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

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3
votes
1answer
119 views

$\nu$ is a Dirac delta

Let $X$ be an locally compact Hausdorff space and $m$ a positive regular Borel probability measure where $m(Y)$ is 0 or 1 for any Borel set of $X$. Does it necessarily follow that $m$ is a Dirac delta?...
4
votes
1answer
166 views

Density argument with Schwartz functions?

I was wondering whether the Schwartz functions are also dense in $$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$ ...
0
votes
2answers
98 views

Strictly convex norm on an infinite-dimensional Hilbert space

Question: Consider the Hilbert space $ H=\ell^2(\mathbb{Z})$. Let ${\rm L}(H)$ be the set of all linear operators on $H$ onto itself. Find a norm $N$ and a domain $DN\subset {\rm L}(H)$ for $N$ ...
7
votes
1answer
273 views

Level sets of weakly differentiable funtions

Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose $$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$ where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
0
votes
0answers
68 views

compact injection

Put: $D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$ Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
3
votes
1answer
102 views

On the complemented subspaces of $L_{p}(p>2)$

M.I. Kadec and A. Pełczyński proved that if $E$ is a subspace of $L_{p}(p>2)$ isomorphic to $l_{2}$, then $E$ is complemented in $L_{p}$. My question is: Is there a constant $C_{p}$ depending only ...
0
votes
0answers
43 views

stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g. $ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$ $ \nabla \cdot u = 0 \; \text{in} \; \Omega$ $ u = 0 \; \text{on} ...
3
votes
1answer
149 views

Convergence in trace

Let $A$ and $B$ be two self-adjoint, positive definite Compact operators on a Hilbert space $\mathcal{H}$. Further, let $A$ be trace class. Define $C_n \equiv AB(\frac{I}{n} + BAB)^{-1}$. Does $\frac{...
3
votes
0answers
99 views

Ideals of $L^1(G)$

I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
0
votes
0answers
81 views

An integral form of sum $\sum_{n\geq 0} \frac{f^{(n)}(0) g^{(n)}(0)}{(n!)^2}$ for two real analytic at 0 functions? Fourier|Taylor series parallels

Thinking of parallels between Fourier series and Taylor series, you might find out that the integral $\frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle$ ...
4
votes
0answers
155 views

A metric on $Homeo([0,1])$

One can define a metric on the set $Homeo([0,1])$ by setting $dist(f,g) =$ measure of support of $f^{-1}g$, that is the measure of the set of points $x$ where $f(x)\ne g(x)$. Was this metric studied ...
2
votes
0answers
133 views

Dual space of vector fields with null divergence

Let us consider $\mathscr D(\mathbb R^3;\mathbb R^3)$ the space of smooth compactly supported vector fields in $\mathbb R^3$ and let us define $\widetilde{\mathscr D}=\mathscr D(\mathbb R^3;\mathbb R^...
1
vote
1answer
99 views

Basis for the Orlicz Space

Does the Orlicz space (https://en.wikipedia.org/wiki/Birnbaum-Orlicz_space) has unconditional Schauder basis? Can we find such a orthonormal basis like the Hermitian polynomials in $L^2$?
3
votes
1answer
92 views

busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras". Let $0\to B\to E\to A\to 0$ be a short exact ...
3
votes
0answers
50 views

Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
2
votes
1answer
71 views

Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$ Let $\phi: S \rightarrow B_2.$ Given that the ...
0
votes
0answers
103 views

Convergence of unitary products on a Hilbert space [migrated]

First: I'm sorry for the basic question--I can move it to Math SE if necessary... Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $\|\cdot\|$ be the ...
2
votes
1answer
97 views

Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...
2
votes
0answers
71 views

Reflexive subspaces of dual spaces

If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...
2
votes
0answers
96 views

Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...
1
vote
0answers
67 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V)$ denotes the set of all linear bounded endomorphisms with operator ...
4
votes
1answer
182 views

Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
3
votes
0answers
63 views

Strengthening of the local smoothing estimates for the free Laplacian

The classical local-smoothing estimates for the free Laplacian asserts that: $$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$ where $B\subset\mathbb{...
0
votes
0answers
80 views

Orthogonal Procrustes problem for sub-spaces?

By Orthogonal Procrustes problem I mean given matrix $A$ and $B$ finding a orthogonal matrix $R$ which most closely maps $A$ to $B$, this has a solution as shown in https://en.wikipedia.org/wiki/...
1
vote
0answers
55 views

Covering rough boundaries of closed sets in manifolds by charts

This question is a little vague, I'm afraid, because I'm not sure I expect there to be a complete answer; but there should be some sort of situations where it is possible. Consider a Riemannian ...
0
votes
0answers
60 views

Order of vanishing of Laplace's equation with potential

Consider the equation $-\Delta u + V u = 0$ with Dirichlet boundary conditions on the bounded domain $\Omega \subseteq \mathbb{R}^n$, where $V$ is a smooth potential. Let $V \leq 0$, and bounded on $\...
4
votes
1answer
112 views

Trivial intersection of kernels

This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one. If $X$ is a separable Banach space, can we find a basic ...
5
votes
2answers
228 views

$id:A\to A^{op}$ is completely positive iff $A$ is abelian

Let $A$ be a $C^*$-algebra and $A^{op}$ it's opposite $C^*$-algebra. Let $id:A\to A^{op}$ be the identity map. $id$ is positive. The claim is: $id$ is completely positive iff $A$ is abelian. I need ...
5
votes
2answers
147 views

Biorthogonal functionals

If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$. What about conversely? If ...
5
votes
0answers
124 views

Weak* continuity of positive parts, again

Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\...
1
vote
1answer
139 views

Does the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$?

I am struggling to know whether the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$. $fg$ has compact support but I can't figure out how I can try ...
9
votes
2answers
158 views

Weak* continuity of positive parts

I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see? Let $(f_\lambda)$ be a net in $l^\infty$ which converges weak* to $f \in l^\infty$. We do not ...
7
votes
1answer
259 views

Abstract result on partitions of unity?

A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...
1
vote
0answers
47 views

Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of $\...
4
votes
0answers
149 views

Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let $d\in\left\{2,3\right\}$ $\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
3
votes
0answers
59 views

On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$: \...
0
votes
0answers
144 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
2
votes
2answers
91 views

Behavior of orbits under small perturbations

Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange. Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...
4
votes
1answer
88 views

Does the Nash inequality hold on manifolds with Lipschitz boundary?

Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...
1
vote
1answer
163 views

analytic continuation argument

In "Pseudo-spectra, the harmonic oscillator and complex resonances" (login required), the author says Sections $2$ and $3$ of this paper concern the operator $Hf(x)=(-\frac{d^{2}}{dx^{2}}+cx^{2})...
7
votes
1answer
190 views

When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ? That is, given $p : ...
4
votes
0answers
125 views

Distributions and functions on the Jacquet module $C_c^\infty(X)_{H,\chi}$

Let $X$ be an $\ell$ space (in the sense of Bernstein-Zelevinski), $H$ be an $\ell$ group which acts on $X$ and $\chi$ be a character of $H$. Denote $C^\infty(X)^{H,\chi}$ the space of locally ...
0
votes
1answer
52 views

Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions? Or are there restrictions on the function space over which this can be defined? For example is the Rademacher complexity defined or has ...
0
votes
0answers
60 views

A point on the absolute value of a bounded linear functional.

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...
3
votes
0answers
75 views

Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
5
votes
1answer
95 views

Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\...
1
vote
0answers
68 views

Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
1
vote
1answer
105 views

Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...
3
votes
1answer
109 views

Abstract Wave Equation and Semigroups

If an operator $A$ on a Hilbert space $H$ generates a strongly continuous semigroup, does then the operator $B$ on $H \oplus H$ given by the matrix $$ B := \begin{pmatrix} 0 & \mathrm{id} \\ A &...
4
votes
1answer
128 views

Functional Calculus of closed operators

I learned that there is a holomorphic functional calculus for closed operators: If $T$ is a closed operator on a Hilbert space, and $f$ is a function that is holomorphic on some open subset $\Omega$ ...