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

learn more… | top users | synonyms (1)

1
vote
1answer
45 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 ...
4
votes
1answer
144 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 ...
1
vote
2answers
60 views

List of tensor product spaces with uniform crossnorms

Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products $H:=\bigotimes_{j=1}^n H^{(j)}$ and ...
8
votes
2answers
127 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 ...
0
votes
1answer
98 views

Convergence of unitary products on a Hilbert space

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 ...
-5
votes
0answers
51 views

Sine, Cosine and Tangent functions [on hold]

Is the input of a Sine, Cosine and Tangent function always an angle?
3
votes
0answers
33 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 ...
5
votes
0answers
64 views

Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only): Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then $$ \left| \int_{\mathbb{R}^3} f - ...
2
votes
1answer
62 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 ...
1
vote
0answers
37 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
51 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 ...
-1
votes
0answers
74 views

how in can to extend the adjoint

Let $T_{a},a\in C$ be a closed operator defined on $D$ subspace of $L^{2}(R)$ onto $L^{2}(R)$ $(T_{a}: D\rightarrow L^{2}(R) )$ with $D$ contains a Schwartz space $S$. ...
1
vote
0answers
63 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
0answers
122 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 ...
2
votes
0answers
85 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 ...
0
votes
0answers
108 views

Are induced Riemannian metrics weakly continuous?

Let $\Omega \subseteq \mathbb{R}^d$ a nice bounded domain. Let $F_n:\Omega \to \mathbb{R}^D$ be a sequence of smooth embeddings* in $W^{1,p}(\Omega,\mathbb{R}^D)$. Assume $F_n \rightharpoonup F$ in ...
7
votes
1answer
168 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 : ...
2
votes
0answers
178 views

Is this continuous linear map weakly compact?

Let $E$ and $F$ be Fréchet spaces, let $U$ be an open subset of $E$, and let ${\mathcal{H}}(U;F)$ be the spaces of holomorphic mappings from $U$ into $F$. Let $\tau_c$ denote the compact-open ...
5
votes
2answers
209 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 ...
0
votes
0answers
74 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 ...
2
votes
0answers
56 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 ...
0
votes
0answers
61 views

Generalized Poincaré Inequality on H1 proof

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
5
votes
2answers
129 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 ...
1
vote
0answers
45 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
102 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 ...
17
votes
6answers
4k views

What is an intuitive view of adjoints? (version 2: functional analysis)

After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations! Again, I can use 'em, ...
4
votes
0answers
126 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 ...
0
votes
0answers
32 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 ...
6
votes
0answers
112 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 ...
1
vote
1answer
130 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 ...
7
votes
1answer
247 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 ...
0
votes
0answers
129 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 ...
5
votes
0answers
153 views

Dual of $BV_0(\Omega)$

It is previously pointed out in Dual or pre-dual of BV that the dual of $BV_c(\Omega)$ (BV functions with essentially compact support in $\Omega$) are so called strong charges, i.e. distributions for ...
3
votes
0answers
63 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 ...
3
votes
0answers
54 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 $\partial U$: ...
2
votes
2answers
87 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 ...
1
vote
1answer
112 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 ...
0
votes
1answer
49 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 ...
2
votes
2answers
302 views

Elements of Minimal Norm on an affine subset of a Banach Space

Let $v_1$ and $v_2$ be two elements of a real Banach space $(X,\lVert\;\rVert)$, each of unit norm. Consider the one-dimensional affine subspace $v_1+tv_2$ for real $t$. Is there a general formula for ...
1
vote
0answers
62 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 ...
0
votes
0answers
54 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 ...
0
votes
1answer
255 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$ where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a ...
3
votes
0answers
72 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 ...
3
votes
0answers
93 views

An estimate for the maximal $C^*$-norm in the group algebra of a free group

Let $F\twoheadrightarrow G$ be an epimorphism of groups, $F$ being finitely generated and free. Let $H$ be its kernel. Consider a lifting $i:G\hookrightarrow F$ of the epimorphism. Every element of ...
3
votes
1answer
107 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 ...
5
votes
1answer
78 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
1answer
102 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 ...
4
votes
1answer
122 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 $\Omega$ ...
7
votes
1answer
227 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...