**0**

votes

**1**answer

51 views

### Norm of derivative of rank one projector

I asked this question on math.stack but I got no answer, so I try here.
Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation}
...

**6**

votes

**1**answer

122 views

### Existence of a measurable map between metric spaces

Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...

**2**

votes

**1**answer

85 views

### Semi-simple Banach algebra

Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?

**2**

votes

**1**answer

76 views

### Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...

**1**

vote

**0**answers

67 views

### An operator factoring through a Banach space containing no copy of $l_{1}$

Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_{1}$, but does not factor through any Banach space $W$ containg no copy of ...

**7**

votes

**1**answer

239 views

### What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?

Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...

**2**

votes

**0**answers

62 views

### Green's functions on linear subspaces and relations to boundary conditions

Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...

**7**

votes

**1**answer

207 views

### Derive an orthonormal system by Riesz basis $\{g(\cdot-\lambda_k),\ \lambda_k\in\mathbb R, \ k\in\mathbb Z\}$

Let $\{g(\cdot-k),k\in\mathbb Z\}$ be a Riesz basis, and let $\varphi\in L^2(\mathbb R)$ be a function defined by its Fourier transform
$$\hat{\varphi}(\xi)=\frac{\hat{g}(\xi)}{\Gamma(\xi)},$$
where
...

**11**

votes

**2**answers

208 views

### Existence of closed operators with arbitrary dense domain of a given banach space

Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., ...

**8**

votes

**1**answer

284 views

### Why should the map $-\Delta^{-1}$ be continuous?

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$
in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...

**8**

votes

**0**answers

204 views

### What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) ...

**9**

votes

**1**answer

96 views

### Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...

**3**

votes

**1**answer

127 views

### A family of convex bodies in Banach-Mazur position

Let $\{K_i\}$ be a family of smooth, origin-symmetric, strictly convex bodies such that $K_i$ converge in the Hausdorff distance (or you may assume $\partial K_i\to \partial K$ smoothly, in the sense ...

**1**

vote

**0**answers

49 views

### Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions:
Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...

**1**

vote

**0**answers

61 views

### When is an injective Fredholm map on $\ell^p$ a diffeomorphism

Consider a family of maps $f_p\colon \ell^p(\mathbb Z,\mathbb R)\to \ell^p(\mathbb Z,\mathbb R)$, $p\ge 2$, where
$$
f_q = f_p\big|_{\ell^q},\qquad \forall\; 2\le q\le p.
$$
Moreover,
$f_p\colon ...

**0**

votes

**0**answers

94 views

### does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...

**5**

votes

**0**answers

83 views

### 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 $m$ ...

**1**

vote

**0**answers

97 views

### The dual of the space of smooth functions that vanish at infinity

Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...

**1**

vote

**0**answers

78 views

### Characterisation of Sobolev spaces using eigenvalues of laplacian

I'm trying to find a reference on a result I thought it was obvious, but since I can't find anything I'm starting to doubt...
I'm looking for a characterisation of the Sobolev space $H^s(\Omega)$ ...

**1**

vote

**0**answers

60 views

### Fréchet differentiability of functional defined by a integral [closed]

I want to prove that if the functional $I: \mathcal{C}^1[t_0,t_f] \rightarrow \mathbb{R}$ defined by
$$
I(x) = \int_{t_0}^{t_f} F(x, \dot{x},t)\,dt
$$
is Fréchet differentiable if $F$ is ...

**5**

votes

**3**answers

250 views

### A space of distributions vanishing on the boundary

The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...

**1**

vote

**0**answers

53 views

### In which sense Daubechies wavelets converge to the Shannon wavelet?

My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in ...

**3**

votes

**2**answers

162 views

### Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; ...

**2**

votes

**1**answer

64 views

### interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...

**0**

votes

**1**answer

102 views

### Unbounded operator [closed]

Is the Laplacian operator $\Delta: C^2([a,b],\mathbb{R}) \to L^2(a,b)$ unbounded? Here $a,b \in\mathbb{R}$, $C^2([a,b],\mathbb{R})$ associated with $L^2$ norm.
If yes, how can we "modify" these ...

**6**

votes

**1**answer

157 views

### Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing?

A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup_{x^{*}\in B_{X^{*}}}\Bigl(\sum_{n=m}^{\infty}\lvert\langle ...

**1**

vote

**0**answers

40 views

### How to define Biharmonic operator for second order sobolev spaces

I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have
$$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : ...

**4**

votes

**1**answer

121 views

### Mixed norm estimate for the heat equation

Consider the inhomogeneous linear heat equation
$$\partial_tu-\Delta u=F$$
on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice ...

**0**

votes

**2**answers

121 views

### “semi-pseudonorm” in references

The following is an excerpt of a note in topological vector spaces.
I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we ...

**0**

votes

**0**answers

44 views

### Composition of upper semi-continuous real valued function with upper semi-continuous matrix valued function

Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if
$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$
where $\preceq$ ...

**1**

vote

**0**answers

98 views

### Homomorphic Commutator? Equation

So I was considering the following functional equation:
Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that
$$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), ...

**7**

votes

**0**answers

169 views

### Verifying Woronowicz’s proof that all $ {\text{SU}_{q}}(2) $’s are isomorphic as $ C^{*} $-algebras, where $ -1 < q < 1 $

This question is related to one that I asked some time ago.
Definition 1. Let $ q \in (-1,1) $. Let $ A $ be a unital $ C^{*} $-algebra and $ (x,y) $ a pair of elements of $ A $. Then define the ...

**9**

votes

**1**answer

197 views

### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...

**3**

votes

**1**answer

106 views

### Lebesgue differentiation theorem holds on locally doubling space?

It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...

**2**

votes

**2**answers

252 views

### “Generalisation” of one-parameter semigroups

Let $(Y,\left\|\cdot\right\|_Y)$ be a Banach space and $A:D(A)\subset Y \to Y$ a closed operator. Studying dynamical systems of the form
\begin{equation}
u'=Au
\end{equation}
quickly leads to the ...

**3**

votes

**1**answer

96 views

### Point-ultraweak limit of *-homomorphisms/cpc order zero maps

Suppose we have the following:
A C*-algebra $A$ and a von Neumann algebra $M$ (we can assume that $M$ is $\mathcal B(H)$).
A sequence of *-homomorphisms $\phi_i\colon A\to M$
an ultrafilter ...

**11**

votes

**1**answer

264 views

### Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a ...

**4**

votes

**1**answer

176 views

### Zeta-Determinant Theorem

Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...

**6**

votes

**3**answers

215 views

### Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = ...

**4**

votes

**1**answer

217 views

### Transform Riesz basis $\{e^{i \lambda_n t}\}_{n\in\mathbb Z}$ to an orthogonal basis

It is well-known that $\{e^{i n t}\}_{n\in\mathbb Z}$ is an orthonormal basis for $L^2(-\pi,\pi)$. A theorem by Kadec (Kadec $1/4$ theorem) studies the perturbed exponential system:
If ...

**1**

vote

**0**answers

62 views

### About optimizing a convex function on a hypercube

Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...

**2**

votes

**0**answers

101 views

### Changing frames of the tangent bundle with Schwartz functions [closed]

Let's consider two global frames $\{v_{1},....v_{N}\}$ and $\{u_{1},....u_{N}\}$ of the tangent bundle $T\mathbb{R}^N$.
Now consider the matrix $\{f_{i,j}\}$that change the frame $\{v_k\}$ to ...

**1**

vote

**1**answer

89 views

### Why $A^*A =A$ implies that $A$ is a C$^*$ algebra (Proposition 5.2.8 of An Invitation to Quantum Groups and Duality by Thomas Timmermann)

I am reading "An Invitation to Quantum Groups and Duality
From Hopf Algebras to Multiplicative Unitaries and Beyond" by Thomas Timmermann.
In the proposition 5.2.8 (page 117) the author provide a ...

**2**

votes

**2**answers

192 views

### Existence of smooth proper functions with bounded derivatives on manifolds

Suppose $M$ is a complete Finsler manifold of finite dimension. Is there always a smooth proper real-valued function $f$ on $M$ so that the norms of the first and second derivative are bounded ? The ...

**1**

vote

**1**answer

96 views

### Spherical decreasing rearrangement on the sphere

On $\mathbb{R}^n$, we have the concept of spherically decreasing rearragement of a function, which means, given a function $f$, one can design a radial and decreasing function $f^*$ such that $\Vert ...

**2**

votes

**1**answer

101 views

### Monotonicity of the integral

Let $R(x)$ be the residual function associated to the normal probability density, i.e.
$$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$
Define
...

**-1**

votes

**1**answer

204 views

### An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...

**6**

votes

**1**answer

325 views

### Uniform definition of $S(\mathbb{R})$ and $S(\mathbb{Q}_p)$

Let $\mathcal{P}=\{\infty, 2,3,5,7,11,\ldots\}$ be the set of primes of $\mathbb{Q}$ and let $\mathbb{Q}_p$ denote the corresponding completions, so in particular $\mathbb{Q}_{\infty}=\mathbb{R}$.
Is ...

**2**

votes

**2**answers

199 views

### Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian ...

**0**

votes

**0**answers

28 views

### Estimate $\int_0^1 g_k(x) d\mu(x)$, where $\{g_k(x)\}_{k=0}^\infty$ is a generic Parseval frame for $L^2(\mu)$

I came across the following integral in some applied problem:
$$\int_0^1 g_k(x) d\mu(x)$$
where $\mu$ is a Borel probability measure on $[0, 1)$, and $\{g_k(x)\}_{k=0}^\infty$ is a generic Parseval ...