All Questions
10,447 questions
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80
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Cyclicity of composition operators
Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ ...
- 5,405
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votes
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116
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Is there a proper term for a "continuum-convex" set?
Let $X$ be a Banach space, and for any compactly supported Borel probability measure $\mathbb{P}$ on $X$, define the mean $\mu_\mathbb{P}$ by $\mu_\mathbb{P}=\int_X x \, \mathbb{P}(dx)$.
I want to say ...
- 708
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votes
0
answers
40
views
Time regularity of traces
I have a question about the time regularity of the traces in one dimension.
Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(...
- 11
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answers
70
views
Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99
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112
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How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
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votes
1
answer
102
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Gronwall-type inequality with nonlinearity
Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$...
user139844
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answers
141
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Extending an unbounded dense linear functional
Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$
Let $V \subset H$ be a dense subspace of $H$
Let $f : V \to \mathbb{C}$ be a unbounded functional linear
My question is:
Is ...
- 433
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votes
0
answers
103
views
A question on the Haar basis for $L_{1}[0,1]$
Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. It is important to know the exact expression of the norm of $\|\sum_{i=1}^{n}a_{i}x_{i}\|$ for all $n$ and all scalars $a_{1},a_{2},\ldots,...
- 3,343
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1
answer
136
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A minimal surface with a local extremum in normal direction is a plane [closed]
I'm currently struggling with concluding a proof and need a hint. So the first part of the exercise
was that given an open subset $\Omega \subset \mathbb{R}^2$ and a harmonic function
$f: \Omega \to \...
- 13
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votes
0
answers
112
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Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
- 1,334
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0
answers
148
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About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
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0
answers
84
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Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
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votes
1
answer
139
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Build an explicit "small perturbation" of the identity satisfying some properties
How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant ...
user139844
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56
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Existence of minimal subset of dual ball such that the intersection of kernels is trivial
Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} ...
- 820
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198
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eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
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0
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176
views
A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
- 227
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answers
165
views
Compact embedding of Lipschitz continuous functions
Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
- 3,388
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answers
109
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Generator problem for reduced group C*-algebra
(Not sure if it is appropriate or not, if no I will delete the post)
Recently I am concerned about the number of generator of $C^{*}_{r}(\mathbb{F}_{k})$, the reduced group algebra of the free group, ...
- 523
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votes
1
answer
162
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Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
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199
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Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?
Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
- 111
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0
answers
109
views
The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
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0
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129
views
Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
- 111
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votes
1
answer
213
views
'Diamagnetic' inequality for negative Sobolev spaces
Let us look at the subspace of smooth complex functions of $L^2(\mathbb{R}^n,\mathbb{C})$, call $H^s$ the Sobolev spaces. By the diamagnetic inequality $\lvert \nabla \lvert\psi\rvert\rvert (x) \le \...
user140442
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72
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Does $L^{\infty}[0,1]$ admits infinitely many densely defined derivations in weak* topology?
To clarify the question. First we define what is densely defined derivation. A densely defined derivation $\delta:D(\delta):\rightarrow L^{\infty}[0,1]$ where $D(\delta)$ is a dense subalgebra( in ...
- 523
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96
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Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
- 101
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1
answer
431
views
Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$
$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\zbar-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$,...
- 11
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197
views
Link between a categorical and an algebraic characterization of (infinite-dimensional) Hilbert space
On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1 latest Arxiv version) offers a characterization of the ...
- 2,655
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85
views
How can we define $\chi_{\Omega}(A)$?
I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic ...
- 1,305
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answers
54
views
Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k
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62
views
An inequality regarding operator concave function
Crossposted from math.SE
Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...
- 141
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0
answers
95
views
Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
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votes
1
answer
357
views
Continuity of point evaluation on space of Hölder functions with $L^p$ norm
Let $L>0$ and $\Omega \subset \mathbb{R}^n$ a bounded Lipschitz domain. Define
$$
B_{\frac12,L}:=\{f \in L^2((0,1) \times \Omega) : \|f(t,\cdot)-f(s,\cdot)\|_{L^2(\Omega)} \leq L|t-s|^{\frac12},~ \...
- 319
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0
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114
views
Analysis, matrix exponential, infimum and limit
I was working in this problem for a long time and I didn't have success.
Someone could help me, please?
The problem:
Let $f: \mathbb{R}^{n^2} \times \mathbb{Z}^{n} \longrightarrow \mathbb{R}$ defined ...
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0
answers
85
views
An amenable operator algebra has the total reduction property
This is from
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
- 101
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134
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The Limit of a Matrix Series
I have an invertible matrix $B$ and a diagonal (not necessarily invertible) matrix $D$ and I'm studying the series as $n\to\infty$ of $\frac{1}{n} B \sum_{k=0}^{n-1}( I - \frac{1}{n} B^T D B)^k B^T D$....
- 111
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109
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Operator algebra on an invariant subset
In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...
- 109
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291
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Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
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0
answers
48
views
Surjectivity of the limiting operator
Consider the operator
\begin{eqnarray*}
K_{n} &:&L^{2}(0,1)\longrightarrow L^{2}(0,1)^{n}, \\
u(x) &\mapsto &A_{n}U_{n}(x)=A_{n}(u(\frac{x}{n}),u(\frac{x+1}{n}),...,u(%
\frac{x+n-1}{n})...
- 617
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101
views
Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?
I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
- 537
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0
answers
46
views
Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
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0
answers
95
views
A property of the Hilbert transform involving the cotangent function
A lemma of a paper by T. Elgindi and I.-J. Jeong (Arch. Rational Mech. Anal. 235 (2020) 1763–1817, Lemma 2.2) states the following:
Let $g(z)=\operatorname{sgn}(z)k(|z|^\alpha)$ with $k$ smooth and $k(...
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0
answers
194
views
Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 1
0
votes
0
answers
171
views
Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?
I know that $\ell_2$ is isomorphic to a subspace of $L_p(0,1)$ for any $1\le p<\infty$. However, I haven't seen anything about $L_\infty$. Is $\ell_2$ is isomorphic to a subspace of $L_\infty(0,1)$?...
- 617
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votes
0
answers
79
views
Compact operators and projective tensor space
I know that the space of all the bounded linear maps between two Banach spaces, denoted by $L(X,Y)$, has a relationship with the projective tensor space of $X$ and $Y$,
$$({X \widehat\otimes_{\pi} Y})...
- 11
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votes
0
answers
72
views
Di Perna-Lions theory for transport equations with an additional integral operator
I'm looking for a reference about some possible generalization of the well-known Di Perna-Lions theory for transport equations (say, on $[0,T] \times \mathbb{R}^d$) of the form
\begin{align}
\...
- 161
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votes
0
answers
161
views
When does a positive operator preserve invertibility
Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...
- 175
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votes
0
answers
120
views
Are bounded maps determined by their images on the algebraic tensor product?
Let $\mathcal A,\mathcal B,\mathcal C$ be von Neumann algebras. Let $F:\mathcal A\otimes\mathcal B\to\mathcal C$ be a bounded linear map. Assume $F(a\otimes b)=0$ for all $a,b\in\mathcal A,\mathcal B$....
- 896
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votes
0
answers
106
views
Extension of a Hilbert basis
The picture below is taken from this paper: http://real.mtak.hu/22877/.
The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
- 617
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votes
0
answers
154
views
When is the heat semigroup Gibbs?
Defining the Laplacian on a region $Ω$ of $\mathbb{R}^d$ with Dirichlet boundary conditions, under what conditions on the region (or any other possible assumptions) is the semigroup it generates Gibbs,...
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0
answers
67
views
Dual of isometric copies into dual Banach spaces
Let $X$ be a Banach space and $X_1\xrightarrow{}X$ isometrically. Under some assumption can we guarantee that $X^*$ contains an isometric copy of $X_1^*$. I am also interested to know if this happens ...
- 1,879