**4**

votes

**3**answers

273 views

### Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.

**1**

vote

**0**answers

31 views

### Pertubations of self-adjoint first order operators

If we consider the self-adjoint operator $L= L_0 + h$ on the appropriate Sobolev spaces of maps from $S^1$ to $\mathbb{R}^N$, say, where $L_0$ is a first order self-adjoint operator and $h$ is a ...

**0**

votes

**0**answers

34 views

### Orthogonal complements of intersections of closed subspaces

Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$.
$\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...

**1**

vote

**0**answers

47 views

### When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?

I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...

**0**

votes

**1**answer

73 views

### Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [on hold]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...

**0**

votes

**1**answer

138 views

### Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question
I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...

**3**

votes

**1**answer

118 views

### What kind of role has Functional Analysis played in Signal Processing? [on hold]

Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?

**-4**

votes

**0**answers

61 views

### Sequences and reflexivity [on hold]

Assume $X$ to be a real reflexive Banach space.
Why are sequential topological notions
topological notions ?
For ex :
sequentially closed set is closed,
sequentially compact set is compact,
and so ...

**2**

votes

**1**answer

161 views

### The following type of Gronwall's inequality holds?

Let $I=[0,b)$, $b< \infty$. Suppose $u$ is a positive bounded measurable function on $I$. $v(s)$ is a positive, smooth function on $I$. Note that $u(b),v(b)$ may be $0$.
Suppose that
$$
u(t) ...

**0**

votes

**0**answers

22 views

### Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only):
Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then
$$
\left| \int_{\mathbb{R}^3} f - ...

**-1**

votes

**0**answers

85 views

### Show: Real roots of a polynomial [on hold]

I´ve got some problems with this task I found in a german script.
Be $\ N \in \mathbb{N}$. Define matrix $\ \Pi := (\pi)_{0\leq i,j \leq N}$ with
\begin{equation}
\pi_{i,j} := \binom{N}{j} p_i^j ...

**0**

votes

**0**answers

110 views

### Heat asymptotics

Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...

**1**

vote

**1**answer

36 views

### Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem.
Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...

**1**

vote

**1**answer

68 views

### Is the residual spectrum of every power bounded operator contained in the open unit disk?

$\newcommand{\cH}{\mathcal{H}}
\newcommand{\CC}{\mathbb{C}}$
Let $\cH$ be a Hilbert space. A linear operator $T: \cH \to \cH$ is said to be power bounded if $\sup_{n \geq 0} \|T^n\| < \infty$.
If ...

**2**

votes

**0**answers

73 views

### Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = ...

**2**

votes

**0**answers

52 views

### Poincare inequality for the measure of Brownian path

I am wondering if the Poincare inequality holds for the Brownian path space.
As the simplest example, let $\{w_t, t \in [0, 1]\}$ be a 1-d standard BM: has independent increments and continuous ...

**1**

vote

**0**answers

76 views

### Fixed point theorem in ordered spaces

Can someone provide a proof or a source containing a proof of the following theorem
Theorem: Let $D$ be a subset of the cone $K$ of partially
ordered space $E,$ $F:D\rightarrow E$ be nondecreasing. ...

**2**

votes

**2**answers

128 views

### constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...

**4**

votes

**1**answer

149 views

### Open problems in Banach spaces, universality

I have gathered a list of universality problems in Banach spaces which have been solved:
1.The non existence of a separable reflexive space universal for the class of separable reflexive spaces.
...

**1**

vote

**0**answers

54 views

### Energy inequalities for Sobolev spaces of negative integer

I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| ...

**4**

votes

**1**answer

159 views

### Does nuclearity pass to un-tensoring?

Let $A$ be a C*-algebra such that $A \otimes_{\min} A$ is nuclear.
Does it follow that $A$ is nuclear?

**1**

vote

**1**answer

69 views

### Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...

**0**

votes

**0**answers

74 views

### Boundedness of heat semigroup on $L^1(\Omega)$

On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...

**-5**

votes

**0**answers

54 views

### Is this set compact? [closed]

Let A denote the set of $m\times n$ matrices whose entries bounded by $1$ in absolute one. Is the set $A$ a compact set in the smooth function space $C^1(R^n,R^m)$ with $C^1$ topology?

**2**

votes

**1**answer

169 views

### Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$.
Let $\epsilon\in[\frac{1}2,1)$.
Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ ...

**0**

votes

**1**answer

80 views

### Question about measure lemma?

"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ ...

**-5**

votes

**0**answers

65 views

### limit exercise which requires ingenuity1 [closed]

Hello im an eleventh grader in the best mathematics high school in my country. I wanted to see if any of you guys can help me solve a limit without integrals and L'Hospital, as i havent learnt them ...

**4**

votes

**0**answers

102 views

### SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, ...

**0**

votes

**0**answers

24 views

### Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...

**0**

votes

**0**answers

69 views

### the inverse of Trace theorem when $p = 2$

I can see there is an answer the inverse for the trace theorem and Image of the trace operator
My question is that
given $f\in H^{1/2}(\partial\Omega)$, is it possible to extended it into $\Omega$ ...

**1**

vote

**0**answers

106 views

### How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
For which $G$ can the ...

**5**

votes

**0**answers

66 views

### Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...

**0**

votes

**0**answers

59 views

### About equivalence of two fractional Sobolev/Hilbert spaces

Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space
$$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac ...

**4**

votes

**1**answer

163 views

### Multivariable function analysis

Sorry if the terms I'm going to use is not professional enough:) This is about the complexity analysis of an algorithm.
Let $\alpha$ be the greatest real root of the polynomial ...

**4**

votes

**0**answers

100 views

### Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...

**2**

votes

**2**answers

260 views

### Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions

Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...

**-2**

votes

**2**answers

139 views

### On Bohr-MollerupTheorem [closed]

In http://mathworld.wolfram.com/Bohr-MollerupTheorem.html, Bohr-Mollerup Theorem is given where it is stated that $\Gamma$ function is the unique log convex function that satisfies ...

**1**

vote

**2**answers

202 views

### A question involving Mazur's Lemma

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."):
"Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...

**1**

vote

**1**answer

144 views

### compact inclusion of domains of unbounded operators

Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold.
Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset ...

**3**

votes

**0**answers

54 views

### Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$

Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$
Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write
...

**2**

votes

**1**answer

135 views

### How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...

**3**

votes

**2**answers

235 views

### Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
This question is strongly linked to
is the space of all borel measures on ...

**0**

votes

**0**answers

18 views

### Weak continuous convergence of operators [migrated]

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$.
Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$.
Let $x_n \rightharpoonup x$ in ...

**11**

votes

**2**answers

258 views

### A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...

**3**

votes

**1**answer

108 views

### Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...

**4**

votes

**1**answer

67 views

### Does this infinite sum arising from separation of variables converge?

This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.
Let $a_k >0$ be an increasing sequence ...

**1**

vote

**0**answers

35 views

### Reference request: Topological space of polygonal chains and its properties [migrated]

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains:
image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License
A polygonal chain can be ...

**1**

vote

**2**answers

109 views

### Sobolev trace map: is the fractional seminorm bounded by just the gradient?

Let $M$ be a compact Riemann manifold. Consider the trace map $T:H^1(M) \to H^{\frac 12}(\partial M)$. Is it always the case that
$$|Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u ...

**3**

votes

**1**answer

127 views

### Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$

(This is a repost of a question from math.SE, http://math.stackexchange.com/questions/1240966/existence-of-state-on-a-c-algebra-satisfying-tauab-ab)
Let $a,b$ be elements of a unital C*-algebra $A$ ...

**4**

votes

**2**answers

148 views

### Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and
g is radially symmetric,
the function $ (0, \infty )\ni t \mapsto g ...