All Questions
13,944 questions
2
votes
0
answers
114
views
Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.
Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.)
...
- 88
2
votes
0
answers
137
views
Invariant linear manifolds for multiplication by the independent variable in L^2 (R)
In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-...
- 31
5
votes
0
answers
417
views
Direct integrals and fields of operators
Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.
Suppose now ...
- 3,897
0
votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
1
vote
0
answers
308
views
Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
2
votes
0
answers
82
views
Description of the norm of certain interpolation space
Dear all,
I suspect that there should be some detailed description of the norm (or of the unit ball) of the following complex interpolation space (for any $0< \theta < 1$): $$\Big(B(\ell_1^n, \...
- 769
7
votes
0
answers
161
views
Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
1
vote
0
answers
114
views
Mappings preserving convex compactness
Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...
- 11
0
votes
0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
3
votes
1
answer
263
views
Asymptotically multiplicative functions and matrices
Hi,
Let $\mathbb{N}_{cop}^2$ denote the set of all pairs of coprime natural numbers. A function $f:\mathbb{C}\rightarrow\mathbb{C}$ is called asymptotically multiplicative, iff $\epsilon_{m,n}:=f(mn)...
- 7,127
3
votes
0
answers
223
views
Extension of positive operators and Bauer-Namioka
When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
- 31
2
votes
0
answers
200
views
Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21
4
votes
0
answers
109
views
rank of a C^1 map
I saw this three star problem in Hirsch ..
If we have open sets $U \subset R^3$ ,$V \subset R^2$ and $f:U \to V$ is $C^1$ and onto...Prove there is at least one point in $U$ where $f$ has full rank
...
- 153
0
votes
1
answer
130
views
Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 800
1
vote
0
answers
52
views
Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\...
- 71
1
vote
0
answers
61
views
Multi-completely monotone functions
Consider a $C^{\infty}$ nonnegative function $f(x,y,z)$, $x,y,z>0$ and let $\lambda f(\lambda x, \lambda y,\lambda z) \equiv f(z,y,z)$ for any $\lambda > 0$ (positive homogenity). Define
$$
g_{...
- 1,329
3
votes
0
answers
383
views
Neglect of Compact Quantum Metric Spaces [closed]
Does anyone have an opinion on Rieffel's theory of compact quantum metric spaces? To me it seems to be a very interesting new area of mathematics. It shows how to generalise complicated geometric ...
- 1,462
0
votes
0
answers
80
views
relationship between different function classes
I was wondering if there is a survey of relationship between several different well-studied function classes ?
ps - The question may be vague but I am looking for something along the lines of - http:/...
- 1
0
votes
0
answers
92
views
Lower bound for double sums with power law decay terms.
This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion.
The motivation to ask here if the inequality below ...
- 2,044
6
votes
0
answers
98
views
Do the translates of integrable function approximate its radial part?
For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (...
- 415
4
votes
0
answers
162
views
Symmetric functions and regularity (II)
My previous question (where $n=2$) was a bit too naive. I think that this one, which is the one being of genuine interest to me, is more involved.
Let $f=\mathbb R^n\rightarrow\mathbb R$ be a ...
- 52.3k
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
0
votes
0
answers
104
views
Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
1
vote
1
answer
210
views
Is the metric obtained by altering the metric of a Hilbert space on a finite-dimensional subspace equivalent to the original one? [closed]
Suppose a Hilbert space W can be written as the direct sum (not necessarily orthogonal) of the closed subspaces H and V, where H is assumed to be of finite dimension. Define a new inner product via
||...
- 2,935
1
vote
0
answers
195
views
Lower semicontinuity of Bregman distances/divergences
For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$:
\begin{...
- 12.7k
2
votes
0
answers
140
views
WLD Banach spaces
Does anyone know of an example of a weakly Lindeloff determined (WLD) Banach space which does not contain c_0 and is not weak Asplund? I believe the example of a WLD, non-weak Asplund space by Argyros ...
- 21
0
votes
0
answers
73
views
A constrained prolongement
Let $\Omega$ be a domain of $R^n$, let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$
I am wondering about the existence of a function $\tilde{\theta} \in W^{2,\infty}...
- 25
2
votes
0
answers
156
views
Holomorphic stability of inverse limit of pre-$C^*$-algebras
Let A be a C*-algebra and let At be a set of dense *-subalgebras of A, stable under holomorphic functional calculus on A, which are also Banach algebras complete with respect to the norms ||$\cdot$||t....
- 613
2
votes
1
answer
168
views
Local supporting points of Lipschitz functions
Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a
Lipschitz function. Say that a point x in X is a local supporting point
of f if there exist x^* in X^* and an open neighborhood U ...
- 31
4
votes
0
answers
102
views
quasinilpotence and finite spectrum II
Let A be a quasinilpotent operator on a Hilbert space and let every
operator of the algebra generated by $A$ and $A^{*}$ have finite
spectrum. Does then follow, that A is nilpotent ?
See also ...
- 2,753
2
votes
0
answers
197
views
Generating cones having no surjections [in operator spaces]
Is this little toy known ?
Let $E$ be some Banach space, and let $K$ be the closed unit ball
of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$
be the natural ...
- 4,060
3
votes
0
answers
130
views
Positive block matrices over tensor algebras
Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form
$$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$
where $a,b$ are ...
- 18.7k
1
vote
0
answers
133
views
Square powers of hemicontinuous operators
Let H be an infinite dimensional real Hilbert space.
A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line
segment of H to the weak ...
- 4,060
0
votes
0
answers
44
views
Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
0
votes
0
answers
45
views
Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
- 111
0
votes
0
answers
16
views
Representing a periodic strip operator as a tensor product of operators
I hope this question is not trivial, but here goes. I want to consider a bounded operator on $\mathcal{H}=\ell^2(\mathbb{Z}\times \{0,...,N-1\})$ that is a discrete Schrodinger like operator.
...
- 633
0
votes
0
answers
36
views
Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
- 109
0
votes
0
answers
29
views
On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
- 765
0
votes
0
answers
49
views
ODE satisfied by a special function
Posted on MSE
Context
I would like to estimate the distribution of the difference of two inverse gaussian variables. The convolution doesn't lead to any special functions according to Mathematica . ...
- 393
0
votes
0
answers
57
views
Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
- 75
0
votes
0
answers
22
views
An auxiliary problem while constructing the system of Jordan sets on a plane
Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$
where $a,b,c, d \in R$, $a<b$, $c<d$.
Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$.
...
0
votes
0
answers
122
views
Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
0
votes
0
answers
50
views
Kirszbraun-like extension of periodic functions
Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...
- 1