**2**

votes

**0**answers

134 views

### Dual space of vector fields with null divergence

Let us consider $\mathscr D(\mathbb R^3;\mathbb R^3)$ the space of smooth compactly supported vector fields in $\mathbb R^3$ and let us define
$\widetilde{\mathscr D}=\mathscr D(\mathbb R^3;\mathbb R^...

**1**

vote

**1**answer

100 views

### Basis for the Orlicz Space

Does the Orlicz space (https://en.wikipedia.org/wiki/Birnbaum-Orlicz_space) has unconditional Schauder basis? Can we find such a orthonormal basis like the Hermitian polynomials in $L^2$?

**3**

votes

**1**answer

97 views

### busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras".
Let $0\to B\to E\to A\to 0$ be a short exact ...

**3**

votes

**0**answers

51 views

### 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 ...

**2**

votes

**1**answer

72 views

### Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...

**2**

votes

**1**answer

104 views

### Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...

**2**

votes

**0**answers

76 views

### Reflexive subspaces of dual spaces

If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...

**2**

votes

**0**answers

99 views

### Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...

**1**

vote

**0**answers

67 views

### Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V)$ denotes the set of all linear bounded endomorphisms with operator ...

**4**

votes

**1**answer

182 views

### Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...

**3**

votes

**0**answers

63 views

### Strengthening of the local smoothing estimates for the free Laplacian

The classical local-smoothing estimates for the free Laplacian asserts that:
$$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$
where $B\subset\mathbb{...

**0**

votes

**0**answers

82 views

### Orthogonal Procrustes problem for sub-spaces?

By Orthogonal Procrustes problem I mean given matrix $A$ and $B$ finding a orthogonal matrix $R$ which most closely maps $A$ to $B$, this has a solution as shown in https://en.wikipedia.org/wiki/...

**1**

vote

**0**answers

58 views

### Covering rough boundaries of closed sets in manifolds by charts

This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible.
Consider a Riemannian ...

**0**

votes

**0**answers

61 views

### Order of vanishing of Laplace's equation with potential

Consider the equation $-\Delta u + V u = 0$ with Dirichlet boundary conditions on the bounded domain $\Omega \subseteq \mathbb{R}^n$, where $V$ is a smooth potential. Let $V \leq 0$, and bounded on $\...

**4**

votes

**1**answer

113 views

### Trivial intersection of kernels

This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one.
If $X$ is a separable Banach space, can we find a basic ...

**5**

votes

**2**answers

234 views

### $id:A\to A^{op}$ is completely positive iff $A$ is abelian

Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive.
The claim is: $id$ is completely positive iff $A$ is abelian.
I need ...

**5**

votes

**2**answers

147 views

### Biorthogonal functionals

If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$.
What about conversely? If ...

**5**

votes

**0**answers

126 views

### Weak* continuity of positive parts, again

Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\...

**1**

vote

**1**answer

141 views

### Does the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$?

I am struggling to know whether the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$.
$fg$ has compact support but I can't figure out how I can try ...

**9**

votes

**2**answers

161 views

### Weak* continuity of positive parts

I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see?
Let $(f_\lambda)$ be a net in $l^\infty$ which converges weak* to $f \in l^\infty$. We do not ...

**7**

votes

**1**answer

263 views

### Abstract result on partitions of unity?

A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...

**1**

vote

**0**answers

48 views

### Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of $\...

**4**

votes

**0**answers

158 views

### Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...

**3**

votes

**0**answers

59 views

### On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...

**0**

votes

**0**answers

145 views

### Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...

**2**

votes

**2**answers

93 views

### Behavior of orbits under small perturbations

Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange.
Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...

**4**

votes

**1**answer

90 views

### Does the Nash inequality hold on manifolds with Lipschitz boundary?

Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...

**1**

vote

**1**answer

171 views

### analytic continuation argument

In "Pseudo-spectra, the harmonic oscillator and
complex resonances" (login required), the author says
Sections $2$ and $3$ of this paper concern the operator $Hf(x)=(-\frac{d^{2}}{dx^{2}}+cx^{2})...

**7**

votes

**1**answer

216 views

### When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ?
That is, given $p : ...

**4**

votes

**0**answers

127 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 ...

**0**

votes

**1**answer

53 views

### Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions?
Or are there restrictions on the function space over which this can be defined?
For example is the Rademacher complexity defined or has ...

**0**

votes

**0**answers

61 views

### A point on the absolute value of a bounded linear functional.

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...

**3**

votes

**0**answers

75 views

### Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...

**6**

votes

**1**answer

131 views

### Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\...

**1**

vote

**0**answers

69 views

### Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...

**1**

vote

**1**answer

105 views

### Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...

**3**

votes

**1**answer

112 views

### Abstract Wave Equation and Semigroups

If an operator $A$ on a Hilbert space $H$ generates a strongly continuous semigroup, does then the operator $B$ on $H \oplus H$ given by the matrix
$$ B := \begin{pmatrix} 0 & \mathrm{id} \\ A &...

**4**

votes

**1**answer

129 views

### Functional Calculus of closed operators

I learned that there is a holomorphic functional calculus for closed operators: If $T$ is a closed operator on a Hilbert space, and $f$ is a function that is holomorphic on some open subset $\Omega$ ...

**5**

votes

**1**answer

211 views

### Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...

**3**

votes

**0**answers

52 views

### Isometry from a representation to the representation tensored with itself

Suppose, the group $ G=S(2^{\infty})$ has a unitary representation $ \pi $ on a separable infinite dimensional Hilbert space $ H $.
(The group $ S(2^{\infty}) $ is the direct limit of the following ...

**0**

votes

**0**answers

33 views

### Rademacher function and weakly p-summable sequence

The function $ r_{n}:[0,1]\rightarrow \lbrace -1,1\rbrace $ be defined by $ r_{n}(t):=sgn(sin(2^{n}\pi t) $ is known as the n-th Rademacher function.
is $ (r_{n}) $ weakly p-summable sequence in ...

**0**

votes

**1**answer

168 views

### Formula for an integration on $\mathbb{Q} \cap [0,1]$

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting:
$$ \...

**3**

votes

**0**answers

66 views

### Eigenvalues of approximations to product-convolution operators

Consider an operator $T: L^2 \mapsto L^2$ of the form $TA = g (h \ast A)$ where $g$ is and $h$ are bounded $C^\infty$ functions.
This operator $T$ can be shown to be Hilbert-Schmidt, hence compact. ...

**5**

votes

**1**answer

119 views

### is the maximal tensor product of compact operators an essential ideal?

I'm searching for a counterexample for $C^*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $...

**1**

vote

**1**answer

77 views

### Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...

**0**

votes

**0**answers

51 views

### Monotone functions in ordered Banach spaces

Let $(X,\preceq)$ be a real ordered infinite-dimensional Banach space and $f:X\to \mathbb{R}$ is a Fréchet differentiable function. It is said to be a monotone non decreasing map if
$$ x\preceq y \...

**3**

votes

**1**answer

125 views

### is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive?

First of all, sorry for my bad english. I tried to find out whether the following statement is true or not:
Let $X$ be a operator system, $B$ a $C^*$-algebra and $f:X\to B$ linear such that $f(1)\ge ...

**3**

votes

**2**answers

288 views

### A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...

**5**

votes

**0**answers

130 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^{**}/...

**3**

votes

**1**answer

177 views

### On construction of a $\mathbb{Q}$ periodic function with Fourier series

Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series:
$$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$
Using ...