All Questions
12,780 questions
15
votes
1
answer
1k
views
Convolution algebras for double groupoids?
There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
- 12.3k
4
votes
0
answers
166
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 ...
- 93
0
votes
1
answer
666
views
A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
- 197
3
votes
1
answer
358
views
Reference Request: Hamiltonian and quantum completeness.
Let $L$ be a differential operator in $L^2(M, dvol)$ wrt to a Riemannian volume form (say). Let us call it quantum complete if it is essentially-self-adjoint. Consider $H$ - the symbol of $L$. It is a ...
- 329
26
votes
3
answers
2k
views
About the category of von neumann algebras
I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category ...
- 357
0
votes
1
answer
717
views
Interpolation of derivatives
If $U$ is an open interval of $\mathbb{R}$ and $f : U \to \mathbb{R}$ is an $L^2(U)$ function with second derivative $f'' \in L^2(U)$ (in the weak sense), is $f \in W^{1,1}(U)$?
EDIT: Removed false ...
- 3
5
votes
1
answer
762
views
Schwartz Kernel theorem for tempred functions
Let $T(R)$ denote the space of tempered functions on the line,
i.e. the smooth functions that give Schwartz function after a
multiplication by any Schwartz function, equipped with the natural
nuclear ...
- 2,649
3
votes
1
answer
613
views
The Levi form of the distance squared function in a non-positively curved Kaehler manifold
Suppose that if $X$ is a complete, simply connected Kaehler manifold with non-positive sectional curvatures. Let $P \in X$ and $h : X \to \mathbb{R}$ be the function defined by $h(x) = dist(P,X)^2$. ...
- 75
7
votes
1
answer
1k
views
Exactness of completed tensor product of nuclear spaces
Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence
of (complete) nuclear spaces, i.e. it is a short exact sequence of
(complete) nuclear spaces, all the maps are continuous, the map $...
- 2,649
5
votes
2
answers
944
views
Realize a homomorphism $\mathcal{C}(X) \to \mathbb{R}$ as an evaluation
Let $X$ be a compact Hausdorff space. It is well-known that every homomorphism $F : \mathcal{C}(X) \to \mathbb{R}$ is the evaluation $f \mapsto f(x)$ at some point $x \in X$. The usual proof is not ...
- 63.1k
23
votes
0
answers
1k
views
Is analytic capacity inner regular?
For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by
$$\gamma(K) := \sup |f'(\infty)|$$
where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
- 2,154
8
votes
3
answers
2k
views
Measures on general topological groups
I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...
- 1,152
5
votes
1
answer
207
views
If self-composition of an analytic function on the disk gives a linear fractional map, is the original function linear fractional?
Suppose $f: \mathbb{D} \rightarrow \mathbb{D}$ is analytic. Furthermore, suppose $f \circ f = g$, and $g$ is a linear fractional map. Does this guarantee that $f$ is linear fractional? I know it would ...
11
votes
1
answer
1k
views
Quasi-nilpotent trace class operators as limits of nilpotents
In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to:
If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
- 31.5k
3
votes
1
answer
571
views
Subspaces of a Sobolev space
For $a \in \mathbb{R}^N\setminus\{0\}, N \ge 2$, and $\lambda \in \mathbb{R}$ let
$$
X_{\lambda,a}=\{u(\cdot+\lambda a):\, u(x)=u(|x|) \in W^{1,2}(\mathbb{R}^N)\}.
$$
Denote by $X_a$ the closure of ...
- 397
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 ...
user5810
1
vote
1
answer
240
views
Conditions for differentiability of minima and minimizers of linear functionals?
Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$.
For every continuous linear functional $F$ on $B$, define
$V(F)=min_{c\epsilon C} F(c)$ and
$S(F)= { \lbrace c \epsilon C :...
- 13
4
votes
1
answer
230
views
A convergence problem about integral operator in the space of representations
This would be a basic problem in representation theory.
Let $G$ be a unimodular real Lie group, $(\pi,V)$ a smooth representation of $G$ in a Frechet space $V$. Let $f$ be a smooth function on $G$. ...
- 2,709
7
votes
0
answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
8
votes
1
answer
656
views
When is the adjoint of a hypoelliptic operator also hypoelliptic?
Suppose that $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.
Recall that $L$ is a hypoelliptic differential operator if for ...
- 329
0
votes
0
answers
169
views
Is degree a "strict -transform" birational invariant for surfaces in the complex projective 3-space?
(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...
15
votes
1
answer
1k
views
Intersection of complemented subspaces of a Banach space
The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here.
Question. Let $X$ be a Banach ...
- 60.6k
2
votes
0
answers
242
views
Core of divergence form operator with unbounded coefficient
Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x)...
- 617
1
vote
0
answers
2k
views
Derivative total variation function
Hello everyone,
I would like some help on proving the following statement:
Let $f\in\mathrm{BV}[a,b]$, i.e. $f$ is of bounded variation and let $T(x)$ be the total variation of $f$ on $[a,x]$ for $x\...
- 11
2
votes
1
answer
544
views
Approximation of the radon-derivative
I am looking for the following statement.
Let $X$ be a topological space and let $\mu$, $\nu$ be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by $...
- 21
5
votes
2
answers
1k
views
Cohomology class of a current
Hello,
i still have a question about positive closed currents. In particular i know that if $X$ is a compact complex manifold and $T$ is a positive closed current of bidegree $(1,1)$ such that
its ...
- 75
12
votes
1
answer
1k
views
Decomposition of positive definite matrices.
It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...
12
votes
2
answers
1k
views
Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 123
0
votes
0
answers
104
views
Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
5
votes
0
answers
1k
views
Generalized Stone Weierstrass theorem
Given a smooth function $f$ on some compact $K$ in the euclidean space $\mathbb{R}^d$, does exist a sequence of polynomial functions $p_n$ such that $p_n$ and all of its derivatives converge uniformly ...
- 367
2
votes
0
answers
262
views
A specific projection and compactness on the Bargmann-Fock space
Let $F_2$ be the Bargmann Fock space defined as the space of entire functions $f$ on $\mathbb{C}$ such that \begin{align*} \int_{\mathbb{C}} |f(z)|^2 e^{- |z|^2} dA(z) \end{align*} ($dA$ is just ...
2
votes
1
answer
362
views
Quasinilpotent example [duplicate]
Possible Duplicate:
Quasinilpotent operator
Do you know any example of a quasinilpotent operator such that every its power is non-compact?
Of course direct sum of nilpotent operators(or Volterra ...
- 21
0
votes
1
answer
438
views
Möbius Transform of a Continuous Possibility Function
In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.
There is ...
- 103
5
votes
2
answers
412
views
Wiener Tauberian Theorem for nonunimodular group
Is there a nonunimodular group for which Wiener's Tauberian theorem is true?
Is a locally compact topological group whose volume grows polynomially with radius always unimodular?
- 415
3
votes
1
answer
740
views
Centralizers in C*-algebra
Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$?
For ...
- 179
2
votes
2
answers
551
views
L^2 basis of class functions on a compact Lie group that are point-wise small
Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...
- 4,466
1
vote
0
answers
305
views
Strong minimum principle for maximal plurisubharmonic functions
Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
- 1,211
32
votes
3
answers
2k
views
How is the Julia set of $fg$ related to the Julia set of $gf$?
Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f \...
- 27.7k
0
votes
0
answers
138
views
Notion of simplicity of a function(al)
Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).
Specifically, intuitively one could ...
- 59
26
votes
4
answers
5k
views
Can $L^{2}$ be represented as a space of functions (not equivalence classes)?
Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
- 4,597
34
votes
1
answer
4k
views
Theme of Isbell duality
Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
- 63.1k
4
votes
0
answers
820
views
Calderón's complex interpolation: what is the corresponding classical theorem?
This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of exact interpolation functor of exponent $\theta$ is ...
- 1,202
0
votes
1
answer
183
views
graph of the size of a complex function [closed]
Hi
Here there are two graphs for two functions from $R^2\mapsto R$.
Is there similar graph for the absolute value of a complex variable function $f:C\mapsto C$ that has the same point (like saddle ...
- 163
21
votes
2
answers
2k
views
Uncertainty principle and Cramer-Rao bound - is there relation?
Just out of curiosity.
The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer ...
- 24.9k
4
votes
1
answer
287
views
Second conjugate operators to operators on $c_0$
I posted my question at MS but unfortunately it is still without a response, so let me ask it here.
We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,...
- 51
7
votes
3
answers
1k
views
Gamma-function analogues for Gauss sums
I have a Gauss sum, which I have to calculate. I have heard that it has an analogues form with the Gamma function, but couldn't find its formula shape. It would be so nice of you to help me and write ...
- 71
1
vote
1
answer
435
views
skoda el-mir theorem
now i'm studying the skoda el-mir theorem about the extension of a positive closed current $T$.
But if $T$ ed $S$ are two positive closed currents on a manifold $X$ such that are equal on $X\setminus ...
- 75
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
8
votes
4
answers
1k
views
Monge Ampere equations
I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
- 1,211
1
vote
1
answer
931
views
Finding a counter example for Minkowski's integral inequality for $p=\infty$ [closed]
Dear All,
As we know that this following Miskowski's integral inequality is true for $1\leq p<\infty$
$
[\int_{S_1}|\int_{S_2}F(x,y)d\mu_1(x)|^pd\mu_2(y)dy]^{1/p} \leq \int_{S_2}[\int_{S_1}|F(x,y)...
- 51