Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
8,572
questions
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 ...
- 173
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
$$\text{(i)}\quad \text{supp}\...
- 411
1
vote
0
answers
53
views
Banach spaces in which every DP-set is a limited set
Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset.
$A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \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^...
- 1,269
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.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 43
4
votes
2
answers
123
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 4,144
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]:
$$||\mu||_0:= \...
- 135
0
votes
0
answers
94
views
When is the image of $T \colon \ell^2 \to \ell^2$ a Gaussian random variable?
In finite dimensions, if $T$ is a linear operator and $x$ is a (centered) Gaussian random variable, then $Tx$ is again a (centered) Gaussian random variable.
Now suppose that $x$ is a (say, centered) ...
- 253
2
votes
0
answers
66
views
Generalizations of elliptic chain complexes
I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....
- 123
1
vote
1
answer
33
views
Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
- 180
1
vote
1
answer
95
views
Spectral theorem for unital $C^{*}$-algebras
Let $A$ be a unital $C^{*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{*}(a)$ be the $C^{*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space ...
- 973
1
vote
1
answer
56
views
The eigenvalues of the product $WD$ for some particular matrices
Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
- 3,583
0
votes
1
answer
138
views
Looking for English version of a paper of Jean Ginibre
I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...
- 69
11
votes
2
answers
434
views
Existence of an open convex set
Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $...
- 135
1
vote
0
answers
140
views
Weakly null sequences in projective tensor products
First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009.
Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
- 1,518
0
votes
0
answers
59
views
Equicontinuity and estimating the modulus of continuity for linear operators on the Schwartz space
Let us consider a collection of continuous linear mappings on the Schwartz space $T_\alpha : \mathcal{S}(\mathbb{R^2}) \to \mathcal{S}(\mathbb{R^2})$ defined by
\begin{equation}
[T_\alpha(\phi)](x,y):=...
- 1,269
0
votes
0
answers
54
views
Two questions about the vector-valued Lipschitz algebra
For a commutative Banach algebra $A$ and for any $0<\alpha<1$, let $\text{Lip}_\alpha(K,A)$ consist of all $A$-valued functions $f$ on a metric space $(K,\text d)$ with the property that $\rho_\...
- 2,078
1
vote
0
answers
106
views
Weak convergence using tensor product
I haven't got to see this argument used in the PhD thesis of [R. Ryan]: Applications of topological tensor products to infinite dimensional holomorphy, doctoral thesis, Trinity College, Dublin (1980), ...
- 103
6
votes
2
answers
343
views
Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$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:...
- 109
-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 ...
- 263
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 $(X, d)$ ...
- 67
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.
$$...
- 3,583
-1
votes
0
answers
28
views
How to prove the convergence of the functions with the norm above the critical point of Kondrakov?
I have a sequence of smooth functions with norm 1 in $L_q$ space. I need to prove this sequence strongly converges to some function. But I lack the compactness theorem since, $q$ is above the critical ...
- 1
3
votes
0
answers
210
views
Dunford-Pettis like properties for Banach spaces of operators
Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$.
Suppose $A$ satisfies the property (RCC) given below:
$$
\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 ...
- 13
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” ...
- 165
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 ...
- 756
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 ...
- 129
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:
$$
\frac{\partial}{\partial\mathbf{x}^T}\int_\...
- 39
0
votes
1
answer
144
views
When some Fourier coefficients are fixed, can we control the extremals of the function?
Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f
:\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?
1- $-n\lneqq f_{\min}$ (where $f_{\...
- 3,583
0
votes
1
answer
51
views
Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$
A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
(1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
(2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
(3) $\phi$ is ...
- 6,216
3
votes
1
answer
349
views
Boyd & Chua 1985: Is the proof of Lemma 2 correct?
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading this article by Boyd and Chua [1], in which they prove the approximability of arbitrary time-invariant (TI) operators ...
- 33
2
votes
0
answers
49
views
A division of real analytic functions
Problem statement
Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$.
Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...
- 559
2
votes
1
answer
87
views
Schauder bases in Banach spaces with a symmetric $k$-FDD
The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into ...
- 3,932
0
votes
0
answers
90
views
Solving a functional equation involving the exponential generating function of the Stirling numbers of the first kind
Let $F(x)=\sum_{n\geq 2m-1} f_n \frac{x^n}{n!}$, and $F_o=\sum_{n\geq m} f_{2n-1} \frac{x^{2n-1}}{(2n-1)!}$ for $m\geq 1$. Suppose
$$
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 $h \in H$...
- 429
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?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 3,583
1
vote
0
answers
48
views
The norm of Sobolev space involving the time
Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help.
Let's assume we have a function
$$
\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 ...
- 2,078
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 ...
- 323
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,
$$S = -\frac{1}{2\pi i} ...
- 407
1
vote
0
answers
121
views
What pre knowledge does Mumford's Tata collections on theta need?
I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...
- 11
2
votes
0
answers
39
views
When can convolutional integral operators be sampled
Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form
$$
f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt,
$$
for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...
- 41
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 ...
- 405