All Questions
10,233 questions
6
votes
1
answer
175
views
Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions
Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
- 407
4
votes
1
answer
259
views
The real and the imaginary part of a vector
In an infinite-dimensional Banah space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x_n\}$, define:
$$
F_r: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} ...
- 307
3
votes
1
answer
428
views
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...
- 3,477
0
votes
1
answer
141
views
Infimum of norms of elements in a hyperplane
In a Banach space X, given a norm one bounded linear functional $f$ and $c\in \mathbb{C}\backslash \{0\}$, define $H = \big\{ x\in X \,\vert\, f(x) = c\big\}$ and $\inf H$ = $\inf_{h\in H} \|h\|$.
Is ...
- 307
2
votes
0
answers
103
views
Schwartz kernel theorem for restricted operators
Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
- 1,171
3
votes
1
answer
219
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
3
votes
2
answers
273
views
Representing measurable map to compact space as a continuous map
Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space
$$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(...
user103549
0
votes
1
answer
241
views
Norm functions induced by convex bodies
Given a centrally symmetric convex body $K$ in the plane (with smooth boundary), it is easy to see that there exists a norm function $g:\mathbb{R}^2\to \mathbb{R}_{\geq 0}$ for which $K$ is the unit ...
- 191
1
vote
1
answer
262
views
Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support
In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the
Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
-2
votes
1
answer
216
views
Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$?
If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that
\begin{...
- 3,477
2
votes
1
answer
171
views
Defining states on von Neumann algebras from filters on the projection lattices
Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for ...
- 2,830
4
votes
2
answers
904
views
Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?
Trying to find and answer to this question, I have encountered two more-studied problems.
The first is to find when a Banach space admits an equivalent uniformly convex norm. The answer is that for ...
- 1,955
3
votes
2
answers
147
views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 300
1
vote
0
answers
95
views
Are the sum and product of nonlinear compact operators compact?
In the 'interactive' proof of Lomonosov's Theorem about hyperinvariant subspaces (in the book Hilbert Space Operators, A Problem Solving Approach), one is asked to prove the compactness of the ...
- 11
1
vote
1
answer
180
views
Conditioning a $\mathrm{C}^*$-algebra state with infinite precision
This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success.
Let $\mathcal{A}$ be a unital $\...
- 1,037
4
votes
0
answers
119
views
Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
0
votes
0
answers
418
views
Spectral theorem for commuting operators
Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
- 3,535
1
vote
0
answers
164
views
Reference on spectral theory of self-adjoint operators
I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self ...
- 13
7
votes
0
answers
162
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
- 1,805
0
votes
2
answers
197
views
Convergence of the infima of convex functions on $\mathbb{R}^m$
Any thoughts on proving the following statement, which is a generalization of the result in convergence of the infima of convex functions from domain $\mathbb{R}$ to $\mathbb{R}^m$ and also Theorem 1 ...
6
votes
3
answers
551
views
Hahn-Banach Theorem for convex polytopes and their supporting hyperplanes
A polytope in $\mathbb R^n$ is the convex hull of a nonempty finite set in $\mathbb R^n$.
Let $C$ be a polytope in $\mathbb R^n$.
We shall say that a hyperplane $H\subseteq \mathbb R^n$
$\bullet$ ...
- 42k
1
vote
0
answers
32
views
Rätz orthogonality and involution
In the Rätz’s sens of orhtoganality, can we find an exemple of an involution u(different to -Id)such that x orthogonal to y then x orthogonal to u(y)
0
votes
1
answer
229
views
Norm equivalence in finite dimensions - is the equivalence "universal" if the dimension is fixed?
I am aware that in a finite dimensional vector space, any two norms are equivalent.
However, I cannot really figure out how "universal" the equivalence constants are.
To be specific, let us ...
- 3,477
0
votes
1
answer
218
views
Intersection of Hilbert spaces with Schauder basis
Let
$H$ be a infinite dimensional, separable, complex Hilbert space,
$\{v_{1_n}\}_{n \in \mathbb{N}}$ be a sequence in $H$,
$V_1=\operatorname{span}\{v_{1_n}\}_{n \in \mathbb{N}}$
$U_1=\overline{V_1}$...
- 433
2
votes
0
answers
164
views
Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
- 873
0
votes
0
answers
132
views
Compact embedding of $H^1(0,+\infty)$
Is the following embedding compact ?
$$H^1(0,+\infty) \rightarrow L^p(0,+\infty), \text{ with } p>1 $$
- 29
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
3
votes
1
answer
166
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,689
1
vote
1
answer
172
views
Banach space valued distributions and test functions
Let $A,B,C$ be Banach spaces and $m\,:\,A\times B\to C$ be a bilinear map such that $\|m(a,b)\|\leq \textrm{const}\,\|a\|\|b\|$. We denote by $\mathcal{S}(\mathbb{R}^d)$ be the standard space of ...
- 552
5
votes
1
answer
199
views
Is the unit ball of $B(H)$ a Baire space (with the SOT)?
Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t_i \to t$ in the strong operator topology if $t_i \xi \to t \xi$ for every $\...
- 733
2
votes
0
answers
137
views
Compact embedding for the space $H^1(0,+\infty)$
It is well known that $H^1(I)$ is compactly embedded in $C(I)$ where $I$ is bounded interval of $\mathbb{R}$, which is not correct for $I$ unbounded.
So, I search about a functional space $Y$ such ...
- 29
0
votes
0
answers
77
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 39
0
votes
1
answer
102
views
Limit of minimizers of a class of functionals
Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ ...
- 1,219
2
votes
0
answers
79
views
Does this variant coincide with the usual Hölder space?
$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$
Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$.
The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of ...
- 825
1
vote
1
answer
138
views
Integration of vector function against vector measure
Let $X,Y,Z$ be Banach spaces and let $m\,:\,X\times Y\to Z$ be a bilinear map such that $\|m(x,y)\|\leq C \|x\|\|y\|$ for some fixed constant $C$. Moreover, let $\mu$ be a Borell vector measure on $\...
- 552
4
votes
0
answers
120
views
Matrix product of entire functions
Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
2
votes
0
answers
122
views
Are the extensions of the expected value, below, finite for all functions in only a shy subset of all measurable functions?
This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions:
Let $(X,d)$ be a metric space. If set $A\subseteq X$,...
- 63
4
votes
0
answers
127
views
What can be possible conditions for the solution of an autonomous ODE to be conservative with respect to the initial data?
Let $F : \mathbb{R}^n \to \mathbb{R}^n$ be a smooth mapping and consider the following autonomous ODE
\begin{equation}
y'(t)=F(y(t))
\end{equation}
with the initial data $y(0)=x \in \mathbb{R}^n$.
...
- 3,477
0
votes
1
answer
79
views
Convergence in sequential Lebesgue spaces
Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
- 1,101
4
votes
0
answers
348
views
A proof of the Gagliardo-Nirenberg interpolation inequality using Jensen's inequality
A brief look at the statement of Gagliardo-Nirenberg interpolation inequality would suggest that there should exist a proof by a clever use of Jensen's inequality. In other words, there should be a ...
- 765
3
votes
1
answer
106
views
First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?
Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by
\begin{equation}
Hf(x)=x^2f(x)-f''(x)
\end{equation}
for any smooth function $f$ compactly ...
- 3,477
2
votes
1
answer
199
views
Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
- 3,477
2
votes
2
answers
228
views
Hardy space inclusion in the right-half plane
I'm looking for an example of a function $u \in H_2$ such that $u \notin H_\infty$, where $H_p$ is the Hardy space on the right-half plane. Since this notation is perhaps not standard, here is a ...
- 427
7
votes
2
answers
419
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 217
7
votes
1
answer
334
views
Extremal problem for 2-dimensional lattices
Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...
- 16.6k
2
votes
2
answers
167
views
LF or LB space that happens to be finite dimensional
Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further ...
- 3,477
-4
votes
1
answer
145
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
- 1
5
votes
1
answer
510
views
A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
1
vote
0
answers
51
views
Error estimates for inhomogeneous semidiscrete PDE
I have the following semidiscrete problem on a meshed domain $U_h$. Let
$V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and
$V_{h\partial}$ be ...
- 235
5
votes
1
answer
358
views
Is there a meaningful interpretation of an $L^i$-space?
Do complex-normed spaces exist? Is there an extension of $p$-norms to $p\in\Bbb C\setminus\Bbb R$?
A while ago I thought of extending $L^p$-spaces to the complex-normed setting. After some discussions,...
- 1,459