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Maximum number of orthonormal vectors contained in an open cone

Let $H$ be a separable Hilbert space, $\Pi:H\to L$ the orthogonal projection to a linear subspace of finite dimension $p$, and $U$ the open cone of vectors $u\in H$ such that $\langle u,\Pi u\rangle&...
Jesús Álvarez's user avatar
3 votes
0 answers
183 views

Is the construction of ring C*-algebra functorial?

Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? ...
Sayan's user avatar
  • 95
0 votes
0 answers
166 views

Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations. My questions are: (a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
grateful's user avatar
0 votes
2 answers
444 views

Sobolev space: probably simple ode....

I am trying to solve for $y(x)$ in terms of $f(x)$ in a convenient space (eg. $\dot{H}^2(\mathbb{T})$-zero mean). Here is the ode: $y(x)+y(x)y'(x)=f(x)$. I think a contraction mapping argument will ...
Rosa's user avatar
  • 9
8 votes
1 answer
381 views

Estimating flat norm distance from a planar disc

Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
Sergei Ivanov's user avatar
0 votes
0 answers
152 views

Need help determining whether a certain map is a $C^\ast$ homomorphism

Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
Clark Chong's user avatar
4 votes
0 answers
59 views

Behaviour of Markov type under uniform homeomorphism of spheres

A metric space $(X,d_X)$ has Markov type $p$ (with $p \in [1,2]$), if, for every stationary Markov chain $\{Z_n\}_{n=0}^\infty$ on $Y$ (a finite space) and every mapping $f:Y \to X$, one has $$ \...
ARG's user avatar
  • 4,432
0 votes
2 answers
415 views

Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$

A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
user avatar
2 votes
2 answers
354 views

A bound on linear functionals over cotype 2 spaces

This is a modification of the somewhat naive question that I asked below. Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
Brad Rodgers's user avatar
  • 2,151
0 votes
1 answer
330 views

Convex sets and projections

Hello! I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
MasterOfOrion's user avatar
1 vote
1 answer
635 views

Closed range for a continuous linear transformation

I have a Banach space $B$ and a continuous linear transformation $F:B \rightarrow B\times B$. One of the induced transformations $F(1):B \rightarrow B$ and $F(2):B \rightarrow B$ into the factors of ...
Chris Leary's user avatar
1 vote
2 answers
294 views

inequality of norms [closed]

Let $X$ and $Y$ be two Banach spaces with norms $\|\|_X$ and $\|\|_Y$ respectively. If $Z=X\times Y$ is also a Banach space with norm $\|\|_Z$ then what is the relation between $\|\|_X,\:\|\|_Y$ and $\...
Sanket A. A. Tikare's user avatar
2 votes
0 answers
117 views

Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...
Santiago's user avatar
  • 197
0 votes
0 answers
146 views

How to bound Haar coefficients in terms of total variation?

I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says: We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ such ...
Dustin G. Mixon's user avatar
1 vote
1 answer
353 views

Separability of the space of bounded continuous maps

Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric ...
Orbicular's user avatar
  • 2,935
0 votes
1 answer
396 views

Characterization of Measureable Sets [closed]

Every countable union of rectangles in R2 is a Lebesgue measurable set. Is the converse true, too? Specifically, I wonder whether the following statement is true: Let A be a set in the unit square ...
Nahpetz's user avatar
  • 99
2 votes
2 answers
181 views

convergence of the coefficients of lacunary series

I just want to find some standard reference to the following result: let $(a_k)_k$ be the sequence of coefficients of a lacunary Fourier series which converges to an $L_1(T)$ function in the sense of ...
António Caetano's user avatar
6 votes
0 answers
257 views

What is the intersection of the closures of left invertible operators and right invertible operators?

From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that $$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = \...
Qingping Zeng's user avatar
5 votes
0 answers
240 views

Linear ODEs in a locally convex vector space

Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...
Allan Yashinski's user avatar
0 votes
0 answers
272 views

L_2-norm representation

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$. I am wondering if one can get nice representation of $L^2$-...
David's user avatar
  • 71
3 votes
0 answers
113 views

inifinite tensor product algebra representation

For a finite integer $N$, let $A_n = \bigotimes^n M_N(\mathbb{C})$. $A_n$ embeds in $A_{n+1}$. Let $A_\infty = \cup A_n$. Are the (complex) irreducible representations of $A_\infty$ known? It is ...
magya_bloom's user avatar
1 vote
2 answers
288 views

Is it possible that the intersection of two nest algebras contains only scalars?

Dear all, I really want to know the answer of the following question. I would appreciate any help. Assume H is a separable Hilbert space, is it possible to find two nests N1, N2 such that the ...
heller's user avatar
  • 61
1 vote
1 answer
226 views

How are real-analytic functions encoded in computer algebra?

I would like to know how are encoded the real-analytic functions on the interval by the computers. When I think in a real-analytic function I just think in a composition of the ''typical'' analytic ...
Umberto's user avatar
  • 105
0 votes
0 answers
150 views

$n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
David's user avatar
  • 71
3 votes
0 answers
498 views

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases). Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
user16007's user avatar
  • 800
1 vote
2 answers
504 views

Do all graphs of C1 functions have Hausdorff dimension 1?

Suppose f is a real-valued function of one variable, and suppose f is of differentiability class C1. My question is, if $\Gamma$ is the graph of f, then must $\dim_H(\Gamma)=1$? If anyone knows of a ...
James McCollum's user avatar
1 vote
0 answers
84 views

Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
konrad's user avatar
  • 11
0 votes
1 answer
503 views

When are operators extended by linearity bounded?

Greetings. Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
Adam Azzam's user avatar
0 votes
0 answers
83 views

Comparison between operators

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any $\varepsilon>0$,...
user45340's user avatar
1 vote
1 answer
367 views

An integral which is related to Biharmonic extension

In my research, I need to evaluate an integral: $$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}\log(|\xi^{2}|+\frac{1}{4})d\xi$$ where $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it ...
wrwrnm's user avatar
  • 11
5 votes
1 answer
514 views

Request for reference: Banach-type spaces as algebraic theories.

Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
Andrew Stacey's user avatar
1 vote
0 answers
153 views

Functional Analysis Generalizations: indeterminated inner product and functions over manifolds

There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
Jose Adsuara's user avatar
3 votes
0 answers
302 views

Dense subalgebras of von Neumann algebras and increasing nets

[Question previously asked on Math.SE] Let $N$ be a von Neumann algebra, and $A$ be a dense $∗$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that: For any $x∈N^+$, there ...
Michael's user avatar
  • 33
1 vote
0 answers
149 views

(localized) L^2 norm of quasimode for Laplacian

Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$: $u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\...
Rocha's user avatar
  • 11
0 votes
2 answers
337 views

Is there a general notion of entropy for the states of a C*algebra?

I've seen some definition of the relative entropy between two states of a C*algebra. However this definitions work only for finite dimensional C*algebras and I don't know if there is a correspondent ...
Camilo Argoty's user avatar
0 votes
0 answers
160 views

Is this function in the weighted Sobolev space $H^{2,-s}$?

I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order $...
Sue's user avatar
  • 1
3 votes
1 answer
235 views

Odd element of L^1 group algebra of the integers

Giving some motivation is hard here, so I'll just ask the question. I want an element $a=(a_n)\in\ell^1(\mathbb Z)$ such that: $\|a\|>1$ a is power bounded (turn $\ell^1(\mathbb Z)$ into a Banach ...
Matthew Daws's user avatar
  • 18.7k
0 votes
0 answers
436 views

cokernels of semi-Fredholm operators

I did not find a reference for the following question, so I will pose it here. I think the answer should be elementary. Let $F:X\rightarrow Y$ be a semi-Fredholm operator between Banach spaces, i.e. $...
Orbicular's user avatar
  • 2,935
1 vote
1 answer
506 views

Bessel sequence, uniformly minimal, separated

Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded ...
MiM's user avatar
  • 11
21 votes
0 answers
877 views

Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
Joel Fine's user avatar
  • 6,247
2 votes
1 answer
214 views

union of Stone-Cech remainders

Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
Douglas Somerset's user avatar
7 votes
1 answer
362 views

Nonexpansive multi-valued maps in $\ell^2$

Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
TCL's user avatar
  • 744
1 vote
0 answers
126 views

Is scalarwise measurability determined by the strong dual?

Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that $E$ and $F$ are separable (real) Banach ...
TaQ's user avatar
  • 3,584
1 vote
2 answers
515 views

continuity of extension of maps along curves

Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ ...
Orbicular's user avatar
  • 2,935
1 vote
1 answer
706 views

Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??

Hi! The Plancerel-Polya inequality can be stated as follows: Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
Philipp's user avatar
  • 979
1 vote
1 answer
663 views

What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?

[Background:] Looking at the powerseries for the gamma-function $ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $ then we can arrive at a decomposition $ \Gamma(1+x) = r(x) + g(x) $ ...
Gottfried Helms's user avatar
2 votes
1 answer
230 views

Completing The Space Sections in a Vectorbundle

Hi there. Assume $(M,g)$ is a Riemanian manifold and $E\to M$ is a vector bundle with a bundle metric $\langle\cdot,\cdot\rangle$. We then have the pre-Hilbert space $H_0:=\Gamma_c^\infty(E)$ of ...
Robert Rauch's user avatar
4 votes
1 answer
474 views

Are these operators defined on 2D surfaces self-adjoint?

My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether ...
QHLIU's user avatar
  • 199
2 votes
0 answers
82 views

Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where $$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ \...
Michael Tinker's user avatar
1 vote
0 answers
34 views

Efficient packing in Gaussian measures on Hilbert spaces

Let $B(0,1)$ be the unit ball in a separable Hilbert space $H$ with a Gaussian measure $\mu$ on it. For a small $r > 0$, can we have $x_i \in B(0,1)$, $0 < r_i \leq r$ and a constant $K > 0$ ...
Madhuresh's user avatar
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