All Questions
3,841 questions with no upvoted or accepted answers
3
votes
0
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85
views
Existence for $-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$
Let $\Omega$ be a smooth bounded domain. Consider the equation
$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$
$$u|_{\partial\Omega} = 0$$
where $f,g$ are smooth functions on $\Omega$ and $\varphi$...
- 251
3
votes
0
answers
650
views
description of dual space of space of Radon measure equipped with topology of weak convergence
Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence":
$$
\mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n ...
- 325
3
votes
0
answers
175
views
Araki's proof of simple connectedness of the restricted orthogonal group
I am trying to understand Araki's proof of the statement that the restricted orthogonal group of a Hilbert space with a unitary structure is simply connected. This proof starts on page 114 of these ...
- 556
3
votes
0
answers
45
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Certain limits of normal completely positive idempotents
Suppose $L_n$ is a sequence of normal completely positive mappings of $B(H)$ into itself of norm strictly less than one with the property that $L_n(L_m(A)) \to L_m(A)$ in the strong operator topology ...
- 365
3
votes
0
answers
109
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Is the Banach space $C(K)$ a $1$-Lipschitz comp-extensor?
Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map $...
- 42k
3
votes
0
answers
180
views
When is a minimal immersion holomorphic?
Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:
$\phi\colon X\to Y$
be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
- 2,818
3
votes
0
answers
141
views
Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials
Let $I$ be a subinterval of $\lbrack 0,1\rbrack$. How well can one hope to approximate it in the $L_1$-norm (on $\mathbb{R}/\mathbb{Z}$) by a trigonometric polynomial of degree $\leq R$, that is, a ...
- 20.2k
3
votes
0
answers
218
views
Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?
The real motion group of $\mathbb R^2$, $M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ ...
- 650
3
votes
0
answers
108
views
Radial Poincare inequality for Gaussian measures
Let $\mu$ be a zero mean Gaussian probability measure on $\mathbb{R}^n$ whose covariance is less than the identity. If $f$ is a $1$-Lipschitz real function on $\mathbb{R}^n$ such that there exists a ...
- 2,772
3
votes
0
answers
97
views
Why c.p.c order zero maps induce morphism between cuntz semigroups
Maybe a naive question:
Right now I am reading the paper ``Completely positive maps of order zero'' written by Wilhelm Winter and Joachim Zacharias. I do not quite understand the last corollary when ...
- 181
3
votes
0
answers
149
views
Classical subspaces of non-atomic Banach lattices
Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $...
user114263
3
votes
0
answers
182
views
Prove a certain function maps to upper half plane
Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in ...
- 31
3
votes
0
answers
422
views
Isometries between subspaces of finite-dimensional vector spaces
I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.
Taking $n \le m$, one ...
- 31
3
votes
0
answers
89
views
Discrete Lions Peetre interpolation
In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\...
user114263
3
votes
0
answers
78
views
Is every weakly Lindelof Banach space a $D$-space?
An open neighbourhood assignment for a topological space $(X, \tau)$ is a map $U: X \to \tau$ such that $x \in U(x)$, for every $x \in X$. A space $X$ is called a $D$-space if for every open ...
- 4,413
3
votes
0
answers
48
views
Questions on "The condition number of a randomly perturbed matrix"
This question is about the two vectors $w'$ and $y$ that are necessary for the argument in section $7$ (page 6) of this paper by Terence Tao and Van Vu,
https://arxiv.org/abs/math/0703307 (that ...
- 2,246
3
votes
0
answers
120
views
Schur property for a sum of Banach spaces
Suppose we have two Banach spaces X and Y each of them having the Schur property (weakly convergent sequences are norm convergent). Does it follows that X+Y has the Schur property? Note that this is ...
- 31
3
votes
0
answers
240
views
About optimizing decay rate of Fourier transforms?
Suppose we have a density function $f(t)$ of a random variable and $f \in C^1(R)$. If characteristic function of $f$ is $\phi_f(x) \asymp O(x^{-\beta})$ and $f$ satisfies some restrictive conditions ...
- 31
3
votes
0
answers
127
views
Max-Plus algebra and hyperplane arrangements
Given an expression in the Max-Plus algebra is it possible to recognize if it represents a continuous piecewise linear (CPWL) function whose polyhedral complex is a hyperplane arrangement?
Or ...
- 2,246
3
votes
0
answers
168
views
Zak transform and VMO
The Zak transform of a function $f\in L^1(\mathbb R)\cap L^2(\mathbb R)$ is defined as follows:
$$
Zf(x,\omega) := \sum_{k\in\mathbb Z}f(x+k)e^{-2\pi i k\omega},\quad (x,\omega)\in Q_0 :=(0,1)^2.
$$
...
3
votes
0
answers
82
views
Proving the existence of a continuous function that satisfy a certain property from a finite version of this property
Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set.
In particular, $M$ can be described by a finite set of polynomial equalities and inequalities.
Let $\delta_0 > 0$ be a ...
- 745
3
votes
0
answers
1k
views
Inner Product on tensor product of Hilbert spaces is unique?
Given two Hilbert Spaces $H$ and $K$, a natural inner product on $H\otimes K$(= vector space tensor product of $H$ and $K$) is given by
$\hspace{.5in}\langle h_1\otimes k_1,h_2\otimes k_2\rangle=\...
- 165
3
votes
0
answers
198
views
Asymptotic stability of eigenvalues by compact perturbations
I need some references concerning the asymptotic stability of eigenvalues by compact perturbations. In [T. Kato, Perturbation theory for linear operators] there are some results concerning stability ...
- 1,329
3
votes
0
answers
367
views
Reference on semigroup theory and fractional heat equation
Consider the Dirichlet problem associated to the classical heat equation $\partial_t u - \Delta u = 0$ and to the fractional heat equation $\partial_t u + (- \Delta)^s u = 0$.
Where can I find a ...
user103450
3
votes
0
answers
275
views
Seminorms on tensor products of affinoid algebras
Let $A \leftarrow C \rightarrow B$ be affinoid $K$-algebras, where $K$ is a non-archimedean field with non-trivial absolute value. Equipping $A$, $B$, $C$ with the supremum seminorms, there is a ...
- 31
3
votes
0
answers
85
views
essential norm versus invertibility
Let $X$ be an infinite dimensional Banach space and $T:X\rightarrow X$ be a bounded linear operator. If $T$ is invertible and $\lVert T\rVert_e=\lVert T\rVert$, is it true that (or when is it true ...
- 83
3
votes
0
answers
153
views
Identifying a Banach space-valued functions' integral notion
In Teschl's book on Mathematical Methods in Quantum Mechanics (https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf) in section 4.1 a notion of an integral for Banach space-valued functions ...
- 203
3
votes
0
answers
278
views
Interchanging direct sum and direct integral
I am interested in the concept of direct integrals. The definition I am concerned with can be found in Hall, "Quantum Theory for Mathematicians" (p. 146). Short version: We have a family of separable ...
- 31
3
votes
0
answers
241
views
Is the shift operator continuous on the real Hardy spaces?
Is the shift operator (or the translation operator) continuous on Hardy spaces $H^p(\mathbb{R}^n)$ (with $0<p\leq 1$)? i.e. given $f\in H^p(\mathbb{R}^n)$, is the following map
\begin{align}
\tau:&...
- 31
3
votes
0
answers
125
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
- 143
3
votes
0
answers
187
views
Families of unbounded operators
Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\...
- 331
3
votes
0
answers
237
views
Who defined the term "Carleman-matrix" and also their properties as they are?
Working for a couple of years with "Carleman-matrices" I became aware that it is difficult to trace down who actually introduced the name for this type of matrices. Wikipedia lets you alone with this ...
- 5,342
3
votes
0
answers
126
views
An identity of operator norms and de Leeuw's theorem
Let $$Hf(x_1,x_2)=p.v.\int_{-\infty}^\infty f(x_1-t,x_2-S(x_1,x_1-t))\frac{dt}{t},$$
$$T_\lambda f(x)=\lim_{\epsilon\to0}\int_{|x-y|\ge\epsilon}e^{i\lambda S(x,y)}(x-y)^{-1}f(y)dy, $$ where $S(x,y)$ ...
- 187
3
votes
0
answers
109
views
Weak convergence of series representing the log characteristic function
Disclaimer. I already asked this question on math.stackexchange.com without any answers or comments as of yet.
In which weak sense does the series representation of the log-characteristic function ...
- 101
3
votes
0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
- 1,959
3
votes
0
answers
52
views
Integral estimate in the Levitan's paper "On expansion in eigenfunctions of the Laplace operator"
Let $D \subset R^m$ be a domain in a $m$-dimensional Euclidean space, $P \in intD$, and $t > 0$ so small that the sphere of radius $t$ centered at the point $P$ sits in $intD$. Let $\phi : D \...
- 63
3
votes
0
answers
89
views
Trace of a weighted composition operator on Bergman space
I am reading a series of papers by Pollicott, Jenkinson and coauthors which make use of the following type of result:
Theorem: Let $\mathbb{D} \subset \mathbb{C}^d$ be a bounded, connected open set. ...
- 6,206
3
votes
0
answers
63
views
Is the collection of Schur convex functions sequentially compact?
We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
- 185
3
votes
0
answers
465
views
Fractional sobolev spaces
On the whole space $\mathbb R^d$, the fractional Sobolev space
$H_s(\mathbb R^d)$ of order $s\in \mathbb R$ can be defined as the subspace of tempered distributions $T$ such that $\mathcal F T \in L^...
- 630
3
votes
0
answers
190
views
Error term in the Euclidean Weyl law
Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that ...
- 381
3
votes
0
answers
64
views
A constraint satisfaction problem on matrix sum involving symmetric group
Given $n\in\Bbb N$ what is the smallest with $m>n$ we need such that there is a non-negative $\epsilon<1$ and $\Phi_i,\Psi_j\in\Bbb C^{m\times m}$ at every $i,j\in[n]$ ($[n]=\{1,\dots,n\}$) such ...
- 13.9k
3
votes
0
answers
210
views
Meromorphic continuation of resolvent of free Laplacian on homogeneous Sobolev space
Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...
- 481
3
votes
0
answers
261
views
Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions
Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set
\begin{equation*}
{\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1)
\end{...
- 31
3
votes
0
answers
269
views
Finite dimensional representation of tensor product
Let $A$ and $B$ be $C^*$ algebras, and let $\pi:A \odot B \to B(H)$ be a $*$-representation of the algebraic tensor product on a finite dimensional Hilbert space $H$. Let $x \in A \odot B$. Since $H$ ...
- 1,769
3
votes
0
answers
169
views
A spanning set for an annihilator set on a Banach space
Let $(z_n)$ be a $H^\infty$-interpolating sequence on the open complex unit disc $\mathbb D$. If $A$ is some Banach space of analytic functions on the disc, denote by $X$ the closed subspace of all ...
- 31
3
votes
0
answers
280
views
Helmholtz-Hodge decomposition
I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
- 1,385
3
votes
0
answers
91
views
The numerical range of a composition of two operators
For a problem I'm working on, I need the following implication. $A,B$ are two closed densely defined operators on a Hilbert space $H$. I'll be a bit vague about the setting, add assumptions at will as ...
- 9,007
3
votes
0
answers
103
views
Extension of the Gagliardo Inequality
The Gagliardo Inequality generalizes Fubini's Theorem: let $f_j$ be $d-1$ non-negative measurable functions over ${\mathbb R}^{d-1}$. Let us form the function
$$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$
...
- 52.3k
3
votes
0
answers
57
views
Integration of Weyl operators multiplied by quasifree state over a symplectic space
I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
3
votes
0
answers
214
views
Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698