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

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5
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1answer
205 views

Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map. Riesz-Thorin gives us that ...
3
votes
0answers
51 views

Isometry from a representation to the representation tensored with itself

Suppose, the group $ G=S(2^{\infty})$ has a unitary representation $ \pi $ on a separable infinite dimensional Hilbert space $ H $. (The group $ S(2^{\infty}) $ is the direct limit of the following ...
0
votes
0answers
33 views

Rademacher function and weakly p-summable sequence

The function $ r_{n}:[0,1]\rightarrow \lbrace -1,1\rbrace $ be defined by $ r_{n}(t):=sgn(sin(2^{n}\pi t) $ is known as the n-th Rademacher function. is $ (r_{n}) $ weakly p-summable sequence in ...
0
votes
1answer
166 views

Formula for an integration on $\mathbb{Q} \cap [0,1]$

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting: $$ \...
3
votes
0answers
61 views

Eigenvalues of approximations to product-convolution operators

Consider an operator $T: L^2 \mapsto L^2$ of the form $TA = g (h \ast A)$ where $g$ is and $h$ are bounded $C^\infty$ functions. This operator $T$ can be shown to be Hilbert-Schmidt, hence compact. ...
5
votes
1answer
114 views

is the maximal tensor product of compact operators an essential ideal?

I'm searching for a counterexample for $C^*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $...
1
vote
1answer
76 views

Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
0
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0answers
51 views

Monotone functions in ordered Banach spaces

Let $(X,\preceq)$ be a real ordered infinite-dimensional Banach space and $f:X\to \mathbb{R}$ is a Fréchet differentiable function. It is said to be a monotone non decreasing map if $$ x\preceq y \...
3
votes
1answer
119 views

is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive?

First of all, sorry for my bad english. I tried to find out whether the following statement is true or not: Let $X$ be a operator system, $B$ a $C^*$-algebra and $f:X\to B$ linear such that $f(1)\ge ...
3
votes
2answers
287 views

A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again. Take the vector space of infinitely ...
5
votes
0answers
122 views

quasi-weakly compact operators, co-ideals of operator ideals, and Banach spaces $X$ with $X^{**}/X$ separable

Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator $\overline{T}\in\mathcal{L}(X^{**}/...
3
votes
1answer
175 views

On construction of a $\mathbb{Q}$ periodic function with Fourier series

Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series: $$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$ Using ...
2
votes
1answer
172 views

Selfadjointness of hamiltonian with 1/x potential

Let us consider the Hilbert space $L^2([0,\infty))$ and operator $H=-\frac{d^2}{dx^2} + \frac{1}{x}$ on the domain of $C^{\infty}_0((0,\infty))$ (smooth functions with compact support away from $0$). ...
3
votes
2answers
123 views

The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant

In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on: If $ \mathbb{H} $ is an RKHS and we denote the ...
1
vote
0answers
66 views

Hilbert Transform and multiplier in $\mathbb{C}(X)$

I found myself trying to solve an equation of that kind : $$ H f= R f, $$ where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles ...
1
vote
0answers
174 views

Can we find an operator that satisfies this equation? [closed]

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$ Is there a function $f$ from $R^{\infty}$...
1
vote
0answers
79 views

extension for a complex operator

Let be $\lambda>0$. Put $$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
0
votes
0answers
81 views

Continuity of solutions of nonlinear elliptic PDEs

Consider the nonlinear 2nd order elliptic PDE $$\sum_{i, j} a_{ij}(x, t) \partial_i\partial_j u + \sum_k b_k(x, t) \partial_k u + c u = F(u), \quad x \in \mathbb{R}^n, t \in [0, \infty).$$ Here $a_{ij}...
7
votes
2answers
212 views

Weak compactness in the James space and its dual

It is known that there are characterizations of weak compactness in most of classical non-reflexive spaces (e.g. $L_{1}$-spaces and $C(K)$-spaces). I wonder whether there are characterizations of weak ...
1
vote
1answer
111 views

The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
0
votes
1answer
67 views

Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates an analyitc ...
1
vote
0answers
47 views

Is the equicontinuous weak-star topology locally convex on the dual of an LF-space?

The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-* topology $ew^*$ on $X'$ coincides with the topology of precompact ...
2
votes
1answer
105 views

embeds in $ L(c_{0},\ell_{1}) $

Let $ c_{0}:=\lbrace x:\mathbb{N}\rightarrow \mathbb{R} :\lim_{j\rightarrow\infty} x_{j}=0 \rbrace$ denote the usual Banach sequence spaces. Given Banach spaces $X,Y$ let $L(X,Y)$ denote the Banach ...
1
vote
0answers
109 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
1
vote
1answer
190 views

When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space. Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
4
votes
3answers
375 views

What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$? I think it would be the permutations of the integers. Is this right?
5
votes
0answers
152 views

Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra. Suppose $\Omega$ is ...
4
votes
1answer
260 views

Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, non-...
2
votes
1answer
160 views

The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
3
votes
2answers
236 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
2
votes
0answers
105 views

Question about continuity in the “complete Skorohod Topology”?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process" https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf In one of the exercises, exercise 8.9 ...
2
votes
0answers
135 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras

Is there a name for the following property of a $C^{*}$ algebra $A$? $$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$ Example of this situation is $A=C(X)$ where $X$ is the ...
2
votes
1answer
93 views

Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? [closed]

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1,...
0
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0answers
49 views

approximating smooth functions by non-smooth ones, in the distribution topology

The classical Stone-Weierstrass theorem gives a necessary and sufficient condition for a class of continuous functions on a compact to approximate a larger class of continuous functions in $C^0$ ...
4
votes
1answer
87 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
9
votes
3answers
516 views

Can the topological algebra of analytic functions be endowed with a norm that defines the natural topology?

Right, so in my research in complex analysis I was puzzled by this question which may have a simple approachable answer that eludes me, but I am truly itching to find out and in need of it so I am ...
1
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0answers
40 views

The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
14
votes
1answer
350 views

Banach-Mazur distance between the cube and the octahedron

The Banach-Mazur distance $d(X, Y)$ between two normed spaces $X, Y$ of the same dimension is defined as $d(X, Y) = \log\inf \|T\| \cdot \|T^{-1}\|$, where the $T:X \to Y$ is a linear and invertible ...
1
vote
0answers
78 views

Lower bound for the $C^*$-unitisation norm?

Let $A$ be a $C^*$-algebra without unit. Its unitisation $A^+$ carries the $C^*$-norm $$\|(a,\lambda)\|_{A^+} = \sup \{\|ax+\lambda x\|_A \;\big|\; \|x\|_A = 1 \},$$ which is the operator norm of $...
1
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0answers
107 views

Regularity of a Dirichlet form

I have a question about Dirichlet form. Let $\Omega$ be an Euclidean domain of $\mathbb{R}^{N}$ and $X=\bar{\Omega}$. The measure $m$ on the Borel $\sigma$ algebra $\mathcal{B}(X)$ is given by $m(A)...
5
votes
1answer
180 views

Injectivity/Surjectivity of $F_A :=\frac{d}{dt} +A(t) $ for a hyperbolic path of matrices $A(t)$ on $H^1 $

I am looking for a reference to the following problem: Given a hyperbolic (no purely imaginary eigenvalues), continuous path of matrices $A(t)$ in $\mathbb{R}$ with hyperolic limits at $\pm \infty $. ...
4
votes
1answer
230 views

Weil's Haar measure construction from below

Weil's construction of a Haar measure on a locally compact group rests on approximating a function from above by sums of translates of another function. I would need to know something similar for an ...
0
votes
0answers
148 views

Johnson's Theorem - Proof (Runde) Clarification

I am reading Runde's Lectures on Amenability. In the proof of Johnson's theorem where he proves "$L^{1}(G)$ is amenable Banach algebra implies $G$ is amenable" : he defines a $L^1(G)$ bimodule action ...
1
vote
0answers
73 views

$C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$$...
5
votes
0answers
145 views

Constructing Extreme Points in Reflexive Banach Spaces

A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...
6
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1answer
181 views

Definitions of Hilbert Bundles

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
0
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0answers
50 views

Bounds on the the spherical harmonics on $S^{p-1}$

The only reference I could find in this regard is for upper bounding the n-homogeneous spherical harmonics on $S^{p-1}$ as in equation 4.29 here, http://www.fen.bilkent.edu.tr/~gurses/...
7
votes
2answers
295 views

“Identity tensor transpose” as a map $M_n \hat{\otimes} M_n \to M_n \overline{\otimes} M_n$

Equipping $M_n$ with its usual operator space structure, $\newcommand{\ptp}{\widehat{\otimes}}$ we can form the projective tensor product of operator spaces $M_n\ptp M_n$. In particular this puts a ...
3
votes
1answer
210 views

How to learn concepts of Functional Analysis which are common in PDE

I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...
1
vote
1answer
87 views

Various limits of the Christoffel Darboux Kernel

In a different thread, we stumbled upon the following question: Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...