Questions tagged [fa.functional-analysis]

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

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2 votes
0 answers
36 views

Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is, $[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$. Now I ...
1 vote
0 answers
31 views

A convergence problem in the space of tempered distributions

Let $K(x):=|x|^{-\alpha}$ be a function on $\mathbb{R}^{n}\setminus\{0\}$ with $0<\alpha<n$. Suppose $\phi$ is a $C^{\infty}_{c}(\mathbb{R}^n)$ function such that $$\text{(i)}\quad \text{supp}\...
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1 vote
0 answers
53 views

Banach spaces in which every DP-set is a limited set

Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset. $A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \lim_{n\to\...
  • 1,518
1 vote
1 answer
67 views

Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?

I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$. I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
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1 vote
1 answer
55 views

Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$. On $\mathcal{L}(E)^2$, we have two equivalent norms: \begin{eqnarray*} N_1(A_1,A_2) &=&\sup\...
  • 1,150
1 vote
0 answers
61 views

Maximal domain of an unbounded linear operator in a weighted Hilbert-space

Let's consider the following (unbounded) linear operator. (So called transport operator in some context.) $$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
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4 votes
2 answers
123 views

Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$ Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$. Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then $$ u(x)...
2 votes
1 answer
129 views

Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm

Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]: $$||\mu||_0:= \...
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0 votes
0 answers
94 views

When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?

In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable. Now suppose that $x$ is a (say, centered) ...
2 votes
0 answers
66 views

Generalizations of elliptic chain complexes

I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....
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1 vote
1 answer
33 views

Discrete uniqueness sets for the two-sided Laplace transform?

Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by $$ Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx. $$ A ...
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1 vote
1 answer
95 views

Spectral theorem for unital $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{*}(a)$ be the $C^{*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space ...
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1 vote
1 answer
56 views

The eigenvalues of the product $WD$ for some particular matrices

Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0 Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is $$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
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0 votes
1 answer
138 views

Looking for English version of a paper of Jean Ginibre

I am in serious need of an English translation for the following paper: Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace, d'après ...
11 votes
2 answers
434 views

Existence of an open convex set

Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$). Can we find an open set $...
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1 vote
0 answers
140 views

Weakly null sequences in projective tensor products

First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009. Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
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0 votes
0 answers
59 views

Equicontinuity and estimating the modulus of continuity for linear operators on the Schwartz space

Let us consider a collection of continuous linear mappings on the Schwartz space $T_\alpha : \mathcal{S}(\mathbb{R^2}) \to \mathcal{S}(\mathbb{R^2})$ defined by \begin{equation} [T_\alpha(\phi)](x,y):=...
  • 1,269
0 votes
0 answers
54 views

Two questions about the vector-valued Lipschitz algebra

For a commutative Banach algebra $A$ and for any $0<\alpha<1$, let $\text{Lip}_\alpha(K,A)$ consist of all $A$-valued functions $f$ on a metric space $(K,\text d)$ with the property that $\rho_\...
1 vote
0 answers
106 views

Weak convergence using tensor product

I haven't got to see this argument used in the PhD thesis of [R. Ryan]: Applications of topological tensor products to infinite dimensional holomorphy, doctoral thesis, Trinity College, Dublin (1980), ...
6 votes
2 answers
343 views

Spectrum of operator involving ladder operators

The ladder operator in quantum mechanics are the operators $$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$ and $$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
0 votes
1 answer
66 views

Minimal set of functions to characterize a distribution

In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
-3 votes
0 answers
71 views

Isometric embeddings of two separable Banach spaces

It is well-known that a separable Banach space is isometrically embedded in $l^\infty$. Consider $X\hookrightarrow Y$ two separable Banach spaces with a continuous embedding $i$. Therefore, there ...
1 vote
0 answers
35 views

Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures. Let $(X, d)$ ...
0 votes
1 answer
63 views

The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$

Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix. $$...
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-1 votes
0 answers
28 views

How to prove the convergence of the functions with the norm above the critical point of Kondrakov?

I have a sequence of smooth functions with norm 1 in $L_q$ space. I need to prove this sequence strongly converges to some function. But I lack the compactness theorem since, $q$ is above the critical ...
3 votes
0 answers
210 views

Dunford-Pettis like properties for Banach spaces of operators

Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$. Suppose $A$ satisfies the property (RCC) given below: $$ \left.\begin{array}{l} (x_n)\subseteq A \textrm{ ...
  • 1,518
0 votes
1 answer
131 views

What does "a universal tree" mean?

It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the ...
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3 votes
0 answers
86 views

Recovering the matrix when the Schur decomposition of its blocks are known

Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and $$E=\left(\begin{array}{cc} G & X \\ X^t & H \end{array}\right)$$ where $G,H,X$ are $m\times m$ matrices. Suppose that $...
  • 3,583
1 vote
0 answers
68 views

Relationship between Beilinson’s resolution of the diagonal and functional analysis

I have been wondering for long enough to embarrass myself on here by asking: is there a reason why Beilinson’s resolution of the diagonal “Coherent Sheaves on Pn and Some Problems of Linear Algebra” ...
5 votes
1 answer
144 views

States "absorbed" by a Haar idempotent on a compact quantum group

Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when $$a\bullet b=b=b\bullet a?$$ Can we say that $b$ absorbs $a$? Can we say ...
0 votes
0 answers
79 views

Finding set of best approximations from a point in $c_0$ to its subspace

Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
2 votes
0 answers
79 views

Reference for Schwartz kernel theorem on vector bundles

In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
1 vote
1 answer
183 views

Sufficient condition for two norms to be equal

Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$. On $\mathcal{L}(E)^2$, we have two equivalent norms: \begin{eqnarray*} N_1(A,B) &=&\sup\left\...
  • 1,150
2 votes
0 answers
99 views

How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]

I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
  • 183
0 votes
0 answers
75 views

Differentiation under the integral sign in higher dimensions [migrated]

Let's say I have a function $f(\mathbf{x},t)$, $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n$. Are there conditions under which the following holds: $$ \frac{\partial}{\partial\mathbf{x}^T}\int_\...
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0 votes
1 answer
144 views

When some Fourier coefficients are fixed, can we control the extremals of the function?

Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold? 1- $-n\lneqq f_{\min}$ (where $f_{\...
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0 votes
1 answer
51 views

Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$

A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that (1) $\phi(x) \ge 0$ for all $x \in \mathbb R$, (2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$, (3) $\phi$ is ...
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3 votes
1 answer
349 views

Boyd & Chua 1985: Is the proof of Lemma 2 correct?

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading this article by Boyd and Chua [1], in which they prove the approximability of arbitrary time-invariant (TI) operators ...
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2 votes
0 answers
49 views

A division of real analytic functions

Problem statement Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$. Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...
  • 559
2 votes
1 answer
87 views

Schauder bases in Banach spaces with a symmetric $k$-FDD

The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into ...
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0 votes
0 answers
90 views

Solving a functional equation involving the exponential generating function of the Stirling numbers of the first kind

Let $F(x)=\sum_{n\geq 2m-1} f_n \frac{x^n}{n!}$, and $F_o=\sum_{n\geq m} f_{2n-1} \frac{x^{2n-1}}{(2n-1)!}$ for $m\geq 1$. Suppose $$ F(x)+F(\frac{x}{1+x})=\frac{2[\ln (1+x)]^{2m}}{(2m)!}+\frac{2[\ln(...
-1 votes
0 answers
71 views

Intersection of Hilbert spaces with Schauder basis plus a fixed singleton

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$. Let $U \subset H$ be a Hilbert subspace of $H$. Let $\{u_n\}_{n \in \mathbb{N}}$ be a Schauder basis for $U$. Let $h \in H$...
3 votes
0 answers
152 views

The essential norm where some Fourier coefficients are fixed

Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$. Q. Let $\phi\in C_{2\pi}$. Is the following statement valid? $$\|\phi\|_2=\inf_{g\in C_{2\...
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1 vote
0 answers
48 views

The norm of Sobolev space involving the time

Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help. Let's assume we have a function $$ \mathbf{u} (\mathbf{x}; t) = \...
1 vote
0 answers
134 views

Could we characterize elements in the second dual by the character space?

Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...
9 votes
1 answer
532 views

Rigorous proof of the pentagon identity

I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra. For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...
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1 vote
1 answer
50 views

Show $A_0 + S$ is invertible, where $S$ is the Riesz projection of bounded $A_0$

Let $A_0$ be a bounded linear operator on a Hilbert space $H$. Suppose $0$ is an isolated point of the spectrum of $A_0$. Let $S$ be the corresponding Riesz projection, namely, $$S = -\frac{1}{2\pi i} ...
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1 vote
0 answers
121 views

What pre knowledge does Mumford's Tata collections on theta need?

I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...
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2 votes
0 answers
39 views

When can convolutional integral operators be sampled

Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form $$ f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt, $$ for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...
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1 vote
1 answer
189 views

Norm continuity of the predual of a von Neumann algebra

Let $M$ be a von Neumann algebra and let $(p_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M_{\ast} $, the predual of $M$. It is well known that $\| p_i f \|_{M_\ast}\to_{i} 0$ for ...
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