All Questions
12,777 questions
6
votes
6
answers
2k
views
Application of bounded spectral theory.
I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are ...
- 2,641
4
votes
0
answers
312
views
Transforming a multivariable integral to make it separable
In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...
- 37.7k
3
votes
0
answers
259
views
Quotient of manifolds by groups and embeddings
Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a ...
- 411
6
votes
2
answers
1k
views
An element of $(L^{\infty})^*$ which does not seem to be a finitely additive abs. cont. measure.
Hi everyone,
I have a question which I am quite stumped on. Consider the linear functional $l(f) = f(0)$ defined on $C([-1,1])$. By Hahn-Banach this linear functional can be extended to one on all ...
- 2,641
4
votes
2
answers
4k
views
Embedding of $BV$ and $L^p$ spaces
An elementary question about Sobolev spaces:
Is there some explicit theorem about embedding relation between spaces $BV(\Omega)$ and $L^p(\Omega)$?
Formulated otherwise: is $BV$ a subset of $L^2$ (i....
- 53
4
votes
1
answer
474
views
Are these operators defined on 2D surfaces self-adjoint?
My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether ...
- 199
5
votes
2
answers
904
views
Are there compact analogues of Cartan's theorems A and B?
Cartan's theorem A says that on for a coherent sheaf ${\mathcal{F}}$ on a Stein manifold X, the fibres ${\mathcal{F}}_x$ over each point x in X are generated by global sections.
I'm wondering if ...
user2529
2
votes
1
answer
2k
views
The normal derivative of the Green's function
I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ ...
- 2,299
17
votes
5
answers
7k
views
A counter example to Hahn-Banach separation theorem of convex sets.
I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed ...
- 2,641
2
votes
2
answers
602
views
Reference for weak*-semigroup
Let $X$ a dual Banach space (there exists a Banach space $Y$ such that $X=Y'$).
A weak* semigroup on $X$ is a semigroup $(T_t)_{\geq 0}$ on $X$ such that, for all $x\in X$, we have $T_tx\to x$ in the ...
- 1,222
5
votes
1
answer
2k
views
Maximum on unit ball (James' theorem).
James' theorem states that a Banach space $B$ is reflexive iff every bounded linear functional on $B$ attains its maximum on the closed unit ball in $B$.
Now I wonder if I can drop the constraint ...
- 455
4
votes
1
answer
822
views
Converse of Picard's Big Theorem?
The celebrated Big Theorem of Picard's is that, in every open set containing an essential singularity of a function $f(z)$, $f(z)$ takes on every value (except for at most one) of $\mathbb{C}$ ...
- 2,019
2
votes
2
answers
1k
views
description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 1,222
2
votes
2
answers
303
views
Characterisation of positive elements in l¹(Z)
Consider the Banach $^* $-algebra $\ell^1(\mathbb Z)$ with multiplication given by convolution and involution given by $a^*(n)=\overline{a(-n)}$.
I would like to find nice necessary and sufficient ...
- 3,184
0
votes
1
answer
261
views
Flat locus of $S_{1}$-morphism
Hi, everybody.
Consider an ${\rm S}_{1}$- morphism $f:X\rightarrow S$ of reduced complex spaces. Assume that $f$ is open (universally open in Alg.geom), equidimensional with $n$-pure dimensional ...
- 435
3
votes
1
answer
235
views
Odd element of L^1 group algebra of the integers
Giving some motivation is hard here, so I'll just ask the question. I want an element $a=(a_n)\in\ell^1(\mathbb Z)$ such that:
$\|a\|>1$
a is power bounded (turn $\ell^1(\mathbb Z)$ into a Banach ...
- 18.7k
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
2
votes
4
answers
222
views
How to compare finite point sets in normed spaces?
I want to define a "distance" between two subsets $A, B$ of a normed space $(V, \|\cdot\|)$ both with (at most) $n$ elements. A straightforward way for me to do this would be to define
$$ d(A, B) := \...
- 21
8
votes
2
answers
1k
views
What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?
Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...
- 7,197
7
votes
2
answers
1k
views
Contour integration problem from probability
Can integrals of the form
$$
\int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x
$$
be computed in closed form using contour integration (or any other ...
- 5,227
6
votes
0
answers
354
views
Ordering of completely bounded maps
Let A be a C*-algebra, let H be a Hilbert space, and let $T:A\rightarrow B(H)$ be a completely bounded (cb) map (that is, the dilations to maps $M_n(A)\rightarrow M_n(B(H))$ are uniformly bounded). ...
- 18.7k
2
votes
1
answer
1k
views
Elliptic regularity on bounded domains
I'm concerned with a generic uniformly elliptic operator $L$ on $\mathbb{R}^n$. If $L$ is uniformly elliptic and I am studying the equation $Lu=f$ then the way I can deduce regularity on $\mathbb{R}^n$...
- 2,641
10
votes
5
answers
4k
views
Orthonormal basis for non-separable inner-product space
Suppose X is an inner product space, with Hilbert space completion H (actually, I'm interested in the real scalar case, but I doubt there's any difference). If H is separable, then so is X, and I can ...
- 18.7k
0
votes
0
answers
320
views
A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
3
votes
2
answers
459
views
Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups
In studying the transformation groups generated by holomorphic vector fields V(z) d/dz on ℂ, I've noticed the (surely well-known) fact that the complex quadratic vector fields:
&...
- 2,858
3
votes
2
answers
6k
views
Need help understanding Riesz representation theorem for Reproducing Kernel Hilbert Spaces
I'd like some help understanding any of the following proofs of Riesz representation theorem -- whichever is simpler -- or in fact any proof of the theorem.
Proof 1: http://nfist.pt/~edgarc/wiki/...
- 661
1
vote
1
answer
263
views
Need help with references on the status of a "Littlewood Problem"
The "Littlewood Problem" in the title asks for a characterization of finite sequences
n1< ...< nk of integers such that zn1+zn2+...+znk≠0
for any complex number z of unit modulus.
Does ...
3
votes
1
answer
572
views
When is a finite matrix a "good" approximate representation of an operator?
I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions
$\rho(r) = \sum_{i=1}^N q_i ...
- 1,890
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 617
7
votes
3
answers
4k
views
infinitely many linear equations in infinitely many variables
Let $(a_{mn})_{m,n\in\mathbb{N}}$ and $(b_m)$ be sequences of complex numbers.We say that $(a_{mn})$ and $(b_m)$ constitute an infinite system of linear equations in infinitely many variables if we ...
6
votes
3
answers
2k
views
Sequential topological vector spaces
Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
- 9,650
1
vote
2
answers
504
views
Do all graphs of C1 functions have Hausdorff dimension 1?
Suppose f is a real-valued function of one variable, and suppose f is of differentiability class C1. My question is, if $\Gamma$ is the graph of f, then must $\dim_H(\Gamma)=1$? If anyone knows of a ...
0
votes
2
answers
337
views
Is there a general notion of entropy for the states of a C*algebra?
I've seen some definition of the relative entropy between two states of a C*algebra. However this definitions work only for finite dimensional C*algebras and I don't know if there is a correspondent ...
17
votes
2
answers
2k
views
Getting a differential equation for a function from a functional equation of its Mellin transform
If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a ...
- 1,412
20
votes
12
answers
9k
views
The role of completeness in Hilbert Spaces
Why do Hilbert spaces have to be complete?
I've been studying (teaching myself about) Hilbert spaces for a while now as they have a habit of popping up in many of the papers I'm come across (I'm a ...
- 661
11
votes
4
answers
2k
views
Is this a $C^{\infty}$ function ?
Let be $(a_n)\in\ell^2(\mathbb N)$ and consider the mapping $f:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ given by
$$
f\Big((a_n)\Big)=(a_n^n).
$$
Question: Is $f$ a Fréchet $C^{\infty}$ function in whole ...
- 2,044
5
votes
3
answers
940
views
Square of an elliptic curve and projective plane
Let's assume one takes $E = \mathbb{C}^* / \langle p \rangle$ an elliptic (Tate) curve over the complex field ($p = e^{2 \pi i \tau}$ where $1, \tau$ are the 2 periods in additive notation; $\Im \tau &...
- 73
26
votes
6
answers
8k
views
prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,063
1
vote
1
answer
307
views
variational formulation: boundedness of the bilinear form
The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to ...
- 343
6
votes
3
answers
3k
views
Why isn't the theorem of approximation applicable in Banach spaces?
Let X be a Hilbert space, A a convex, closed subset of X. Then there exists for every x in X exactly one best approximation in A, that is there exists a y in A such that || x-y || = d(x,A) = inf { || ...
- 61
2
votes
2
answers
356
views
Coefficients of holomorphic functions defined by Borel probability measures on the unit disc
Let be $\mathcal M(\partial\mathbb D)$ denote the set of all Borel complex probability measures on $\partial\mathbb D$ (unit circle in the complex plane). Define a mapping $\Phi:\mathcal M(\partial\...
- 2,044
13
votes
0
answers
564
views
Symmetric (extended) Haagerup tensor product
Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...
- 18.7k
5
votes
4
answers
3k
views
Minimizing the modulus of a polynomial around a circle
I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...
4
votes
2
answers
519
views
Factorization through $\ell_{1}$ and operator ideals
Recently, I bumped into the class of operators that factor through $\ell_{1}(X)$ for some set $X$. For now, $X$ is a set with arbitrary cardinality but if it leads to a more concrete answer to my ...
- 1,848
3
votes
1
answer
362
views
Cartesian product of test function spaces
Mini introduction
Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i \in \...
5
votes
3
answers
2k
views
Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way
I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on $\...
- 7,197
4
votes
3
answers
2k
views
Coprimality and squarefree numbers
As observed on Mathworld, "Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of ...
- 3,612
4
votes
2
answers
676
views
Basis for L_infty(R)
Let $V$ be the Banach space of bounded sequences of reals with the sup norm. Does there exists a subset $B$ of $V$ such that
Linear Independence: For all functions $c$ in $\mathbb{R}^B$, if $\sum_{b ...
user5810
2
votes
3
answers
891
views
Fourier Transforms restricted to mass shell
Hello,
I am stuck with the following (hopefully not too trivial) problem.
I want to know, if the map
$${\cal D}(\mathbb{R}^2)\to L^2(H_m,d\Omega_m)\qquad f \mapsto \hat{f}|_{H_m}$$
has dense range.
...
- 23
28
votes
5
answers
3k
views
Continuous + holomorphic on a dense open => holomorphic?
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D_1$ and $D_2$.
Let ...
- 43.2k