All Questions
10,051 questions
3
votes
1
answer
254
views
gluing along a real analytic manifold
hi,
I have a general question. Assume we have a real analytic $n-$dim. manifold $X$ and $M$ a real analytic compact submanifold of $X$ (of dimension less that the dimension of $X$, say $k < n$). ...
- 89
6
votes
2
answers
320
views
Integration under functional sign
Let $f(x,y)$ be some bounded with its derivatives continuous function on $\Omega \times \overline{\Omega}$, where $\Omega$ is a domain in $\mathbb{R}^n$. Let $f(\,\,\cdot\,,\,y) \in \mathcal{E}(\Omega)...
- 1,329
1
vote
1
answer
780
views
An asymptotic series for the digamma function
As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number.
$$
\psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}}
$$
$B_n$ is the first Bernoulli numbers.
How ...
- 1,551
8
votes
2
answers
2k
views
(sharp)Garding's inequality and inequality with lower bounds
The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part $$P_{2m}...
5
votes
1
answer
391
views
About the quantum spectrum of a certain potential.
Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum ...
- 3,541
4
votes
2
answers
644
views
Does there exists a necessary condition for Lp multiplier?
Let $1 \leq p \leq 2$. A measurable function $m(\xi)$ is called a $L^p(R^n)$ ($L^p$ for convenience) multiplier, if $$\|m(D)\varphi\|/\|\varphi\|_{L^p} \leq C , \varphi \in L^p
$$ for some constant $C$...
- 425
3
votes
0
answers
254
views
Ways to establish equality of measures on locally compact spaces
Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...
- 1,329
1
vote
1
answer
350
views
Strong convergence in reflecxive Banach space
Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f_n \rightarrow f$ weakly in $X$, and $\|f_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly ...
- 425
11
votes
1
answer
1k
views
Stone-Weierstrass analogue for $L^p$
Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...
- 109k
3
votes
1
answer
303
views
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
1
vote
1
answer
318
views
Uniformly continuous functions and Borel hierarchy in the compact-open topology
Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}_u(\...
- 7,817
1
vote
0
answers
1k
views
Can you prove the monotonicity of the function (or find a counter example)?
Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative ...
- 11
4
votes
3
answers
729
views
Inequality of von Neumann for more than two contractions
Good morning,
I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...
- 758
4
votes
4
answers
435
views
Must Neuman Elliptic operator has discrete spectrum ?
It is well known that the Neuman eigenvalue problem has discrete spectrum and the eigen values are
nonnegative and can be arranged in a nondecreasing order of magnitude.
Do we need any smoothness ...
- 41
0
votes
2
answers
146
views
representation of compact supported distribution
Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
- 9
5
votes
2
answers
1k
views
Polar decomposition in C*-algebras
A very nice feature of W*-algebras is the following:
once you have an element $a$ of a W*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$.
It seems that it carries over to AW*-...
- 537
2
votes
1
answer
235
views
ODE for functions with values in locally convex TVS
Given an ODE for a function $u \in C^1(I,V)$, where $V$ is some locally convex TVS (topological vector space) and $I \subset \mathbb{R}$, i.e.
$\frac{d}{dt} u = f(t,u)$
for some function $f: I \...
- 403
3
votes
1
answer
177
views
If $A \subset X'$ annihilates only $0$, then $A$ is dense
Let $X$ be a Banach space with continuous dual space $X'$ with norm topology. Let us regard the following property of $X$:
Property: Any linear subset $A \subset X'$ that satisfies $\bigcap_{\alpha\...
- 5,327
1
vote
2
answers
394
views
When LCS is isomorphic to subspace of some function space?
Updated: Following Michael's suggestion, I rephrase the question slightly.
Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function ...
- 101
0
votes
2
answers
2k
views
The exponent of self-adjoint operator
If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^k$ is self-adjoint for all positive integer $k$? (I have already known that the conclusion ...
- 1,441
3
votes
2
answers
362
views
Invariant subspaces for compact restrictions
Suppose $Y$ is a closed hyperplane in $X$, so we can write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\alpha_ny_n+\beta_nx_0$, for any $...
- 31
5
votes
1
answer
600
views
Closed operators and duality
Usually we would define a "densely defined, closed operator" on a Banach space $E$ to be a linear map $T:D(T)\rightarrow E$, where $D(T)$ is a dense subspace of $E$, and the graph of $T$, $G(T)=\{ (x,...
- 18.7k
3
votes
1
answer
502
views
Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
- 11.5k
6
votes
1
answer
879
views
Bochner's theorem, in stages
Bochner's theorem (for the real line version) asserts an infinite tower of inequalities, as a positivity condition. Taking each one, what do they mean, in an elementary fashion (at least at the start)?...
- 275
2
votes
1
answer
3k
views
The fourier transform of homogeneous distribution and related topics
When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in $R^{n}\...
- 1,644
4
votes
1
answer
360
views
Density character of $\ell_\infty(\kappa, S)$
Let $S$ be an uncountable set. Consider the subspace $\ell_\infty(\kappa, S)$ of $\ell_\infty(S)$ formed by all functions with support of cardinality at most $\kappa$ (here $\kappa<|S|$). Certainly,...
- 537
0
votes
0
answers
395
views
The ratio of two strictly increasing functions
Given:
\begin{equation}
f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c}
K \\\
i \\
\end{array} \right) \left(-1-\frac{1}{ar}\right)^i
\end{equation}
\begin{equation}
f_2(a)=\sum_{i=1}^{k^*-1} ...
2
votes
1
answer
348
views
Atomic enveloping von Neumann algebra
Let $A$ be a $C^*$-algebra. If the second dual of $A$, which is the enveloping von Neumann algebra of $A$, is atomic, can we deduce that $A$ is an ideal in its second dual ?
- 195
0
votes
0
answers
607
views
partial differential equations with mixed boundary conditions
hi,
does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?
actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(...
- 89
7
votes
6
answers
2k
views
Fractional Leibniz formula
Let $T=(-\Delta)^{1/2}$.
Can we have estimates, similar to the one below
$$
\| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1}g\|_p,
$$
hold in $L^p$, where $...
- 1,644
5
votes
2
answers
465
views
A name for a weak topology
Let $V$ be a real vector space and let $V'$ be the algebraic dual of $V$, i.e. the space of all the linear functionals $V\to\mathbb{R}$. Then there exists the weakest topology $\tau$ which makes all ...
- 3,749
1
vote
2
answers
409
views
Does these commutator estimates bound in $L^{2}$
According to the basic rules of symbolic caculus,$[a(x,D),x_{j}]=-ia^{j}[x,D]$.So we have $[(1-\triangle)^{\frac{1}{2}},x_i]=\partial_i(1-\triangle)^{-\frac{1}{2}}$ which is $L^2$ bounded.
It's also ...
- 1,644
2
votes
3
answers
941
views
The topology of $C_0^\infty(M) $
I have read definitions in my PDE book as follows: If $M$ is a smooth paracompact manifold, the space of all linear functional on $C^\infty(M)$ is denoted by $E'$ and the space of all linear ...
- 1,441
4
votes
3
answers
464
views
What classes of functions are closed under all rescalings?
Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables ...
- 81
2
votes
2
answers
665
views
Finite imensional subspaces of $L^\infty.$
This is the question I had meant to ask when I asked this question.: Is there a concise characterization of finite dimensional subspaces of $L^\infty?$ (that's what the discussion with @fedja was ...
- 96.4k
5
votes
3
answers
512
views
Finite dimensional subspaces of $L^1.$
This question is motivated by my discussion (via comments) with @fedja regarding this earlier question. In any case the question is whether there is any concise characterization of finite dimensional ...
- 96.4k
6
votes
2
answers
487
views
Induction theorems for finite-dimensional complex representations of infinite groups
Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
- 976
15
votes
2
answers
1k
views
Is zero a hydrogen eigenvalue?
This question has been bugging me for some time.
Take the hamiltonian for the hydrogen atom: $$\hat{H}=-\frac{1}{2}\nabla^2-\frac{1}{r},$$ acting on (a domain contained in) $L^2(\mathbb{R}^3)$. It is ...
- 2,358
2
votes
1
answer
164
views
Second derivative seminorm of function composition
Consider the seminorm $\| f \|^2 = \int_{-\infty}^{\infty} dx f''(x)^2$
for $f:\mathbb{R}\rightarrow \mathbb{R}$ in the Sobalev space $W^{k,2}(\mathbb{R})$.
Can we put some upper bound on the ...
2
votes
4
answers
411
views
A Fractional Linear Transformation Class Property
Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and $f'(x)>0$...
- 81
5
votes
1
answer
508
views
Projections which are not completely bounded
There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand ...
- 181
5
votes
2
answers
404
views
Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?
The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...
- 181
7
votes
1
answer
588
views
A characterization of Lagrange multiplier. Where to find a proof?
Let $F,G\in C^1(\mathbb{R}^n,\mathbb{R})$. Assume for
$s\in(s_0-\varepsilon,s_0+\varepsilon)$,
\begin{align}
E(s) = \min F\quad\mbox{subject to}\quad G=s
\end{align}
is achieved at some $x(s)\in\...
- 305
3
votes
1
answer
3k
views
Approximate point spectrum
I have a question concerning the relation between the approximate point spectrum and the spectrum of an operator.
Let $T$ be a bounded linear operator of a complex Hilbert space $H$. The approximate ...
- 758
2
votes
2
answers
1k
views
Lebesgue integral with respect to vector measures?
Good evening,
I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense ...
- 758
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
3
votes
1
answer
598
views
is a non-invertible operator a boundary point of the group of invertible operators?
Good evening,
I have a question concerning non-invertible operators.
Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence ...
- 758
14
votes
2
answers
4k
views
What is a good reference that compact resolvent implies Fredholm operator?
Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i.e. $A$ can be ...
- 191
1
vote
1
answer
775
views
Weak* continuity of linear maps
I consider a linear map $T\colon X^*\to Y^*$, where $X^*$ and $Y^*$ are duals of Banach spaces. I would like to know if I can deduce that $T$ is weak* continuous (I consider the weak* topologies on ...
- 113
25
votes
2
answers
4k
views
Dual of the space of Hölder continuous functions?
Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?
- 1,256