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Diffusion processes in wide generality

It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality. Hard question: What are the most general structures on which one may define something ...
Tom LaGatta's user avatar
  • 8,512
0 votes
0 answers
93 views

Infinite limit in all points

Do there exist a Banach space (possibly nonseparable) $X$ and a mapping $F: X\to X$ such that $$ \lim_{x\to a} \|F(x)\| = +\infty \quad \forall a\in X\quad? $$
Gulnara Sharafutdinova's user avatar
0 votes
1 answer
222 views

Bounding near the boundary for a Sobolev function.

Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
alext87's user avatar
  • 3,217
1 vote
1 answer
227 views

Are these ideals in rings of operators on Hilbert space unique?

Suppose that, for every Hilbert space $H$, we have a subset $I(H) \subseteq B(H)$ of bounded linear operators on $H$, and that together all $I(H)$ form a two-sided ideal, in the sense that whenever $h ...
Chris Heunen's user avatar
  • 3,937
4 votes
0 answers
94 views

Algebraic conditions of separability

Let $X$ be a real vector space (without any norm), and $Y$ be a convex subset of $X$, $0\notin Y$. The goal is to find a hyperplane $L$ passing through 0 such that $Y$ lies in a closed halfspace ...
Fedor Petrov's user avatar
1 vote
0 answers
180 views

iterated traces for sobolev functions

It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose ...
Mircea's user avatar
  • 2,041
9 votes
1 answer
395 views

Is there a coalgebraic characterisation of the hyperfinite II_1 factor?

Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...
David Corfield's user avatar
1 vote
0 answers
178 views

Inequalities between self-adjoint operators

Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$...
Jesús Álvarez's user avatar
1 vote
1 answer
111 views

Log-nonexpansive functions: terminology and references

During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important. (Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
Suvrit's user avatar
  • 28.6k
3 votes
0 answers
188 views

Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?

This question is related to the following question Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)? A couple of authors have observed that composing a ...
jzadeh's user avatar
  • 265
2 votes
0 answers
53 views

Topology of fibers of operators under C^{\infty} convergence

A smooth family of maps $f_t : L^2 (\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is given, with $t \in (0,1]$. Suppose that when $t \rightarrow 0$ the family restricted to balls of given radius converges ...
Alessandro Gentile's user avatar
1 vote
1 answer
138 views

Estimating norms of derivatives

Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\...
Viktor Bundle's user avatar
2 votes
2 answers
317 views

Bibliography for topologies defined by a family of seminorms

Hello I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource. Thank you very much.
Learner's user avatar
  • 143
6 votes
0 answers
639 views

Hilbert subspaces of indefinite inner product spaces

Let $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$. Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] ...
Ady's user avatar
  • 4,060
0 votes
1 answer
234 views

A property of "Schwartz" quadratic forms

Consider $K(x, y)$, $f(x)$ Schwartz functions and $g(y)$ a tempered distribution. Suppose $$K(x, y) = K(y, x)$$ Define $$h(t) = \int f(x - t) K(x, y) g(y - t) dx dy$$ It appears to me $h(t)$ is a ...
Vanessa's user avatar
  • 1,368
2 votes
0 answers
176 views

A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# ...
Beni Bogosel's user avatar
  • 2,222
0 votes
0 answers
68 views

can we say fixed point existance of a set valued map over a compact set is homotopy invariant?

Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
behrad mahboobi's user avatar
1 vote
1 answer
254 views

Extending linear operators to multi-linear ones

Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that $$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
user avatar
0 votes
1 answer
221 views

Sort-of extension of Young inequality to arbitrary measures

Hello folks, Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $. The Young inequality may be ...
Seaking's user avatar
1 vote
0 answers
215 views

Classification of Self similar sets

I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are ...
ViperRobK's user avatar
0 votes
0 answers
184 views

Extension of closed linear functionals...

If f is a closed linear functional defined on a dense subspace of a Banach space X, and, consider f1 which is an extension of f to X, is there a way to show that f1 is also closed without invoking the ...
Abhi. A's user avatar
  • 55
1 vote
0 answers
153 views

The existence of the solution of the perturbed KdV Equation(semi-group operator)

Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$,$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$ I want to use ...
89085731's user avatar
1 vote
0 answers
29 views

Counting variables to look for invariances/range conditions

A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...
icurays1's user avatar
  • 203
10 votes
0 answers
609 views

Asymptotic non-distortion of the separable Hilbert space

By the work of E. Odell and Th. Schlumprecht, we know that the separable Hilbert space $\ell_2$ is arbitrarily distortable. But I don't know if an "asymptotic" version of their result is true. To ...
Pandelis Dodos's user avatar
8 votes
0 answers
196 views

Parametrizing derivations from the algebra of smooth functions on a manifold to its dual

$\newcommand{\Der}{\operatorname{Der}}$ $\newcommand{\Real}{{\mathbb R}}$ (Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
Yemon Choi's user avatar
  • 25.8k
0 votes
0 answers
143 views

description of a convex set of functions

Hi everyone, I have a question about the characterization of a set of functions. Let $\Phi$ a set containing all the functions $\phi(x): \mathbb{R}_+\rightarrow \mathbb{R}_{+}$ that satisfy the ...
Higgs88's user avatar
  • 69
7 votes
1 answer
286 views

a.e. convergence of the powers of an operator built from rotations

Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by $$T(f)(x)=1/2(f(x+a)+f(x+b))$$ For which values of $a,b$ do we have almost ...
coudy's user avatar
  • 18.7k
4 votes
0 answers
487 views

Convolutions and Toeplitz Operators

Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$. Let be $...
Leandro's user avatar
  • 2,044
8 votes
0 answers
605 views

convergence rate in Wiener's approximation theorem

Wiener has the following fantastic results about approximations using translation families: Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
gondolier's user avatar
  • 1,839
0 votes
0 answers
301 views

Lifting of product of a Banach algebra

Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product. A lifting of $T$ is ...
BigBill's user avatar
  • 1,222
3 votes
0 answers
84 views

Application and relevance of Sobolev gradients

The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka I would like to learn how strong the impact of Sobolev ...
shuhalo's user avatar
  • 5,327
4 votes
0 answers
167 views

Relationship between sequential compactness of a convex set and its extremal points

Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a ...
anonymous's user avatar
0 votes
1 answer
297 views

Continuity of cylindrical functions.

Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the ...
Jonas T's user avatar
  • 455
4 votes
1 answer
221 views

existence of charaterization of amenable groups by complementation?

Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$. Do you know a charaterization of discrete amenable ...
BigBill's user avatar
  • 1,222
11 votes
0 answers
310 views

Combinatorial Hilbert spaces

Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...
David Feldman's user avatar
1 vote
0 answers
76 views

h-oscillating function

I need help understanding the following condition: $u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
Rocha's user avatar
  • 11
2 votes
1 answer
265 views

Multiple ergodic averages with varying number of terms

Hi. I've been stuck on the following question for some time. Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \...
nonameisfinetoo's user avatar
1 vote
0 answers
125 views

base change for distributions

For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
Rami's user avatar
  • 2,649
6 votes
0 answers
354 views

Ordering of completely bounded maps

Let A be a C*-algebra, let H be a Hilbert space, and let $T:A\rightarrow B(H)$ be a completely bounded (cb) map (that is, the dilations to maps $M_n(A)\rightarrow M_n(B(H))$ are uniformly bounded). ...
Matthew Daws's user avatar
  • 18.7k
2 votes
0 answers
146 views

Subspace where an operator is positive

Given a self-adjoint operator $\hat{T}$ on a Hilbert space $\mathcal{H}$, and assuming it has a basis of eigenvectors $\{\phi_n\}$ such that $\hat{T}\phi_n=\lambda_n\phi_n$, one can consider the ...
Emilio Pisanty's user avatar
3 votes
0 answers
251 views

What is the origin of the metrization problem for compact convex sets?

The following is an ``old question in analysis:'' Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable? Here perfectly normal means ...
Justin Moore's user avatar
  • 3,547
1 vote
0 answers
125 views

Isomorphisms of group extensions arising from antisymmetric forms

Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
Ollie's user avatar
  • 1,411
2 votes
0 answers
114 views

non-closed weak graph limit of symmetric operators

Hi Everyone, I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
Tlas's user avatar
  • 21
1 vote
0 answers
149 views

Banach spaces with simple best approximate solutions

Let $\langle V,||.||\rangle$ be a Banach space such that: $\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$ $\;\;$ that ...
user avatar
1 vote
1 answer
263 views

Need help with references on the status of a "Littlewood Problem"

The "Littlewood Problem" in the title asks for a characterization of finite sequences n1< ...< nk of integers such that zn1+zn2+...+znk≠0 for any complex number z of unit modulus. Does ...
Quotient Group's user avatar
1 vote
0 answers
136 views

Boundedness of Integral

Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point. Define the integral $$ Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta $$ and ...
pil's user avatar
  • 233
2 votes
1 answer
194 views

Is there any result discribing the value of the correlation of a measurable function of `$X$` and itself: `$corr(f(X),X)$` ?

Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$? BACKGROUND The background of asking the value of $...
J.Xie's user avatar
  • 23
5 votes
0 answers
104 views

Regularity of simplices, part deux

This question is directly inspired by Pietro Majer's question and my answer to it. One can define a simplex, and the dihedral angles thereof in an infinite dimensional Hilbert space (one has to take ...
Igor Rivin's user avatar
  • 96.4k
1 vote
0 answers
119 views

Particular types of basis on a normed vector space of finite dimension

Is it true that on every normed vector space $V$ of dimension $n$ there exists a basis of norm $1$ vectors $v_i$, such that $\|\sum_{i=1}^n\epsilon_iv_i\|\geq 1$ for all possible combinations of $\...
Daniel's user avatar
  • 31
3 votes
1 answer
280 views

An analogue of an old proposition

For the absolute value $|C|=(C^*C)^\frac{1}{2}$ and the Hilbert-Schmidt norm $\parallel C\parallel_{HS}=(trC^*C)^\frac{1}{2}$ of the operator $C$. The following inequality is shown by Araki et al in ...
Russel's user avatar
  • 223