is a cheap rip-off of this question, but I am genuinely interested in an answer.
Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not MF?
Let us ...
- 1,786
5
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205
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Steklov averages in PDE: what to do when we have time-dependent elliptic operator
One may have an equation (with boundary conditions omitted below)
$$u_t - Au = f$$
$$u(0)=u_0$$
which has a weak solution $u \in L^2(0,T;V) \cap C([0,T];H)$ in the sense that
$$-\int_0^T \int_\Omega u(...
- 91
5
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295
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Convergence of convex combinations in topological vector spaces
I am studying certain quadratic forms on $L^0(m)$ equipped with the topology of (local) convergence in measure which in general is not locally convex. I am also interested in the situation where $m$ ...
5
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244
views
Tensorization of Orlicz norm?
Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...
- 1,269
5
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364
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Version of Stone Weierstrass for functions not vanishing at infinity
I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
- 51
5
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104
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Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws
Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital $*$-...
- 9,330
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376
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Non-linear positive map
In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...
- 421
5
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answers
141
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Rate of convergence of Riemann sum of quasi-regular functions
The following result is well-known (I consider the 3-dimensional case only):
Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then
$$
\left| \int_{\mathbb{R}^3} f - \...
- 53
5
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answers
428
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Koopman representation, weakly compact action, Ozawa Popa
Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
- 193
5
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254
views
A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
5
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620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
5
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178
views
Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?
Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability ...
- 2,306
5
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211
views
Infinitesimal Generator of Billiard Flow
The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
- 9,853
5
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206
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On a variant of Eidelheit's theorem
A theorem of Eidelheit from 1940's asserts that two Banach spaces $X$ and $Y$ are isomorphic if and only if $L(X)$ and $L(Y)$, the algebras of all bounded linear operators, are isomorphic as Banach ...
- 51
5
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237
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Is Akcoglu's theorem for power bounded positive operators still an open problem?
I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.
" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...
- 1,594
5
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0
answers
250
views
Estimating singular values of integral operators
I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb R}...
- 1,017
5
votes
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answers
236
views
Discrete versus Continuous Hilbert Transform
Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform $\...
- 16.2k
5
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answers
161
views
$L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?
Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \...
- 4,010
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161
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Are the integer index finite depth irreducible subfactors Kac-coideal?
Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra?
In other words, of the following form (...
5
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answers
98
views
The regularity of Dirichlet form in Besov space
Let $C_{0}(\mathbb{R}^n)$ denote the set of continuous functions in $\mathbb{R}^n$ with compact support, $C_0^\infty(\mathbb{R}^n)$ denote the set of infinite differentiable functions in $\mathbb{R}^n$...
- 369
5
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913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
5
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328
views
Are inclusions from spaces of $C^\infty$ sections into spaces of $C^k$ sections homotopy equivalences?
[EDIT: The answer to my original question was obviously no, as user56365 pointed out. Here is what I should have asked.]
For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre ...
- 648
5
votes
0
answers
2k
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Denseness of finite rank operators in $\mathcal{B}(X,Y)$
Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
https://math.stackexchange.com/questions/...
- 107
5
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answers
133
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Banach spaces admitting no proper quasi-affinity
I am interested in examples of Banach spaces $X$ satisfying the following two conditions:
(1) Every (continuous linear) injective operator $T:X\to X$ with dense range is surjective.
(2) $X$ is ...
- 4,461
5
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0
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286
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$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?
For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...
- 1,051
5
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0
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179
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Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?
Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
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146
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Special elements in $L^{\infty}(G)^*$
Let $G$ be a locally compact group. Let $M(G)$ denote the measure algebra and $L^1(G)$ denote the group algebra on $G$. Then $M(G)$ acts on $L^1(G)$ by convolution. So by duality $M(G)$ acts on $L^1(G)...
- 306
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426
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Do all $L^{\infty}(\mu)$ spaces have the Grothendieck property?
Consider $L_{\infty}(\Omega,\Sigma,\mu)$, where $(\Omega,\Sigma,\mu)$ is any measure space. Does it it have the Grothendieck property? If the measure space is localizable, then it is true. The ...
- 149
5
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154
views
When is an inner derivation a Fredholm operator?
Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by ...
- 777
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answers
148
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Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
- 6,351
5
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105
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Strictly convex renormings making power bounded operators into contractions
Let $X$ be a Banach space and let $T$ be a power bounded linear operator on $X$ (i.e. $\sup_{n\ge0}\|T^n\|_{op}<\infty$). We can of course define an equivalent norm $\|\cdot\|'$ on $X$ so that $\|T\...
5
votes
0
answers
394
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construction of heat kernels for non-compact manifolds with boundary
Recently, I am studying heat semigroup for noncompact manifolds with boundary.
In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be ...
- 199
5
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308
views
"Contraction mapping principle"
Are there any applications of the following fact?:
Let $X$ be a complete Hausdorff semi-metric space with a collection of semi-metrics $\{d_\alpha(\cdot,\cdot)\}_{\alpha\in A}.$
Further let $f:X\to ...
- 61
5
votes
0
answers
133
views
Series representation for unbounded perturbations of semigroup generators
Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then $A+B$...
- 3,388
5
votes
0
answers
569
views
Argmax of random walk vs of Brownian motion
Consider a random walk on $\mathbb{Z}$ with triangular drift and jumps that are standard normals. That is,
$$
\begin{cases}
RW_{t+1} = RW_t - d + \epsilon_t, \quad t \geq 0,\\
RW_{t-1} = RW_t - d + \...
- 1,069
5
votes
0
answers
428
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
5
votes
0
answers
543
views
Moduli of smoothness, Besov spaces, and Sobolev spaces
For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is
\begin{equation}
\omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})}
\end{equation}
where $\Omega_{rh}=\{...
- 3,322
5
votes
0
answers
274
views
Reference request: The relationship between norm and trace forms on an Albert algebra
I am interested in either a nice reference, or some clarification.
Overview: I am considering $J_3(\mathbb{O})$, the Jordan algebra of $3\times 3$ self adjoint octonionic matrices. This algebra is a ...
- 133
5
votes
0
answers
598
views
Do the banded operators check the invariant subspace problem?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...