All Questions
10,233 questions
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About vector valued measure algebras
Let $G$ be a locally compact group and $A$ be a Banach algebra. By $L^1(G,A)$ and $M(G,A)$ we denote the $A$-valued group, and measure algebra.
Is $M(G,A)$ a Banach algebra (with convolution as the ...
- 2,118
4
votes
1
answer
418
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Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 43.2k
5
votes
2
answers
276
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Dilation of bounded linear operators
Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
- 51
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1
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44
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Identify maxima for 2-Dimensional Function without knowing cross-derivative
I am trying to proof the uniqueness of a maximum for a two-dimensional function (well behaved, twice differentiable, domain $R^2$, etc.), yet cannot compute the exact derivatives or the Hessian.
I ...
- 103
1
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0
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70
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Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 39
1
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1
answer
164
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Complex interpolation of subspaces
Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\...
- 1,879
5
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1
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256
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Is $k(A, B) = \text{Tr}[(A^{1/2} B A^{1/2})^{1/2}]$ a positive definite kernel?
Let $\mathbb{S}_n$ denote the set of $n \times n$ symmetric positive semidefinite matrices. I am trying to figure out whether $k: \mathbb{S}_n \times \mathbb{S}_n \to \mathbb{R}_+$ defined as:
$$k(A, ...
- 204
2
votes
1
answer
77
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Measurability of random function with values in $C(K,E)$
Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random ...
2
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2
answers
281
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Most general reverse Hölder inequality for polynomials
Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981
5
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1
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205
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Continuity of the extension of a tracial state with respect to the strong operator topology
Problem: Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\...
- 85
3
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1
answer
100
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Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?
Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
- 315
1
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1
answer
103
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Are there PDEs in which Hessian appears in the weak formulation
Before stating the question, I would like to first use an example for the type of formulation that I'm interested in.
Suppose we consider the continuity equation $\partial_t \rho + \mathrm{div}( \rho ...
- 820
8
votes
1
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245
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Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
- 81
3
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0
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198
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On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
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120
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Smooth vectors of unitary representations and reducibility
A representation $(\pi,V)$ of a locally compact group $G$ is unitary if $V$ is a Hilbert space endowed with a $G-$invariant bilinear form. In general, a unitary representation can't be expressed as a ...
- 103
1
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0
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120
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Is a discrete harmonic function bounded below on a large portion of $\mathbb{Z}^2$ constant?
In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates ...
- 543
5
votes
2
answers
665
views
Separate continuity implies (joint) continuity
I believe that the following fact is true and I am looking for a reference.
Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V$, $W$ be Fréchet spaces.
...
- 21.8k
3
votes
1
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226
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$\tau$-measurable operator
Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
- 85
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2
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166
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Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions
Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by
$$
F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}.
$$
It is clear that $F$ is strictly ...
- 6,853
1
vote
1
answer
180
views
Shape, shift and scaling retrieval of a sampled function
Let $f(x)$ be some unknown continuous square-integrable function defined on the interval $[0,1]$.
Suppose we have $i=1,...,n$ samples $f_i$ of $f$ of the following form:
$$f_i(x)=a_i*f(x+b_i)+c_i$$
...
- 230
2
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0
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114
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A maximum principle in $\mathbb{R}^N$
Let $\delta > 0$ and define
$$
H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N.
$$
By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
- 167
2
votes
1
answer
369
views
For a Banach space $X$, can we find a reflexive (or weakly sequentially complete) space $Y$ such that $X\subset Y$?
It could be a naive question. Probably, it is not true.
However, this question makes sense in the setting of function spaces.
For example, for $L_\infty (0,1)$, we have $L_p(0,1)\supset L_\infty (0,1)$...
- 617
1
vote
0
answers
64
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Sequential Hölder-norm for functions in $H_{\alpha}([0,1]^{d})$?
I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even ...
- 149
1
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1
answer
126
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Any example of a multi-valued monotone maximal operator without subdifferential?
Is there an example of a multivalued maximal monotone operator that is not the convex subdifferential of a proper convex lower semicontinuous? Besides, among these type of operators, are there any ...
- 395
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56
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Whether $d_x(t) := \|P_t(x)-x\|_H$ is increasing in $t$ where $P_t:H \to H$ is the proximal operator of a convex function
Let $H$ be a Hilbert space (e.g Euclidean $\mathbb R^n$), and fix a proper convex function $f:H \to (-\infty,+\infty]$. Given any $t \ge 0$, let $P_t:H \to H$ be the proximal operator of $f$ at level $...
- 6,853
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votes
0
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92
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Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
1
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1
answer
117
views
On the existence of a countable dense family in "increasing" pointwise convergence
Suppose that $(X,d)$ is a locally compact metric space and $\mu$ is a $\sigma$-finite Radon measure on the Borel sigma-algebra of this space. I am aware that if $(X,d)$ is separable and $\mu$ has full ...
2
votes
2
answers
290
views
Domain of Schrödinger operators
Let $S$ be a Schrödinger operator on $\mathbb{R}$, $Su=-u''+Vu$ with $V\geq1$ continuous and going to $+\infty$ at infinity (you can think of it as $x^2+1$). I wondering which assumptions do I have to ...
- 41
2
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1
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529
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Difference in tracial and finite von Neumann algebras
A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful ...
- 163
2
votes
1
answer
455
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Existence of conditional expectations map onto subalgebras
Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existence of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any ...
- 163
0
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0
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70
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Follow-up question regarding real singular matrices with additional details
After my question whose answer turned out to be false, I re-examined the course of my proof, which is actually seperate from the one in my question, and found out that there's another condition, at ...
- 286
1
vote
0
answers
385
views
Densely defined and closed operator
Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be ...
- 85
1
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0
answers
63
views
$L^2$ norm of a kernel with a variable width
Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
3
votes
1
answer
521
views
Is the set of real matrices with at least one real logarithm closed under multiplication?
Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
- 286
1
vote
0
answers
166
views
About a weak$^*$ convergent net
Let $G$ be a locally compact abelian group and $A$ be semisimple commutative Banach algebra such that $A^{**}$ has Radon-Nikodym property. Denote by $\Gamma$ and $M(G)$ the dual group and the measure ...
- 2,118
5
votes
2
answers
330
views
Functional equation of bounded analytic functions
Let $\mathbb{D}:=\{z \in \mathbb{C}|~|z|<1\}$. Consider $f,~g \in H^{\infty}(\mathbb{D}):=\{f\in\text{ Hol}(\mathbb{D}): \|f\|_{\infty}:= \sup_{z \in \mathbb{D}} |f(z)| < \infty\}$ such that $f^...
- 149
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3
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1k
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Condensed Pontryagin duality
Has Pontryagin duality been extended to condensed abelian groups? The obvious approach being to define $\hat M$ as the internal hom to the circle group. Is it true that $\hat{\hat M}=M$ with this ...
user473423
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0
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106
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Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 286
7
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80
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Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator
Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
- 808
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0
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141
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Support function of the intersection of a hyper-ellipsoid and a Euclidean ball
Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where
$$
E(r) := E \cap B_2^d(r)...
- 6,853
9
votes
4
answers
2k
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How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074
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135
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Reverse Sobolev inequality for family of holomorphic functions
Denote by $P_m$ the space of polynomials of degree $m$ in a single complex variable $x$. This is a "Reverse Sobolev inequality":
Theorem. Let $U \subset \mathbb{C}^2$ be an open domain and $...
- 981
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0
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143
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Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
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0
answers
62
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Proving the uniqueness of the solution to a functional equation involving integral
Consider the functional equation
$$
g\left(a\right) = \int_0^1 \frac{e^{c(a,h)+f(h)}}{1+e^{c(a,h)+f(h)}}dh
$$
and this holds for all $a$. $g(a)$ and $c(a,h)$ are known functions on a continuous ...
0
votes
0
answers
168
views
Completely continuous maps from projective tensor products into $c_0$
Let $E$, $F$ be two Banach spaces and $E\mathbin{\hat{\otimes}}_{\pi}F$ denote their projective tensor product.
For each unit norm $\xi\in E$ and $\gamma\in F$, let's define
$$
J_{\gamma}:E\to E\...
- 2,605
1
vote
1
answer
252
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GNS Representation — A theorem from Thirring’s book
After the GNS representation for $C^{*}$-algebras is presented in Thirring's book Quantum mathematical physics, the author states the following theorem.
The Spectral Theorem: For any given Hermitian (...
- 1,305
2
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0
answers
73
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RKHS lying in another RKHS
Suppose $H_1$ and $H_2$ are reproducing kernel Hilbert spaces such that $H_1 \subset H_2$. For $f \in H_1$, when can I bound $\|f \|_1$ with $C\|f\|_2$ (for some $C$)? Is there a relationship between ...
- 93
3
votes
1
answer
451
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Topological vector spaces in direct sum
A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow.
This question had emerged as an offshoot of a bigger ...
- 603
2
votes
1
answer
60
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Specific estimation of the norm for a linearly transformed function in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$
According to the standard definition, $\mathcal{S}_0^{\beta}(\mathbb{R})$ is a subspace of smooth functions on $\mathbb{R}$ with the property that
\begin{equation}
\lvert x^k f^{(q)}(x) \rvert \leq CA^...
- 3,477
1
vote
0
answers
85
views
Spectral projection with height less than $\lambda$
Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$...
- 1,879