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

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106 views

A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology. Is there a continuous map ...
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56 views

Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain $$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$ $$u|_{\partial\Omega} = 0$$ where $a$ is real-valued and satisfies $C_1 \leq a(r) \leq C_2$ ...
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41 views

Searching for conditions?

I have this operator $$Au(t)=\int_0^1 G(t,s) f(s,u(s)) ds$$defined from $H^1_{0}$ to $H_0^1$ and satisfy the problem: $$\begin{cases} -(Au)''(t)=f(t,u(t)), t\in[0,1]\\Au(0)=Au(1)=0\end{cases}$$ Where ...
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1answer
102 views

Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that $$ ||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ? $$ where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
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85 views

Explicit formula for Bergman kernel on the unit ball

On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...
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1answer
132 views

Multiplication of generalized functions

I would like to know if there is any associative algebra $( A(\mathbb{R}), +, \cdot )$ such that: EDIT: Fan Zheng found an inconsistency in my requirements. Therefore I changed the function space in ...
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1answer
159 views

Asymptotic behaviour of eigenvalues

If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
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1answer
125 views

Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...
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91 views

Weak (Sobolev) derivative and the Frechet derivative (chain rule) [closed]

Let $A: H^s(\Omega) \to H^1(\Omega)$ be a bounded linear map. Let $u \in H^s(\Omega)$. Let $f:\mathbb{R} \to \mathbb{R}$ be nonlinear and Lipschitz such that $f(u) \in H^s(\Omega)$. Is it possible to ...
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2answers
214 views

Does this C*-algebra embed into a simple nuclear C*-algebra?

Let $\mathcal K$ denote the C*-algebra of compact operators, and fix an embedding $\phi_n:M_n(\mathbb C) \to \mathcal K$ for each $n\in \mathbb N$. Define the C*-algebra $A := \{(a_n)_{n=1}^\infty ...
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52 views

Fractional Poincare inequality on closed manifold

Let $u \in H^{\frac 12}(M)$ on a compact closed Riemannian manifold. Can someone refer me to a source where the inequality $$\lVert u - \bar u \rVert_{L^{2^*}} \leq C|u|_{H^{\frac 12}}$$ is proved, or ...
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34 views

Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
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80 views

Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space

Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map ...
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1answer
119 views

Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?

I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that $$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...
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40 views

Interpolation of Banach spaces: theta = 0, 1

Let $A_1 \subset A_0$ be Banach spaces with continuous embedding. Is $(A_0, A_1)_{i, \infty} = A_i$ for $i =0, 1$, with equivalent norms? Here, $(\cdot, \cdot)_{\theta,p}$ denotes the $K$-method of ...
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100 views

A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$

The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
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31 views

Topological tensor products of spaces of holomorphic functions of slow growth

Let $X$ be a Banach space, $M$ be a complex manifold, and $\Omega$ a relatively compact domain in $M$. We consider the space $\mathcal{A}^{-\infty}(\Omega, X)$ of $X$-valued holomorphic functions of ...
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53 views

Hilbert scales of covariance operators

Assume we have 2 covariance operators(positive definite trace class) $S$ and $T$ on Hilbert space $\mathcal H$ with corresponding eigenpairs $\{e_j,\lambda_j\}$ and $\{f_j,\lambda_j\}$. Assume that ...
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70 views

Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1. Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...
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121 views

Generalising the $L^2$-Norm to Compact Quantum Groups

For a compact quantum group (in the sense) of Woronowicz, we have a noncommutative $C^*$-algebra replacing ${\mathbb C}(G)$. This $C^*$-algebra is endowed with a linear functional generalizing the ...
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58 views

Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems? $$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...
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39 views

Causal (Volterra type) differential equation with local Lipschitz condition

Consider the equation $$ u'(t) = (Fu)(t) $$ where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type) nonlinear operator. It means that the value of $(Fu)(t_0)$ ...
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52 views

Hilbert c*-module over approximately finite c*-algebra

Is there a construction of a Hilbert c*-module over an approximately finite c*-algebra using Hilbert c*-modules over finite algebras? How do we get a Hilbert c*-module over an inductive limit of such ...
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1answer
110 views

Identifying the weak limit of a gradient (Bochner spaces)

Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f_n \to f$, $f_n(0)=f(0)=0$ and $f_n$ Lipschitz with Lipschitz constant ...
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79 views

How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?

We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$ By Helson-Kahane-Katznelson-Rudin Theorem, it follows that, "Let $F$ be a function on $\mathbb C$ and if ...
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92 views

Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two ...
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87 views

Normal points of an operator and discrete eigenvalues

Let $\mathcal{H}$ and $\mathcal{L}({\mathcal{H}})$ denote a separable Hilbert space and the set of bounded linear operators on it respectively. As a graduate student entering the field of ...
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68 views

Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?

The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question: Given ...
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1answer
130 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
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2answers
117 views

Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & ...
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82 views

Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} ...
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1answer
184 views

Is the space of Radon measures a Prohorov space?

Consider the spaces $C_c(\mathbb{R})$ of compactly supported continuous functions equipped with the inductive limit topology and the Banach space $C_0(\mathbb{R}) = \overline{C_c(\mathbb{R})}^{\, ...
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1answer
167 views

Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully). Suppose $g$ is holomorphic on ...
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207 views

On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$ with $s_0$ a zero of the Riemann Zeta function in the critical strip. This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
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107 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
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4answers
599 views

Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...
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1answer
195 views

Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality $$ |\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...
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1answer
446 views

Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?

It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...
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1answer
107 views

Approximation Property: Decomposition

This thread originated from MSE: Approximation Property: Decomposition Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional ...
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379 views

Homemorphism between $X$ and $\mathcal{P}(\mathcal{P}(X))$

Let $X$ be a topological space, $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$. Endow the latter with the weak* topology. I was wondering whether there exists a (nontrivial) ...
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57 views

On Properties Of Lusternik-Schnirelmann Category

I have this part of proof from "Analysis and Topology in Nonlinear Differential Equations" book page 292: I don't see how we find that $cat(\Omega)\leq cat( N_{\varepsilon}\cap ...
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72 views

Approximation Property: Characterization

Problem Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$. Suppose it has the approximation ...
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69 views

Compact Approximation

This thread originated from MSE: Compact Approximation This is meant as lemma for: Approximation Property Given a Banach space $E$. Denote compact domains by $\mathcal{C}$. Denote compact ...
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87 views

Convergence in distribution of random measures

Let $M$ denote the space of real Radon measures on $\mathbb{R}$ as the topological dual of $C_c(\mathbb{R})$ equipped with the inductive limit topology (for possibly unbounded Radon measures) or ...
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116 views

Harmonic extension of $L^\infty$ function is in $L^\infty$?

Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of $$-\Delta v(x,y) = 0$$ $$\partial_\nu v = 0$$ $$v(x,0) = ...
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89 views

Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
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1answer
286 views

Non-reflexive Banach space s.t. X,X*,X**,… are separable

Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?
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1answer
222 views

Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element? Let $n(A)$ be the infimum of such ...
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2answers
165 views

Quantum Field theory - integral notation

I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists. I'm a student of mathematics, and I understand the notion of a ...
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0answers
81 views

Spectrum of convolution operator

This question was asked already on Stack Exchange under http://math.stackexchange.com/q/1114095 . It might be not on a research level, but as it could not be answered on Stack Exchange, I hope for ...