All Questions
3,602 questions with no upvoted or accepted answers
4
votes
0
answers
255
views
Lower semi-continuity of integration
I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
- 5,405
4
votes
0
answers
136
views
Davies' definition of elliptic operators in "Heat Kernels and Spectral Theory"
I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
- 5,407
4
votes
0
answers
115
views
point-wise approximation of the identity in hereditary Lindelof spaces
Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...
- 4,058
4
votes
0
answers
166
views
Is this property an isomorphic characterization of $\ell_1(\Gamma)$?
Let $\Gamma$ be an infinite set. Then every $(x_i)_{i\in\Gamma}\in \ell_1(\Gamma)$ has at most a countable number of components $x_i\neq 0$.
As a consequence, every separable subspace $M$ of $\ell_1(\...
- 4,461
4
votes
0
answers
143
views
A point concerning Fremlin's example on Borel sets in non-separable Banach spaces
Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...
- 4,058
4
votes
0
answers
131
views
Algebraic sum of relative commutants in a finite von Neumann algebra
Let $(M,\tau)$ be a finite von Neumann algebra with faithful normal tracial state $\tau$. Suppose that $A \subset M$ is a finite-dimensional abelian unital subalgebra--say $A = W^*(p_1,\dots, p_n)$ ...
4
votes
0
answers
322
views
Compactness of semigroups of one-dimensional diffusions
I have a question about semigroups of one-dimensional diffusions.
Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as
$$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$
It is ...
- 721
4
votes
0
answers
238
views
Does Novikov condition imply BMO martingale?
Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $...
- 195
4
votes
0
answers
116
views
Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?
As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, ...
- 3,612
4
votes
0
answers
198
views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
- 411
4
votes
0
answers
211
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
4
votes
0
answers
256
views
One-parameter unitary group preserving invariant domain of infinitesimal generator
Let $\mathcal{H}$ be a separable Hilbert space (e.g. $L^{2}(\mathbb{R}^{d}))$, and let $\mathcal{D}_{1}\subset\mathcal{H}$ be a dense subspace (e.g. $\mathcal{S}(\mathbb{R}^{d})$). Suppose that an ...
- 873
4
votes
0
answers
120
views
Reductive Operator Problem
In the 1972 paper ''An equivalent Formulation of the Invariant Subspace Conjecture'' Dyer, Pedersen, and Porcelli announce the following result:
The Invariant Subspace Problem has a positive ...
- 355
4
votes
0
answers
113
views
Index of glued operator
Suppose $X_1$ is a manifold which has a tubular end $\mathbb R^+\times Y$, and $X_2$ is a manifold which has a tubular end $\mathbb R^-\times Y$. Here, $X_1,X_2$ are orientable manifolds and $Y$ is a ...
- 1,761
4
votes
0
answers
141
views
algebraic connectivity of a tree
Suppose that $T$ is a tree with $n$ vertices and $L$ is the Laplacian matrix of $T$ and $0=\mu_1 \leq \mu_2 \leq \cdots \leq \mu_n$ are laplacian eigenvalues.
I think the multiplicity of $\mu_2$ can ...
- 221
4
votes
0
answers
176
views
Distributional PDE solutions as topological linear duals of PDE solutions
Let
$$
P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)
$$
be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
- 19.8k
4
votes
0
answers
102
views
Sub-quadratic Kolmogorov-Arnold?
The Kolmogorov-Arnold representation theorem says, essentially, that when computing a continuous function, the only multivariate function you really need is addition. (Somewhat) more precisely, it ...
- 3,979
4
votes
0
answers
100
views
A bounded operator associated with a principal bundle
Assume that $(X, B, G)$ is a $G$- principal bundle where $G$ is a compact topological or Lie group.
The normalized Haar measure of (each fiber of ) $X$ is denoted by $\mu$.
The space of continuos ...
- 356
4
votes
0
answers
125
views
Properties of solution to Schrödinger equation
Given a Schrödinger equation with, let's say continuous, periodic potential
$$-y''(x)+V(x)y(x)=\lambda y(x)$$
where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
- 501
4
votes
0
answers
119
views
Index of a subfactor of a full $II_1$ factor
On pg. 151 of "Coxeter Graphs and Towers of Algebras" by F.M. Goodman, P. de la Harpe, and V.F.R. Jones (1989), it is stated that there is no known example of a full $II_1$ factor having a subfactor ...
- 229
4
votes
0
answers
84
views
Can a spectral projection fail to preserve a closed invariant subspace of its parent operator?
Let $X$ be a complex Banach space, and let $T:X\longrightarrow X$ be a bounded linear operator. Let $\sigma_1$ and $\sigma_2$ be disjoint compact subsets of $\mathbb{C}$ for which $\sigma_1\cup \...
- 778
4
votes
0
answers
164
views
A modern reference for the "Intermediate Derivatives Theorem"
In the book Non-Homogeneous Boundary Value Problems and Applications I by Lions and Magenes, the Intermediate Derivative Theorem is stated as follows:
Intermediate Derivative Theorem: Let $X\subset ...
- 1,247
4
votes
0
answers
144
views
Embedding of $\ell_2$ in $L^p([0,1])$
Let $(g_n)_{n\geq 1}$ be a sequence of i.i.d. complex Gaussian random variables on $[0,1].$ Then it is easy to see that the map $j:\ell_2\to L^p([0,1])$ defined as $je_n=[E(g_n^p)]^{\frac{1}{p}}g_n,n\...
- 455
4
votes
0
answers
263
views
Approximately inner conditional expectations of $II_{1}$ factors
In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. ...
- 7,067
4
votes
0
answers
81
views
Amenable Banach algebras -- homological characterization
Recall from Theorem VII.2.19 of Helemskii's monograph "The homology of Banach and Topological Algebras" that amenability of a Banach algebra $A$ is equivalent to any of the conditions below:
(i) ...
- 375
4
votes
0
answers
311
views
Some elementary decay estimates of solutions to the heat equation
Preliminaries: Let $u$ be the solution of the Cauchy problem for the heat equation with initial datum $u_0 \in L^1 \cap L^p$. Then I know that the following estimates hold:
$$\Vert u(t,\cdot)\Vert_{L^...
- 303
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
4
votes
0
answers
92
views
Simultaneous representations of elements of projective tensor products
Let $E,F$ be Banach spaces and consider the projective tensor product $E \widehat\otimes F$. If $\tau \in E \widehat\otimes F$ with $\|\tau\|<1$ then by definition we can find $(x_n)\subseteq E$ ...
- 18.7k
4
votes
0
answers
70
views
Estimate the composition of a bounded multiplier with a trace class operator
Let $T$ be a trace class operator on $\ell^2 (\mathbb{N})$. Let $A$ be a multiplier on $\ell^2 (\mathbb{N})$ defined by a sequence $a=(a_n)_{n\in\mathbb{N}}$ in $\ell^{\infty} (\mathbb{N})$. That is, ...
- 151
4
votes
0
answers
184
views
Tensor product of bornological spaces and linear functionals
It is easy to see that the dual space of a bornological space $V$ (i.e. the space of bounded linear functionals) may be zero (just take any vector space with the power set as bornology). Hence in ...
- 9,853
4
votes
0
answers
89
views
How can I can derive an explicit bound for the solution of the poisson's PDE?
i need some help on this question
Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with
$\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
- 41
4
votes
0
answers
585
views
Dual of the space of all bounded functions, $B(X, \mathbb{R}).$
Let $X$ be a non compact separable metric space. Denote
by $B(X, \mathbb{R})$ the set of all bounded real functions
endowed with the sup norm, this is a Banach space. Denote by
$C_b(X,\mathbb{R})\...
- 757
4
votes
0
answers
140
views
Is there any accepted single-word that means "partial function"?
When I'm explaining things involving partial functions, I usually end up stumbling over my words, like so: "Suppose $f : A \rightarrow B$ is a function, uhh, sorry I mean a partial function, and ...
- 3,793
4
votes
0
answers
123
views
Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
- 1,616
4
votes
0
answers
361
views
Spectral mapping theorem
Rudin's book contains in chapter 10 a spectral mapping theorem for (self-adjoint) unbounded operators that respects the point-spectrum, in the sense that he shows $f(\sigma_p(T))=\sigma_p(f(T))$ for ...
- 305
4
votes
0
answers
161
views
Sigmoid functions in a particular function set
Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}....
- 1,902
4
votes
0
answers
118
views
Smoothness in von Neumann algebra of measurable functions
Let $A=L^{\infty}(M)$ be an algebra of essentially bounded measurable function on manifold $M$. Let $D$ be a first order elliptic differential operator acting on some hermitian bundle $S$ over $M$ (...
- 9,330
4
votes
0
answers
298
views
Operator topologies
Let $L(H)$ be the space of bounded operators on some Hilbert space.
We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT).
...
- 41
4
votes
0
answers
128
views
Positive and Negative parts of functions in Schrodinger $U_{\Delta}^{p}$ spaces
Let $H$ be a Hilbert space. For $1<p<\infty$, define the atomic space $U^{p}(\mathbb{R};H)$ as follows. We say that $a(t)$ is a $U^{p}$ atom if $\{t_{k}\}$ is a partition of $\mathbb{R}$, $a_{k}\...
- 873
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
4
votes
0
answers
269
views
Algebras and $\sigma$-algebras associated to random variables
Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^...
- 1,411
4
votes
0
answers
110
views
Banach space admitting a unique subsymmetric basis but not a symmetric one
I have two quick questions:
It can be shown without too much trouble (using methods from Altshuler/Casazza/Lin, 1973) that any Lorentz sequence space admits a unique (up to equivalence) subsymmetric ...
- 1,591
4
votes
0
answers
265
views
Does the Cauchy–Schwarz inequality imply 2-positivity?
Recall the following generalisation of Cauchy–Schwarz.
Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...
- 355
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
4
votes
0
answers
322
views
Cauchy-Riemann Operators and Selberg Zeta Function
The determinant of hyperbolic Maaß-Laplacian operator on arbitrary tensors and spinors can be written in terms of Selberg zeta function. Is there a corresponding formula for the determinant of the ...
- 989
4
votes
0
answers
609
views
Does every separable Banach space have a Markushevich–Auerbach basis?
Let $X$ be a separable Banach space and $X^*$ be its dual, let $\{x_i\}$ be a sequence in $X$ with dense linear span and such that there exists a sequence $\{x_i^*\}$ in $X^*$ satisfying $x_i^*(x_j)=\...
- 902
4
votes
0
answers
114
views
Coming up with a represenation for sum of functions in the Fourier algebra
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com.
Let $G$ be a discrete group.
Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
- 161
4
votes
0
answers
271
views
Concentration of infinite-dimensional Gaussian measure
I have the question about finding the subspace of concentration of a Gaussian Measure. More precisely:
$\textbf{Question:}$ Assume we have a separable Hilbert space $\ell_2$ with Borel $\sigma$-...
- 537
4
votes
0
answers
171
views
quasi-nilpotent part of a dual operator
Definitions and notation.
Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as
\begin{equation*}H_0(T):=\left\{...
- 1,591
4
votes
0
answers
172
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 ...
- 41