All Questions
12,778 questions
0
votes
1
answer
426
views
Lower bounds for partial sums of multiplicative functions
The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
$$F(s)=\...
- 2,480
8
votes
1
answer
749
views
Is $SU(\infty)$ amenable?
We can write the finitary special unitary group $SU(\infty)$ as the direct limit
$\varinjlim SU(n)$ of ordinary special unitary groups. These groups $SU(n)$ are compact, thus amenable. In other ...
- 606
1
vote
1
answer
293
views
Basic sequences
Nowadays, we know that there exist Banach spaces without unconditional basic sequences. Do we know if something a bit milder holds? Namely, is that true each non-reflexive Banach space contains a ...
- 113
4
votes
0
answers
102
views
quasinilpotence and finite spectrum II
Let A be a quasinilpotent operator on a Hilbert space and let every
operator of the algebra generated by $A$ and $A^{*}$ have finite
spectrum. Does then follow, that A is nilpotent ?
See also ...
- 2,753
3
votes
1
answer
3k
views
Infinite dimensional vector spaces with compact unit ball
Let $X$ be an infinite dimensional vector space over a field $\mathbb{K}$. Suppose that $(X,\|\cdot\|)$ is a complete normed vector space, in the sense that any Cauchy sequence is convergent. Suppose ...
- 2,044
-1
votes
1
answer
2k
views
Reducibility (or not) of algebraic curves [closed]
[ I am a bit clueless about why this question is getting downvotes!? I put it up with a genuine serious interest and I don't seem to be making any egregious error either - apart from those unsure ...
- 3,541
-1
votes
1
answer
934
views
Domain and exponential of self- adjoint operator
Let $A$ be a self - adjoint operator on a Hilbert space $\mathcal{H}$ and let $D(A)$ be its domain. If $\psi \in D(A)$ then $exp(-itA) \psi \in D(A)$ iff $A$ is bounded ?
Thank ...
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 ...
- 11
28
votes
2
answers
1k
views
Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
- 5,898
19
votes
2
answers
5k
views
Is there an infinite-dimensional Banach space with a compact unit ball?
A popular pair of exercises in first courses on functional analysis prove the following theorem:
The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.
My ...
- 11.4k
2
votes
1
answer
199
views
Uniqueness of free complements
Let $A,B$ be subfactors of a II$_1$ factor $M$ with $A*B\simeq M$. That is, $A$ and $B$ are freely independent with respect to the trace and $M\simeq A\vee B$. We'll call $B$ a free complement for $A$ ...
- 1,411
2
votes
1
answer
287
views
Is a certain composition of harmonic forms again harmonic?
Let $(X,\omega)$ be a compact Kahler manifold, and let $\alpha$ and $\beta$ be smooth $(1,1)$-forms on $X$ that are harmonic (with respect to $\omega$). I can consider each of my $(1,1)$-forms as an ...
- 4,343
2
votes
1
answer
265
views
A Volterra-type equation
Consider the following integral equation
$\phi(x) = f(x) + \frac{1}{x}\int_0^x N(x,y)\phi(y)\;dy$,
where $f$ and $N$ are continuous and bounded functions. Are solutions $\phi$ of the above equation ...
- 595
0
votes
0
answers
268
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
- 225
5
votes
1
answer
641
views
Characterizing invertible nonnegative matrices with bounded sums
Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
- 28.6k
12
votes
2
answers
547
views
Balls in spaces of operators
I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.
Is ...
- 121
7
votes
0
answers
140
views
integral transforms defined by polygons
Are there any literature on operators of the form
\[ (Tf)(x) = \int K(x,y)f(y), dy\]
where $K(x,y)$ is the characteristic function of a polygon in $\mathbb{R}^2$. I would like to know if the spectrum ...
- 22.8k
1
vote
1
answer
224
views
Can symmetrizing a contraction increase the speed of convergence?
Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
- 199
28
votes
3
answers
4k
views
A separable Banach space and a non-separable Banach space having the same dual space?
I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is ...
- 6,034
2
votes
1
answer
942
views
A singular value inequality
Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
$$\left|s_{1}\...
- 173
20
votes
2
answers
4k
views
Ideals of the ring of smooth functions
The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
- 101
0
votes
2
answers
2k
views
Motivation behind defining the Ramification Divisor
I would like to understand what exactly is the motivation for defining the notion of a ramification divisor of a function.
As I see the definition,
If $f$ is a meromrophic function between two ...
- 3,541
6
votes
2
answers
293
views
quasinilpotence and finite spectrum
Let A be a quasinilpotent operator on a Hilbert space and let $A^{*}A$ have finite spectrum.
Does then follow, that A is nilpotent ?
- 2,753
3
votes
2
answers
1k
views
Sequences of linear combinations of measures
Let $X$ be a Polish space. Let $J\in\mathbb{N}$.
Let $\lbrace a^n_1\rbrace_n,\dots,\lbrace a^n_J\rbrace_n$ be $J$ sequences of reals.
Let $\lbrace \mu^n_1\rbrace_n,\dots,\lbrace \mu^n_J\rbrace_n$ be ...
user12713
2
votes
0
answers
327
views
Generalizations of Kato-Rosenblum theorem?
The Kato-Rosenblum theorem says that if $H_0, H$ are self-adjoint operators on a Hilbert space such that the difference $H-H_0$ belongs to the trace class, then the strong limit of $\exp(itH)\exp(-...
- 21
6
votes
1
answer
2k
views
Polynomials are dense in weighted $L^2$ space
Hi,
It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight $x^{\alpha-1}e^{-x}/\Gamma(\alpha)\;...
- 1,765
7
votes
4
answers
3k
views
Classification compact Riemann Surfaces
I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too?
In other words, is the complex structure ...
- 251
3
votes
3
answers
5k
views
Show that holomorphic functions are infinitely differentiable without complex analysis [duplicate]
Possible Duplicate:
Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?
Is there a way to show that holomorphic ...
- 131
3
votes
2
answers
2k
views
Domain of Holomorphy
I found the following definition of domain of holomorphy in several places.
Def1: A connected open set $\Omega$ in the n-dimensional complex space ${\mathbb{C}}^n$ is called a domain of holomorphy ...
- 1,563
34
votes
1
answer
3k
views
tr(ab)=tr(ba), part 2.
This is a Banach space version of Andre Henriques' question
Trace Question
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
- 31.5k
6
votes
2
answers
2k
views
How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
- 71
2
votes
1
answer
1k
views
A question about Ahlfors's proof of modular function being a covering space of the twice punctured plane
I have a question about Ahlfors's proof of modular function being a covering space of the twice punctured plane .See Ahlfors' complex analysis, second edition, page 272. You can either explain or ...
- 1,471
4
votes
1
answer
1k
views
Harmonic functions on the plane
I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it ...
- 103
35
votes
2
answers
9k
views
tr(ab) = tr(ba)?
It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace ...
- 43.2k
7
votes
3
answers
1k
views
Reference on semigroup theory and parabolic PDEs
Recently started to study semigroup theory. My background is equivalent to the first three chapters of the Jack Hale's book "Asymptotic behavior of dissipative systems".
Looking for a reference to an ...
user11178
6
votes
2
answers
2k
views
Continuity of a convolution (Version 2)
Hello,
This problem bothers me for some time. Suppose that
$\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support);
$\psi$ is ...
2
votes
3
answers
302
views
Holomorphic map from a neighborhood in $\mathbb C$ to S^3
Does there exist a holomorphic map from a neighborhood of $\mathbb C$ to $S^3 \subseteq \mathbb C^2$?
- 778
3
votes
0
answers
455
views
Morphism of von Neumann Algebras
Hello,
Is there a counterexample to the following statement:
let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ?
( a $W^* $-...
- 663
4
votes
1
answer
3k
views
Dense sets in the space of continuous functions
Let $X$ be a compact metric space, and
let $C(X)$ be the Banach space of continuous real-valued function on $X$, with
the maximum norm.
Suppose $S\subset C(X)$ is a set of functions with the ...
- 43
12
votes
1
answer
468
views
Status of the compact AR problem?
The so-called "compact AR Problem" reads:
Is every compact convex set in a metrizable topological vector space an absolute retract?
It is open according to the chapter by T. Banakh, R. Cauty and ...
- 5,575
4
votes
1
answer
3k
views
General form of Rouche's Theorem
Let $\Omega$ be the interior of a compact set $K$ in the plane. Suppose $f$ and $g$ are continuous on $K$ and holomorphic in $\Omega$, and $|f(z)-g(z)|<|f(z)|$ for all $z\in K-\Omega$. Then $f$ and ...
- 75
1
vote
2
answers
536
views
Transition matrix for holomorphic vector bundles [closed]
If we are given the local trivialities of a holomorphic vector bundle then by definition we can write down the transition matrix of that vector bundle.
In some very natural situations, we are not ...
- 11
2
votes
1
answer
276
views
Conformal Extension from a closed set to open
Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
- 541
3
votes
0
answers
236
views
Is it possible to "approximate" a compact set of the plane by compact sets with smooth boundary?
Hi,
Let $K \subseteq \mathbb{C}$ compact. Suppose that $K$ is connected, and that the boundary of $K$ is a simple, closed, piecewise $C^1$ curve. Denote by $A(K)$ the set of all functions holomorphic ...
- 31
2
votes
0
answers
290
views
Consequence of Modified Young's inequality
Let $f\in L^1(\mathbb R^n)$. Define operator $T_f(g)=|f|\ast g$ for functions $g$ on $\mathbb R^n$. The set of measurable functions $f$ on $\mathbb R^n$, such that $T_f$ is bounded from $L^p(\mathbb R^...
- 415
3
votes
4
answers
1k
views
$Aut(\mathbb{CP}^n)$ [..especially $n=1$ and $n=2$..]
I am confused and curious about the meaning of the $Aut(\mathbb{CP}^n)$.
Is what is called the "linear automorphism group" of $\mathbb{CP}^n$ the same as $Aut(\mathbb{CP}^n)$? It somehow seems to me ...
- 3,541
9
votes
1
answer
682
views
A differential inequality needed to prove a theorem about odd-dimensional souls
I need a solution to this problem (which is really a calculus problem) in order to prove a rigidity result for open nonnegatively curved manifolds with odd-dimensional souls:
Suppose that $f,g:\...
- 93
5
votes
1
answer
503
views
Approximating high-dimensional integrals by low-dimensional ones
This question is motivated by the following naive one: suppose we have a nice subset $X$ of some Euclidean space, say a polyhedron, and a nice $\mathbb{R}$-valued function $f$ on this subset, say a ...
- 23.5k
2
votes
2
answers
1k
views
Sufficient conditions to the existence of a weakly convergent subsequence from a Cauchy sequence in a (merely) normed space
Bonsoir/bonjour à toutes et à tous.
The title has it all, but... We know (as a consequence of the Eberlein-Šmulian theorem) that any bounded sequence, $\{x_n\}_{n \;\! \in \;\! \mathbb{N}}$, in a (...
- 10.5k
3
votes
0
answers
237
views
Monotonicity of a certain parametric integral
I would like to ask for some help (hints, ideas) in solving the following problem:
Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that
if we define for any $x\in\mathbb{R}...
- 696