All Questions
3,600 questions with no upvoted or accepted answers
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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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176
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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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165
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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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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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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:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
- 111
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109
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The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
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129
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Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
- 111
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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
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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
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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
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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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95
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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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114
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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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85
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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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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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48
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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
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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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95
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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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171
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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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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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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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161
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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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120
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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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106
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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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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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67
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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
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129
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Certain decompositions of decomposable Banach spaces
Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
- 1,453
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144
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Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
- 1,879
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93
views
Regularity of semilinear parabolic PDE in the whole space
I need regularities in Holder space of the following parabolic PDE:
$$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
- 1,399
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39
views
Topology on upper semicontinuous functions and Baire one functions
Which is the standard topology on upper semicontinuous functions and on functions of Baire class one, in both cases defined on a compact real interval?
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247
views
Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
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votes
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answers
46
views
Independence of variables predicted by the generator
Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...
- 449
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votes
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164
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Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
- 101
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67
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Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 617
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239
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Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
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53
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Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
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answers
81
views
What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
- 6,853
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104
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Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 61
0
votes
0
answers
150
views
Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
0
votes
0
answers
153
views
Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$
How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
- 99
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134
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The set of bounded lipschitz functions over a compact is barrelled but not a neighborhood of zero?
I recently learned that Banach spaces are barrelled, i.e any convex, balanced, absorbing and closed subset is a neighborhood of zero (wikipedia). I'm having trouble understanding why the following ...
0
votes
0
answers
231
views
Definition of tensor product of dense subspaces of Hilbert spaces
Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...
- 1,305
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0
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44
views
Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?
Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
- 6,853
0
votes
0
answers
101
views
Example of a ‘weak’ ideal which is not an ideal?
For Hilbert spaces $H$ and $K$, let $V=B(H,K)(H \neq K)$. A sub space $I$ of $V$ is called an ideal of $V$ if $$IV^*V+VV^*I \subset I$$ and $I$ is called weak ideal of $V$ If $$\text{span}\{IV^*V-VV^*...
- 1,115
0
votes
1
answer
414
views
Sufficient conditions for an asymptotic compactness
This question relates a theory of Mosco convergence.
Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$.
A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
- 721
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votes
0
answers
300
views
Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 205
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votes
0
answers
90
views
Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
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votes
0
answers
227
views
Average gap between zeros on the critical strip of the Riemann Zeta Function
Are there any approximations of the average gap between 2 successive zeros along the Riemann zeta functions critical strip up to the nth zero? If so, is it hypothesized that this average gap converges ...
- 307