**0**

votes

**0**answers

107 views

### Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq ...

**6**

votes

**1**answer

190 views

### Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...

**3**

votes

**1**answer

231 views

### Is $L(\ell_2,\ell_2)$ dense in $L(\ell_2,c_0)$?

Let $\ell_2:=\{ x:\mathbb{N} \to \mathbb{R}: \sum_{j=1}^\infty x_j^2<\infty\}$ and
$c_0:=\{ x:\mathbb{N} \to \mathbb{R}: \lim_{j\to\infty}x_j=0,\, \sup_{j\in\mathbb{N}}|x_j|<\infty\}$ denote the ...

**0**

votes

**0**answers

154 views

### Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...

**3**

votes

**2**answers

153 views

### Is every Montel locally convex vector space compactly generated?

Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that $X$ is a semi-Montel space if every ...

**1**

vote

**0**answers

47 views

### Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon ...

**3**

votes

**1**answer

114 views

### Has uniform ellipticity implications on the spectrum?

Let $X$ be a complete Riemann surface with a smooth metric, and $L$ a line bundle on it also equipped with a smooth metric; associated to this data there is a Laplace-Beltrami operator $D_L$ acting on ...

**4**

votes

**1**answer

90 views

### A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued ...

**2**

votes

**0**answers

116 views

### Two isomorphic Gelfand triplets, is there a problem?

For $j=1, 2$, let $V_j \subset H_j$ be a dense and continuous embedding with $V_j$ a Banach space and $H_j$ a Hilbert space. Identify $H_1 = H_1^*$ and $H_2 = H_2^*$ (using Riesz representation) so ...

**7**

votes

**0**answers

134 views

### What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...

**1**

vote

**1**answer

135 views

### Estimates on evolution operator

Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert ...

**12**

votes

**1**answer

251 views

### Discrete subsets in the topology of pointwise convergence vs. metrisability

While reading Arkhangel'skii's Topological function spaces, I encountered an unexpected application of Martin's Axiom. This is Theorem II.5.20:
Assume $\mathsf{MA}+\neg \mathsf{CH}$. Let $X$ be a ...

**5**

votes

**0**answers

102 views

### A Banach space with all Hilbertian subspaces complemeneted

Assume that $X$ is a Banach space in which every Hilbertian subspace is complemented (let's say that all the projections are uniformly bounded). What can we say about $X$? It has to be K-convex. By ...

**13**

votes

**2**answers

375 views

### What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$

Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...

**0**

votes

**1**answer

130 views

### Relationship between LlogL and Hardy spaces

I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in ...

**1**

vote

**0**answers

74 views

### Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb ...

**3**

votes

**1**answer

166 views

### Is there a non-compact Poulsen simplex?

A Choquet simplex is a closed, convex and metrizable subset of a locally convex Hausdorff topological vector space in which every point is a barycenter of a unique probability measure supported on the ...

**5**

votes

**0**answers

191 views

### Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space.
We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...

**4**

votes

**1**answer

141 views

### Infinite dimensional subspaces of $L^1$

Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case ...

**11**

votes

**1**answer

351 views

### What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...

**5**

votes

**1**answer

108 views

### The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the ...

**2**

votes

**1**answer

89 views

### ODE system has zero as the only solution?

Let $V \subset H$ be a continuous, compact and dense embedding with $V$ and $H$ Hilbert spaces.
Let $\beta_j:[0,T] \to \mathbb{R}$ be functions for each $j$, and let $v_j$ be a basis of $V_0$.
...

**0**

votes

**0**answers

37 views

### How to generalize balanced and absorbing sets to R-modules?

I'm looking for generalizing the notions of balanced set and absorbing set. The goal is using them for analyzing topological R-modules with R being a unit ring.
It's easy to generalize balanced and ...

**2**

votes

**1**answer

96 views

### Eigenvalues and Compact Resolvent

For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...

**1**

vote

**0**answers

54 views

### Inversion of Fourier transform of a multivariate gamma distribution in polar form?

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...

**0**

votes

**0**answers

42 views

### eigenfunctions for integral operator with difference domains

I have a continuous kernel $k(x,y)>0$ and a given positive number $\lambda$, now I define different integral operators $T(f(x))=\int _{y_1}^{y_2}k(x,y)f(y)dy$ by varying $y_1$ and $y_2$ and keep ...

**2**

votes

**0**answers

78 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

**3**

votes

**1**answer

90 views

### $C_0$ semigroups on parameterized Banach spaces or moving domains

Is there any literature corresponding to one or two-parameter semigroups such that eg. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$???
I ...

**0**

votes

**1**answer

87 views

### Scalar Measure Associated to a Positive Operator Valued Measure

Suppose that $X$ is a compact metric space, $\mathcal{H}$ is a Hilbert space, and that $A: \mathcal{B}(X) \rightarrow \mathcal{B}(\mathcal{H})$ is a Positive Operator Valued Measure (POVM), where ...

**4**

votes

**1**answer

122 views

### On the induced matrix norm $\| \cdot \|_{2,\infty}$

The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^m, \| \cdot \|_q)$ is given by
$$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} ...

**2**

votes

**0**answers

56 views

### question about a genralized Skorokhod topology

Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
...

**3**

votes

**0**answers

93 views

### Convergence in energy of bounded (semi)subharmonic functions

Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...

**1**

vote

**0**answers

66 views

### Semigroups on Banach Lattice

Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that
$$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$
Where $X_+$ denotes the positive ...

**0**

votes

**0**answers

94 views

### ellipsoids have spherical section

I want to prove that "For any $(2k-1)$-dimensional ellipsoid $E$ ,there is a $k$-flat $L$ passing through the center of $E$ such that $ E \cap L$ is a Euclidean ball. I see a proof for it in the book ...

**3**

votes

**0**answers

78 views

### Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?

Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of
$$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$
where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...

**2**

votes

**2**answers

98 views

### First order pde with characteristics [closed]

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...

**1**

vote

**0**answers

53 views

### Reference needed for Hilbert-Schmidt result regarding basis of $V \subset H$

I am seeking a reference that says:
If $V \subset H \subset V^*$ is a Gelfand triple with all spaces Hilbert spaces and if $V \subset H$ is a compact embedding, then there is a basis of $V$ which ...

**0**

votes

**0**answers

121 views

### extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...

**2**

votes

**1**answer

98 views

### Characterization of a subset of [0,1] $III$

I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to ...

**0**

votes

**0**answers

59 views

### Showing that a particular function from a locally compact Hausdorff group $ G $ to a $ C^{*} $-algebra $ A $ is Bochner-measurable

Suppose that we have the following data:
A $ C^{*} $-algebra $ A $.
A locally compact Hausdorff group $ G $.
A strongly Borel mapping $ \alpha: G \to \text{Aut}(A) $, the automorphism group of $ A ...

**0**

votes

**1**answer

130 views

### A distributional normal derivative for functions in $H^1(\Omega)$

Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...

**5**

votes

**0**answers

148 views

### Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...

**3**

votes

**0**answers

91 views

### Density of function spaces

Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an ...

**6**

votes

**1**answer

402 views

### Is there a generalized Birkhoff ergodic theorem?

Is there a Birkhoff ergodic theorem for two measure preserving transformations $T$ and $S$ where $S\circ T= T \circ S$ so that $\frac{1}{n+1}\frac{1}{m+1}\sum_{i=0}^{n}\sum_{j=0}^{m}f \circ T^{i}\circ ...

**11**

votes

**1**answer

879 views

### Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow.
In non-Hausdorff topology it is standard to ...

**2**

votes

**0**answers

98 views

### Examples for Markov generators with pure point spectrum

I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...

**3**

votes

**2**answers

219 views

### Generalized Hardy-Littlewood-Sobolev Inequality

The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb ...

**4**

votes

**1**answer

137 views

### Partial Isometries Satisfying Cuntz-like Relations

I have a situation where I have a family of partial isometries, $S_i$, for $i=0,...,N-1$, on a Hilbert space $\mathcal{H}$ such that the adjoint maps, $S_i^*$ for $i=0,..,N-1$ are also partial ...

**7**

votes

**1**answer

285 views

### Counting norms on an infinite dimensional vector space

It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...

**3**

votes

**1**answer

228 views

### Characterization of a subset of [0,1] $II$

My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...