All Questions
12,780 questions
30
votes
4
answers
4k
views
Elementary applications of Krein-Milman
This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...
1
vote
2
answers
696
views
Open problems in SCV
Does anyone know a recent survey in SCV? I am interested in famous problems such as The Union Problem, The Local Steiness Problem, The Intersection Problem, The Open Immersion Problem etc.
31
votes
1
answer
2k
views
Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?
Here's a research problem, which I think interesting.
Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n \...
- 10.1k
5
votes
1
answer
250
views
How does pseudoconvexity restrict the topology?
A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a ...
- 12.1k
11
votes
1
answer
504
views
Do ultrapowers of classical Banach spaces have unconditional bases?
I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$.
Since the ...
- 125
12
votes
4
answers
916
views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 7,609
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 $L^2$-...
- 71
12
votes
6
answers
1k
views
Functions holomorphic on a region minus a Cantor set
Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is ...
- 367
7
votes
1
answer
331
views
States/functionals on crossed product C*-algebras
Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
- 1,017
11
votes
1
answer
702
views
Kuiper's theorem via approximation
Kuiper's theorem says that the unitary group $U(H)$ of a separable infinite dimensional Hilbert space $H$ is contractible, if it is equipped with the norm topology.
Let's suppose, I do not know this ...
- 7,637
8
votes
2
answers
2k
views
when a pseudo-differential operators to be compact?
In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...
- 1,644
11
votes
4
answers
2k
views
generalisation of Cauchy-Riemann equations to 3D
Hi, harmonicity in 2d is preserved under mappings that satisfy the Cauchy-Riemann equations.
What about 3D? What conditions should a mapping satisfy to preserve harmonicity?
is there a general ...
- 111
1
vote
1
answer
887
views
How to explain the condition (C) in critical point theory?
Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f.
How to see the meaning of " $\|\...
- 11
-1
votes
2
answers
407
views
Almost isometric subspaces of $\ell_p$
1) Given $p\in (1,\infty)$.
2) Let us fix two, non-isometric subspaces $X,Y\subseteq \ell_p$ isomorphic to $\ell_p$.
3) Are there an $\varepsilon\in (0,1)$ and an isomorphism $S\colon X\to Y$ such ...
- 537
4
votes
1
answer
882
views
What is the domain of the "average operator"?
I can try to define an averaging operator for functions, namely let
$$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$
by
$$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$
whenever the limit ...
- 225
7
votes
3
answers
498
views
Sums of unitaries with small norm in full group $C^*$-algebras
Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minimal in the sense that ...
- 2,361
3
votes
1
answer
656
views
entire functions of one complex variable with prescribed value and order.
In the complex plane, say $a_n \rightarrow \infty$ and $d_n$ and $A_n$ are arbitrary complex numbers. can we find an entire function with $f(a_n)=A_n$ with order $d_n$? (here "order" means $f(z)-A_n$ ...
- 637
2
votes
1
answer
722
views
How to study the nonregular part of a finite branched holomorphic covering?
A finite branched holomorphic covering is a holomorphic map $f : V \to W$
between holomorphic varieties $V$ and $W$ such that
$f$ is a finite branched covering (in the topological sense)
There is a ...
- 1,111
3
votes
1
answer
503
views
When an AW*-algebra is a W*-algebra
In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence:
It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$.
...
- 537
4
votes
1
answer
293
views
Deforming Fredholm sections
Suppose we have a Fredholm section S, (the differential is Fredholm at the 0-set) of some
Banach vector bundle over X, transverse to the 0-section, with Fredholm index 1 and such that the 0-set of ...
- 187
3
votes
1
answer
831
views
Are Lefschetz thimbles holomorphic manifolds?
I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function
toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the ...
- 229
2
votes
1
answer
624
views
The perturbed KdV Equation
I'm now studying KdV Equation$$u_t-6uu_x+u_{xxx}=0$$To solve the initial-value problem,we can use method of Lax pair,so we can alter the original problem to the problem of solving out $u$ in the ...
- 61
29
votes
2
answers
1k
views
Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense
Let $M$ be a compact complex connected [but not necessarily kähler] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^*)^n \to M$$ such that the image is open. Is the image necessarily ...
- 7,338
3
votes
1
answer
254
views
gluing along a real analytic manifold
hi,
I have a general question. Assume we have a real analytic $n-$dim. manifold $X$ and $M$ a real analytic compact submanifold of $X$ (of dimension less that the dimension of $X$, say $k < n$). ...
- 89
6
votes
2
answers
320
views
Integration under functional sign
Let $f(x,y)$ be some bounded with its derivatives continuous function on $\Omega \times \overline{\Omega}$, where $\Omega$ is a domain in $\mathbb{R}^n$. Let $f(\,\,\cdot\,,\,y) \in \mathcal{E}(\Omega)...
- 1,329
1
vote
1
answer
780
views
An asymptotic series for the digamma function
As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number.
$$
\psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}}
$$
$B_n$ is the first Bernoulli numbers.
How ...
- 1,551
8
votes
2
answers
2k
views
(sharp)Garding's inequality and inequality with lower bounds
The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part $$P_{2m}...
5
votes
1
answer
391
views
About the quantum spectrum of a certain potential.
Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum ...
- 3,541
3
votes
1
answer
243
views
$\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$
The following series evaluation
$\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$
seems attractive to me, and has a proof related to the evaluation of $\zeta(2)$.
Does this ...
- 17.6k
4
votes
2
answers
644
views
Does there exists a necessary condition for Lp multiplier?
Let $1 \leq p \leq 2$. A measurable function $m(\xi)$ is called a $L^p(R^n)$ ($L^p$ for convenience) multiplier, if $$\|m(D)\varphi\|/\|\varphi\|_{L^p} \leq C , \varphi \in L^p
$$ for some constant $C$...
- 425
3
votes
0
answers
254
views
Ways to establish equality of measures on locally compact spaces
Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...
- 1,329
1
vote
1
answer
350
views
Strong convergence in reflecxive Banach space
Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f_n \rightarrow f$ weakly in $X$, and $\|f_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly ...
- 425
11
votes
1
answer
1k
views
Stone-Weierstrass analogue for $L^p$
Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...
- 109k
3
votes
1
answer
303
views
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
1
vote
1
answer
318
views
Uniformly continuous functions and Borel hierarchy in the compact-open topology
Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}_u(\...
- 7,817
1
vote
0
answers
1k
views
Can you prove the monotonicity of the function (or find a counter example)?
Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative ...
- 11
4
votes
3
answers
729
views
Inequality of von Neumann for more than two contractions
Good morning,
I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized ...
- 758
4
votes
4
answers
435
views
Must Neuman Elliptic operator has discrete spectrum ?
It is well known that the Neuman eigenvalue problem has discrete spectrum and the eigen values are
nonnegative and can be arranged in a nondecreasing order of magnitude.
Do we need any smoothness ...
- 41
0
votes
2
answers
146
views
representation of compact supported distribution
Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
- 9
5
votes
2
answers
1k
views
Polar decomposition in C*-algebras
A very nice feature of W*-algebras is the following:
once you have an element $a$ of a W*-algebra $M$, and $a=u|a|$ (the polar decomposition), then $u\in M$.
It seems that it carries over to AW*-...
- 537
2
votes
1
answer
235
views
ODE for functions with values in locally convex TVS
Given an ODE for a function $u \in C^1(I,V)$, where $V$ is some locally convex TVS (topological vector space) and $I \subset \mathbb{R}$, i.e.
$\frac{d}{dt} u = f(t,u)$
for some function $f: I \...
- 403
6
votes
1
answer
409
views
Mandelbrot set and analytic functions such that $f(az)=f(z)^2+c$
It is well known that the function $f(z)=2\cos(\sqrt {-z})$ (or more accurately the entire function $f(z)=2\sum_{n=0}^\infty \frac{z^n}{(2n)!}$) satisfies such a functional equation, i.e. $f(4z)= f(z)^...
- 3,630
3
votes
1
answer
177
views
If $A \subset X'$ annihilates only $0$, then $A$ is dense
Let $X$ be a Banach space with continuous dual space $X'$ with norm topology. Let us regard the following property of $X$:
Property: Any linear subset $A \subset X'$ that satisfies $\bigcap_{\alpha\...
- 5,327
1
vote
2
answers
394
views
When LCS is isomorphic to subspace of some function space?
Updated: Following Michael's suggestion, I rephrase the question slightly.
Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function ...
- 101
2
votes
3
answers
896
views
connectedness of the complement of the zero set of a polynomial $P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$
I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?).
Is it possible to ...
- 229
0
votes
2
answers
2k
views
The exponent of self-adjoint operator
If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^k$ is self-adjoint for all positive integer $k$? (I have already known that the conclusion ...
- 1,441
3
votes
2
answers
362
views
Invariant subspaces for compact restrictions
Suppose $Y$ is a closed hyperplane in $X$, so we can write $X=Y\oplus[x_0]$. Let $y_n$ be a normalized basis of $Y$. Define an operator $S:Y\to Y\oplus[x_0]$ by $Sy_n=\alpha_ny_n+\beta_nx_0$, for any $...
- 31
5
votes
1
answer
600
views
Closed operators and duality
Usually we would define a "densely defined, closed operator" on a Banach space $E$ to be a linear map $T:D(T)\rightarrow E$, where $D(T)$ is a dense subspace of $E$, and the graph of $T$, $G(T)=\{ (x,...
- 18.7k
3
votes
1
answer
502
views
Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
- 11.5k
6
votes
1
answer
879
views
Bochner's theorem, in stages
Bochner's theorem (for the real line version) asserts an infinite tower of inequalities, as a positivity condition. Taking each one, what do they mean, in an elementary fashion (at least at the start)?...
- 275