All Questions
9,780 questions
0
votes
1
answer
208
views
The pth power of a distance function is twice continuously differentiable, for $p>2$?
Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...
- 9
2
votes
0
answers
266
views
Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$
Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of $L^2(0,...
- 171
3
votes
1
answer
99
views
Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$
Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| \,...
- 235
3
votes
0
answers
146
views
Variational Principle for a System of Differential Equations
I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v \...
- 516
1
vote
1
answer
2k
views
spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]
Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(...
- 11
2
votes
1
answer
547
views
Equivalent references for Schwartz's book of the distribution theory
Hello,
It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like
$$
\dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \...
- 1,649
3
votes
2
answers
642
views
Localization of Laplacian eigenfunction on the unit square?
Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
1
vote
2
answers
734
views
Quantum Error Correction
One can correct the errors in a quantum channel iff the coherent information of the input state is not reduced by the channel. This is analogous to sending quantum entanglement through a channel. If ...
- 29
0
votes
0
answers
113
views
Existence of a mapping in a nonseparable Banach space
Do there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that
$$
\forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad (...
- 1
5
votes
1
answer
171
views
purely point spectrum as compactness of orbits
Let $U$ be a unitary operator in a complex separable Hilbert space $H$. Assume that for any vector $x$ its orbit $\{x,Ux,U^2x,\dots\}$ is precompact in $X$ (i.e. closure is compact). Then there exists ...
- 109k
-2
votes
1
answer
146
views
a measure convolution equation
My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...
- 367
7
votes
1
answer
1k
views
Banach spaces with a certain separability property
In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset $...
- 11.4k
2
votes
0
answers
807
views
Why groups that admit Folner Sequences are amenable
I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...
- 21
1
vote
1
answer
172
views
Extension of power bounded operators over a finite subspace
Suppose $Y$ is a Banach space and $X$ is a finite-dimensional subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above and below, in other words ...
- 6,287
5
votes
0
answers
425
views
Reflexive-saturated Banach spaces
Say that a Banach space $X$ is strongly saturated by reflexive subspaces if every closed subspace $Y\subset X$ contains a further reflexive subspace $Z\subset Y$ with $\mbox{dens }Y=\mbox{dens Z}$. If ...
- 387
0
votes
1
answer
130
views
Extending affine maps defined on weakly closed sets to the whole topological space
Given $C$ a weakly closed convex subset of a (real) Banach space $B$, with $0\in C$ and $\varphi:C\longrightarrow \mathbb{R}$ weakly continuous, with $\varphi(0)=0$, can we extend $\varphi$ to a $\...
- 647
0
votes
0
answers
178
views
$2$-normed Spaces
Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper ...
- 1
2
votes
0
answers
71
views
Error bounds for eigenvalue expansion of the Mathieu equation
The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...
- 259
1
vote
0
answers
101
views
Discrete J-method of interpolation [closed]
The discrete $J$ method is, given Banach spaces $A_0$ and $A_1$:
The interpolationn space $[A_0, A_1]_\theta$ is defined by: $a \in [A_0, A_1]_\theta$ if and only if $a$ can be written as $a=\sum_{...
- 11
3
votes
1
answer
352
views
Integral Equation with "convolution"
I've got the following problem I'm working on which is related to some of my research:
Solve:
$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$
for f, given $G$ which has whatever smoothness ...
- 31
1
vote
0
answers
80
views
Sobolev embedding on warped product
Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...
user73978
3
votes
0
answers
381
views
Extension divergence-free, curl-converging vector field
Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
- 3,425
3
votes
2
answers
1k
views
Spectral decomposition for an arbitrary linear combination of position and momentum operators
Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i $\frac{...
- 195
1
vote
0
answers
90
views
Da Prato's notion of Symmetric Operator
For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...
- 379
2
votes
0
answers
272
views
Continuity of multiplicative character
Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
- 61
-2
votes
1
answer
665
views
weak convergence
I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x_n$ is a non negative ...
- 779
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 1,649
3
votes
0
answers
221
views
Comparison of Hardy's inequality and Sobolev's inequality
Both, Hardy's inequality and Sobolev's inequality, are estimates that compare the Laplacian of a function and to the function itself, admittedly in a slightly different fashion.
Still they seem to be ...
- 337
1
vote
1
answer
338
views
On the generalization of the Mittag-Leffler function and fractional derivative
The Mittag-Leffler function $E_{\alpha}(x)$ has an important property:
$$
\frac{\partial^{\alpha}}{\partial x^{\alpha}} E_{\alpha}(x^{\alpha}) = E_{\alpha}(x^{\alpha}).
$$
I tried to find an ...
- 1,329
1
vote
1
answer
188
views
Is sequential completeness of LCS strictly stronger that Riemann integrability of curves?
$\def\RiemInt{\,\text{-}\,\lower.5mm\hbox{$^{^{\rm Riem}}$}\kern-1mm\int}$
This is essentially a reformulation of this MSE question which has not received any answers for about three weeks. To ...
- 3,584
1
vote
1
answer
342
views
Compact sets in TVS
Let $K$ be a compact subset of a Hausdorff topological vector space. Is it true that
$\bigcap_{n\in \mathbb{N}}\frac{1}{n}K$ is either empty of or is a set consisting of the origin only?
- 11
5
votes
0
answers
179
views
Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?
Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
1
vote
0
answers
341
views
spaces of smooth functions with bounds on partial derivatives
EDIT: As there were no takers at all... I have added below a possible approach I came up with...
I would like to ask the following elementary but tricky question about the density of spaces of smooth ...
- 1,338
10
votes
0
answers
508
views
Tensorial decomposition of $B(H)$
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
- 101
2
votes
1
answer
695
views
Is the F.T of $\operatorname{Span}(\mathscr S(\mathbb R^2)\otimes\mathscr D(D))$, $D\subset\mathbb R^2$ dense in $L^2(\mathbb R^4)$?
Let $K$ be the real vector space generated by elements $f$ in $\mathscr S(\mathbb R^2,\mathbb R)\otimes\mathscr D(D, \mathbb R)$, where $D$ is any bounded subset of $\mathbb R^2$. Let $\hat K$ be the ...
- 417
7
votes
1
answer
737
views
Question about projections on a Hilbert space
Sorry for the vague title, I can't think of a better one that isn't overly long.
Suppose that $S$ is a commuting set of projection operators on a Hilbert space. I'll introduce the following notation: ...
- 391
1
vote
0
answers
98
views
Topological tensor products of spaces of holomorphic functions of slow growth
Let $X$ be a Banach space, $M$ be a complex manifold, and $\Omega$ a relatively compact domain in $M$. We consider the space $\mathcal{A}^{-\infty}(\Omega, X)$ of $X$-valued holomorphic functions of ...
1
vote
2
answers
296
views
The operator preseving two disjoint dense operator ranges invariant
Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space. Suppose that $\mathcal{H}_0\subset\mathcal{H}$ which is a dense proper subspace is the range of some bounded linear operator $T$. ...
- 11
12
votes
0
answers
284
views
Star-shaped Folner sequence
Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that $...
- 4,432
3
votes
0
answers
119
views
Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?
Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of
$$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$
where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...
- 107
-1
votes
1
answer
1k
views
relation between inclusion and embedding [closed]
Assume that $X$ and $Y$ are two Banach spaces, now we have that $X$ is included in $Y$, in the sense that $\forall a\in X$, we have $a\in Y$. Then can we get that $X$ is embedded in $Y$, namely, $\...
- 331
4
votes
1
answer
251
views
generator of Dirichlet form coincide with the absolute part of the "Laplacian"
Let M be an Riemannian manifold with the Dirichlet form $$\varepsilon (u,v) =-\int_M \langle \nabla u,\nabla v \rangle$$ for $u,v \in W^{1,2}_0(M)$. Let $\Delta^M:D(\Delta^M) \to L^2(M)$ denote the ...
- 199
2
votes
1
answer
222
views
Arveson index and curvature
Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E_0$ semigroups. I was reading the standard paper of Arveson, but is lost and yet to get intuition about it. An ...
- 421
2
votes
0
answers
238
views
Fractional Derivatives [closed]
How far these Theories of "Fractional Derivatives" be rigorized ? I have few books and references on Fractional Differential Equations etc (mainly they stress on Applied Mathematics parts and similar ...
user20510
8
votes
0
answers
221
views
Density of odd and even eigenstates of an integral operator
Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
- 81
0
votes
0
answers
54
views
Left introversion operators associated to function spaces on semigroups
I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
- 1
2
votes
0
answers
155
views
Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)
Denote by $\mathbb{E}(g)$ the solution of the PDE
$$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$
$$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$
Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...
- 155
1
vote
0
answers
78
views
Related to derivative of Modified Bessel I function wrt the order
I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...
2
votes
2
answers
711
views
Are there good inequalities on the norm?
It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning ...
- 1,528
0
votes
0
answers
135
views
uniform continuity of a function in ultrametric spaces
Consider $[0,1]$ with the metric $d_1(x,y)=\left\{\begin{array}{cc}
0&x=y,\\
\max\{x,y\}&x\ne y.
\end{array}\right.$. Moreover let $(M,d_2)$ be an ultrametric space. Let
$f:(M,d_2)\to([0,1],...
- 1