Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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3
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51 views

Steklov averages in PDE: what to do when we have time-dependent elliptic operator

One may have an equation (with boundary conditions omitted below) $$u_t - Au = f$$ $$u(0)=u_0$$ which has a weak solution $u \in L^2(0,T;V) \cap C([0,T];H)$ in the sense that $$-\int_0^T \int_\Omega ...
4
votes
1answer
137 views

Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections ...
2
votes
0answers
47 views

Power-spectrum of quasi-periodic functions

From Scholarpedia: Quasiperiodic oscillation is an oscillation that can be described by a quasiperiodic function, i.e., a function $F$ of real variable $t$ such that ...
0
votes
1answer
113 views

Convex cones: strict separation

Consider two closed convex cones $A$ and $B$ in $\mathbb{R}^3$. Assume that they are convex even without zero vector, i.e. $A \setminus \{0\}$ and $B \setminus \{0\}$ are also convex (it helps to ...
1
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0answers
92 views

A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
11
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1answer
256 views

Are free positive operators equivalent to almost-commuting operators?

Set $A:=C_0((0,1]) * C_0((0,1])$ (the free product C*-algebra), with canonical generators $a,b$ (positive contractions). Does there exists some $\gamma>0$ such that, for any $x,y \in A$ if $x^*x=a$ ...
5
votes
0answers
129 views

Is there a decomposition strengthening of the Sauer-Shelah Lemma?

Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates ...
9
votes
3answers
531 views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? ...
5
votes
1answer
190 views

Power's Theorem for irreducible representations

Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power ...
2
votes
0answers
70 views

Constructing an additive set function from on a non-additive one

repost from math.se. I was trying to generalize some results from measure theory to functions that are "almost" measures but not additive. Then, I thought it could be interesting to do this in a ...
3
votes
1answer
281 views

“bornophagic” in cited references

Could anybody come up with a cited reference for the following concept? A subset $B$ of a topological vector space $X$ is called "bornophagic" if, for every bounded $A\subset X$, there exists ...
3
votes
1answer
160 views

Norm on space of metrics

I recently heard a differential geometry talk where the speaker constructed a one-parameter family of metrics $g(t)$ on a smooth manifold and said that $g(t)$ is real analytic in the Banach space ...
7
votes
1answer
217 views

Can we recover a topological space from the collection of Borel probability measures living on it?

Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
4
votes
1answer
68 views

Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, ...
3
votes
0answers
59 views

Semi-continuity of the dimension of the null space

Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...
9
votes
2answers
243 views

Traces of operators in nuclear spaces

I am currently reading up on nuclear spaces in Yarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states: Let $F$ be a nuclear ...
7
votes
1answer
438 views

Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
7
votes
2answers
249 views

Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
10
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1answer
181 views

Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm. If $X$ is compact, Atiyah-Jänich proved that ...
3
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0answers
75 views

How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?

For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
0
votes
0answers
135 views

Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
4
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0answers
181 views

Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...
0
votes
1answer
65 views

Norm of derivative of rank one projector

I asked this question on math.stack but I got no answer, so I try here. Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation} ...
6
votes
1answer
130 views

Existence of a measurable map between metric spaces

Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto. Is ...
2
votes
1answer
93 views

Semi-simple Banach algebra

Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?
2
votes
1answer
99 views

Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$. Here, $F$ denotes the ...
1
vote
1answer
119 views

An operator factoring through a Banach space containing no copy of $l_{1}$

Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_{1}$, but does not factor through any Banach space $W$ containg no copy of ...
7
votes
1answer
257 views

What is $\hat{A}=\{[\pi]:\pi$ is a irreducible representation of $A$} ( $A$ is a $C^*$-algebra)?

Let $A=\{f:[0,1]\to M_2(\mathbb{C}): $f continuous and $ f(0)=\begin{pmatrix} f_{11}(0) & 0 \\ 0 & f_{22}(0) \end{pmatrix}\}$ be a $C^*$-algebra with pointwise multiplication, involutions and ...
2
votes
0answers
74 views

Green's functions on linear subspaces and relations to boundary conditions

Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...
7
votes
1answer
210 views

Derive an orthonormal system by Riesz basis $\{g(\cdot-\lambda_k),\ \lambda_k\in\mathbb R, \ k\in\mathbb Z\}$

Let $\{g(\cdot-k),k\in\mathbb Z\}$ be a Riesz basis, and let $\varphi\in L^2(\mathbb R)$ be a function defined by its Fourier transform $$\hat{\varphi}(\xi)=\frac{\hat{g}(\xi)}{\Gamma(\xi)},$$ where ...
11
votes
2answers
213 views

Existence of closed operators with arbitrary dense domain of a given banach space

Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., ...
8
votes
1answer
285 views

Why should the map $-\Delta^{-1}$ be continuous?

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$ in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
8
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0answers
211 views

What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) ...
9
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1answer
102 views

Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be the corresponding Cameron-Martin Hilbert space (also known as ...
3
votes
1answer
129 views

A family of convex bodies in Banach-Mazur position

Let $\{K_i\}$ be a family of smooth, origin-symmetric, strictly convex bodies such that $K_i$ converge in the Hausdorff distance (or you may assume $\partial K_i\to \partial K$ smoothly, in the sense ...
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0answers
51 views

Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions: Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
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0answers
66 views

When is an injective Fredholm map on $\ell^p$ a diffeomorphism

Consider a family of maps $f_p\colon \ell^p(\mathbb Z,\mathbb R)\to \ell^p(\mathbb Z,\mathbb R)$, $p\ge 2$, where $$ f_q = f_p\big|_{\ell^q},\qquad \forall\; 2\le q\le p. $$ Moreover, $f_p\colon ...
1
vote
1answer
188 views

does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$. I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...
5
votes
0answers
92 views

Convergence of convex combinations in topological vector spaces

I am studying certain quadratic forms on $L^0(m)$ equipped with the topology of (local) convergence in measure which in general is not locally convex. I am also interested in the situation where $m$ ...
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0answers
111 views

The dual of the space of smooth functions that vanish at infinity

Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...
1
vote
0answers
84 views

Characterisation of Sobolev spaces using eigenvalues of laplacian

I'm trying to find a reference on a result I thought it was obvious, but since I can't find anything I'm starting to doubt... I'm looking for a characterisation of the Sobolev space $H^s(\Omega)$ ...
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0answers
66 views

Fréchet differentiability of functional defined by a integral [closed]

I want to prove that if the functional $I: \mathcal{C}^1[t_0,t_f] \rightarrow \mathbb{R}$ defined by $$ I(x) = \int_{t_0}^{t_f} F(x, \dot{x},t)\,dt $$ is Fréchet differentiable if $F$ is ...
5
votes
3answers
268 views

A space of distributions vanishing on the boundary

The revised question After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let $$\mathcal ...
1
vote
0answers
55 views

In which sense Daubechies wavelets converge to the Shannon wavelet?

My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in ...
3
votes
2answers
163 views

Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as $$ L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}), $$ where $$ L^2_k (\mathbb{R}^2; ...
2
votes
1answer
68 views

interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
0
votes
1answer
110 views

Unbounded operator [closed]

Is the Laplacian operator $\Delta: C^2([a,b],\mathbb{R}) \to L^2(a,b)$ unbounded? Here $a,b \in\mathbb{R}$, $C^2([a,b],\mathbb{R})$ associated with $L^2$ norm. If yes, how can we "modify" these ...
6
votes
1answer
161 views

Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing?

A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup_{x^{*}\in B_{X^{*}}}\Bigl(\sum_{n=m}^{\infty}\lvert\langle ...
1
vote
0answers
40 views

How to define Biharmonic operator for second order sobolev spaces

I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have $$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : ...
4
votes
1answer
126 views

Mixed norm estimate for the heat equation

Consider the inhomogeneous linear heat equation $$\partial_tu-\Delta u=F$$ on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice ...