Questions tagged [fa.functional-analysis]

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

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5
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124 views

Second dual $X^{**}$ of ternary $C^*$-ring $X$ is again ternary $C^*$-ring?

Recall that a ternary $C^*$-ring is a complex Banach space $X$, equipped with a associative ternary product $[.,.,.]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle ...
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273 views

Hereditarily primary Banach spaces

A Banach space $X$ is said to be prime if every infinite dimensional complemented subspace is isomorphic to the space $X$. The space $X$ is primary if it has an infinite dimensional subspace $Y$ such ...
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2answers
227 views

How to prove the second Korn inequality?

$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\ \begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\|_{...
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1answer
123 views

limit of Riemann-Stieltjes sums as an integral on $\mathscr{H}$

I was reading Leon Takhtajan's book on quantum mechanics and, at some point, he states the J. von Neumann Theorem. The first part of this theorem is as follows. For every self-adjoint operator $A$ on ...
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24 views

Rabinowitz global bifurcation

Do you have any (modern) references to recommend to learn the proof of the so called Rabinowitz Global Bifurcation Theorem? The original paper is not self-contained and I don’t find it very ...
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2answers
239 views

Surjective linear isometries on $\ell_\infty(\mathbb{N})$

In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
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101 views

$L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs: $$u_t - \Delta u = f$$ $$\partial_\nu u = 0$$ $$u|_{t=0} = u_0$$ where $f \in L^p(0,T;L^r(\Omega))$ and $...
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1answer
256 views

Integral on level sets

Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
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141 views

Relation between the spectrum of 𝐷𝐴 and 𝐴 where 𝐴 is a bounded linear self-adjoint positive operator and 𝐷 is a constant diagonal positive matrix

Let $D$ be a $3\times 3 $ constant real positive-definite diagonal matrix and $\Omega\subset\mathbb{R}^3$ be a bounded lipschitz domain. Denote by $(L^2(\Omega))^3$ the set of square integrable ...
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24 views

Homogeneity of coupling of 0-1 stochastic process

We use $2^n$ to denote the set of binary strings of length $n$; $2^{\leq n}$ to denote binary strings of length smaller than or equal to $n$; $\emptyset$ to denote empty string; $|\rho|$ to denote ...
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41 views

Are there references to functional variants of the Unbounded Knapsack Problem?

Looking for a version of the following problem, extended to solutions in $\ell^{\infty}(\mathbb{N})$ Unbounded Knapsack Problem $ \max_{x_1,...,x_n} \sum_{i=1}^n v_ix_i$ $\text{ subject to }$ $\sum_{...
4
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1answer
147 views

Haar integral of rational function of unitaries

I'm trying to compute the following Haar integral over the unitary group: $$ \int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU. $$ Is there anything known about the ...
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191 views

Is the derivative the unique operation on points in the plane that preserves convexity?

Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$ Such that the ...
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1answer
177 views

When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?

I am studying properties of the two-parameter Mittag-Leffler function. $$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$ I am particularly interested in recurrences and ...
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1answer
151 views

About uniform continuity

Is there a definition Df(g) of uniform continuity of g, without using the notion of metric? Let $(E,d_E)$ and $(F, d_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$ We must have : Df$(f)$ ...
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1answer
179 views

Asymptotic cone

Let $S$ be a subset in a real vector space $\mathbb{R}^n$. Define the asymptotic cone $S\infty:=\{y\in\mathbb{R}^n\mid\textrm{there exists a sequence }(y_k,\varepsilon_k)\in S\times\mathbb{R}^+\textrm{...
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1answer
196 views

Equivariant implicit function theorem

Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). ...
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1answer
173 views

Probabilistic interpretation of derivative of a Dirac delta function

Consider $g : \mathbb{R}^d \mapsto \mathbb{R}$ defines some surface $\Sigma$ in $\mathbb{R}^d$. Then I can define a random variable $X_1$ with support only on $\Sigma$ by using a pdf of the form $$p_1(...
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1answer
115 views

Abelian twisted reduced group C*-algebra

Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
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92 views

Quasidiagonal C*-algebras

Let $A$ be a nuclear $C^*$-algebra satisfying UCT condition. Then under what assumptions $A$ is quasidiagonal?
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1answer
97 views

Application of Green function for non linear PDE [closed]

In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$. Is the same thing hold for ...
3
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1answer
105 views

Direct sum of multiplier algebras

Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\bigoplus_{i \in I}^{c_0} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \lim_{i \in I} \|a_i\| = 0\...
1
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1answer
121 views

Invariant distributions for iterated random variables (stochastic dynamical systems)

This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...
3
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72 views

Complex Interpolation between Schatten Classes

Let $H$ be a complex separable Hilbert space, and let $S^p = S^p(H)$ be a Schatten-$p$ class. We denote complex interpolation between two Banach spaces $X, Y$ by $[X, Y]_\theta$. Then, is it true that ...
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90 views

Extending Ky Fan's eigenvalues inequality to kernel operators

--Migrating from MSE since it might fit better here-- Base result The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as: $$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
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0answers
58 views

Regularity of solution to first order time dependent variational problem

Consider the following first order evolution problem over some regular bounded domain $\Omega\subset\Bbb R^d$ $$\frac{\partial\phi}{\partial t}(\mathbf{x},t) +\vec V(\mathbf{x},t)\cdot \nabla\phi(\...
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1answer
103 views

Conditions that make a dual space strictly convex

We know that a normed space $X$ is said to be strictly convex if $\Vert tx+(1-t)y\Vert<1$, $\forall t\in(0,1)$, $\forall x,y\in X$ with $\Vert x\Vert=\Vert y\Vert=1$ and $x\neq y$. For example if $...
3
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2answers
320 views

On the operator $f\to xf'/f$

I'm interested in the following operator $T$, close relative of the standard logarithmic derivative: $$f(x)\to Tf(x)=\frac{\text{d}(\log {f})}{\text{d}(\log {x})}=\frac{xf'}{f},$$ where $f$ is an ...
3
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0answers
65 views

Algebra properties regarding Gevrey spaces: closed under multiplication

In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
3
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1answer
114 views

Must a Schauder basis for $W^{1,p}_0(\Omega)$ be oscillatory?

Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may ...
4
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0answers
96 views

Delta distributions that are smooth on strata of a singular manifold

This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\...
2
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1answer
104 views

A "uniform continuity" type condition on a Hammerstein integral equation

I asked the following question on MathStackExchange, but I have not received the answer that I'm looking for. Although it may not be a research-level question, I thought I could ask it here. I'm ...
1
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1answer
72 views

Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
3
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1answer
103 views

Contractivity of Neumann Laplacean

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on Semigroups of Linear operators I found on many places properties of the Neumann Laplacean. In W....
4
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0answers
87 views

What are the "local degrees of freedom" in the space of smooth functions?

Let $C^k$ be the set of $k$th-order smooth real functions $f:\mathbb{R}\to\mathbb{R}$, and $C^\infty$ the set of smooth real functions. One can specify an $f\in C^k$ by specifying all its derivatives ...
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1answer
104 views

Range of discrete Fourier transform for binary strings

$\DeclareMathOperator\C{\mathbb{C}}$Let X be the set of all $n$-bit binary strings, $x=(x_1,\ldots x_n)$ where where $x_i\in\{-1,1\}$. Now consider the discrete fourier transform $F$, which maps ...
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0answers
55 views

Empirical estimation of Brenier map from data

Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
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67 views

Perturbed Gram matrix

Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix $$\sum_{t=1}^T(x_t ...
2
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0answers
83 views

From some priori estimates can we estimate higher Sobolev norm?

Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that $$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$ where $u|_{\partial\Omega}=\phi$. Can we ...
5
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0answers
107 views

Commutator of pseudodifferential operator and multiplication operator

Cross-post from math.sx. Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the ...
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0answers
58 views

Regularity results for non uniform elliptic equation

I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance, $$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
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0answers
162 views

Complete set of orthonormal functions on $W^{2,2}([0,1]^2, \mathbb{R}^2)$

Consider $L^2([0,1],\mathbb{R})$. Then, $$1, \sqrt{2} \cos(2 \pi j x), \sqrt{2} \sin(2 \pi j x ), \quad j =1,2,\ldots$$ is a Schauder basis on $L^2([0,1], \mathbb{R})$. I am curious, how does this ...
0
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0answers
31 views

Existence of solutions to semi-linear equations over semi-infinite strip

Let $\Omega\subset\mathbb{R}^2$ be a semi-infinite strip defined as $\Omega=I\times [a,\infty)$. Let $L$ be an elliptic differential operator and $f\in C(\mathbb{R})$ (with appropriate growth rate to ...
14
votes
1answer
711 views

Spectrum of matrix involving quantum harmonic oscillator

The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator $$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$ Fix two numbers $\alpha,\beta \...
2
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0answers
133 views

On weak Hahn-Banach smoothness

Let us recall Phelp's property-$U$: A subspace $Y\subset X$ is said to have property-$U$ if every $y^*\in Y^*$ has unique norm preserving extension over $X$. $Y$ is weak Hahn-Banach smooth if $y^*$ ...
3
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0answers
88 views

A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$

Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball. Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta ...
10
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1answer
191 views

Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)

Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...
12
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0answers
167 views

UMD constant of finite dimensional spaces

For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
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0answers
43 views

What is the relationship between barycenters in the Arens-Eells sense and barycenters in the optimal transport sense

Setup: Let $X$ be a complete pointed metric space. Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{...
2
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0answers
73 views

Dimension of critical set of p-harmonic function

Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$. Question: What is the Hausdorff dimension of the critical ...

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