**4**

votes

**1**answer

153 views

### Reference for a Heat Process in a Wedge

I would like to ask about an explicit suggestion/reference for the following type of heat processes:
Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...

**9**

votes

**1**answer

90 views

### A generalisation of $C_0$-semigroups

A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an ...

**6**

votes

**1**answer

281 views

### Embedding of real trees into $\ell_1(\Gamma)$

It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space ...

**1**

vote

**0**answers

135 views

### On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $

This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) ...

**1**

vote

**1**answer

124 views

### Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?

Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...

**2**

votes

**1**answer

155 views

### Is the set of entire functions Borel in the space of analytic functions?

$\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, I ask whether $S$ be a Borel set in the topological space $(\Omega,\scrT)$ in the following ...

**25**

votes

**1**answer

698 views

### Stone-Weierstrass theorem for holomorphic functions?

The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called
Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...

**3**

votes

**0**answers

57 views

### A measurable implicit function with differentiated arguments

I have encountered the need for an unusual implicit function theorem, about which I know very little. I would appreciate it if someone could help me with a few pointers.
The setup is as follows. Let ...

**5**

votes

**1**answer

199 views

### $C^{*}$-correspondences viewed as generalized endomorphisms

I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...

**0**

votes

**1**answer

92 views

### gradient descent in space of functions

Differential equations of the form
$$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$
can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, ...

**1**

vote

**1**answer

101 views

### example of an $\ell_1$-saturated Banach space without an unconditional basis

Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}_p$ which is complementably $\ell_p$-saturated but ...

**11**

votes

**3**answers

675 views

### Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham.
Solution of an invariant subspace problem of K. T. Smith and
P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in
finding ...

**6**

votes

**0**answers

86 views

### Categorical description of dense homomorphisms of topological algebras

Let $A$ and $B$ be topological associative algebras (no matter, in which sense, for example, over $\mathbb C$, with identity, and with separately continuous multiplication). Let us say that a ...

**4**

votes

**0**answers

164 views

### Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...

**2**

votes

**0**answers

59 views

### How sensitive are the n-th step transition probabilities of the simple random walk to a small perturbation of an infinite graph?

Suppose that $G$ is an infinite, locally finite, connected graph.
Fix a vertex $o$ in the graph and for each $n$ and $x$ let $p(n,o,x)$ be the probability that a simple random walk (at each step a ...

**0**

votes

**0**answers

34 views

### Besov regularity of càdlàg functions?

Let $D(\mathbb{R})$ be the space of functions from $\mathbb{R}$ to $\mathbb{R}$ that are right continuous with left limits (also referred to as càdlàg functions). $D(\mathbb{R})$ is often called the ...

**4**

votes

**1**answer

374 views

### A continuous path between two Sobolev functions

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in ...

**2**

votes

**1**answer

124 views

### How do we know the map is $w^{*}$-continuous?

I am reading a paper by David Blecher, which contains the following:
" If $T: Y \to Z$ is a surjective isometric module map between $W^{*}$-modules over $M$, then $T$ is unitary. Also, $T$ is a ...

**1**

vote

**0**answers

222 views

### The closure of a set of closed points

Let $X $ be a compact non-Hausdorff topological space with the following property: for every infinite subset of closed points, say $\{x_i\}_{i \in I}$, there exists $j\in I$ such that $x_j$ is in the ...

**10**

votes

**3**answers

558 views

### Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$

Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...

**7**

votes

**2**answers

148 views

### Characterizing when matrices are 'dissipative'

An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...

**1**

vote

**2**answers

232 views

### Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...

**4**

votes

**1**answer

138 views

### An interpretation for filters of subspaces in Banach spaces

Let $X$ be a separable infinite-dimensional (real or complex) Banach space.
Call a collection $\mathcal{F}$ of closed subspaces of $X$ a filter if it is nonempty, does not contain $\{0\}$, is closed ...

**2**

votes

**1**answer

173 views

### Can we Characterise Rings of Continuous Functions?

Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...

**0**

votes

**0**answers

54 views

### Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...

**4**

votes

**2**answers

269 views

### Extending inequality for $\ell^p$ from integer $p$ to real $p$

Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are two decreasing sequences of positive numbers such that
$a_1<b_1$ and
$$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ ...

**2**

votes

**0**answers

85 views

### Has every Lusin vector space a stronger Polish vector space topology?

Let $X$ be a topological vector space or even a locally convex space such that its (vector space) topology is Lusin, i.e. there is some stronger Polish topology. Does there also exist a stronger ...

**0**

votes

**1**answer

71 views

### On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...

**3**

votes

**1**answer

130 views

### A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...

**3**

votes

**0**answers

73 views

### Lower semi-continuity of the Hellinger-Fisher-Rao distance

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
...

**12**

votes

**2**answers

339 views

### When is the closed unit ball in a smaller Banach space closed in a larger Banach space?

Recently I saw an interesting lemma:
For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in ...

**1**

vote

**0**answers

67 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...

**8**

votes

**3**answers

359 views

### Thin large subspaces of $\ell^N_1$

Consider a sequence $V_N$ of subspaces of $\ell^N_1$ so that $\dim V_N = N- n$ and $n$ is $\mathsf{o}(N)$. Is it true that these spaces are "thick" (unofficial terminology), i.e. are there constants ...

**2**

votes

**0**answers

88 views

### A modification of Minty's trick?

I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$
where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? ...

**9**

votes

**2**answers

319 views

### Completeness of nonharmonic Fourier Series

I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system ...

**4**

votes

**0**answers

93 views

### Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...

**0**

votes

**0**answers

48 views

### Left introversion operators associated to function spaces on semigroups

I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...

**3**

votes

**2**answers

149 views

### Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$
Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
...

**4**

votes

**0**answers

333 views

### Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...

**0**

votes

**0**answers

32 views

### How the diffusion in the unit ball induce the boundary process on the boundary directly?

This is Example 1.2.3 from Fukushima, Masatoshi, Oshima, Yoichi and Takeda, Masayoshi's book "Dirichlet Forms and Symmetric Markov Processes".
In this example, we define a Dirichlet form in the unit ...

**3**

votes

**0**answers

134 views

### Proof without distributions

I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...

**5**

votes

**1**answer

98 views

### Is the Feichtinger's algebra $(S_0(\mathbb{R^d}),||\cdot||_{S_0})$ reflexive?

The Feichtinger's algebra
$S_0(\mathbb{R^d})=M^{1,1}(\mathbb{R^d}):=\{f\in L^2(\mathbb{R^d}):V_g(f)\in L^1(\mathbb{R^{2d}})\}$, where
$V_g(f)(x,\omega)$ is the short-time Fourier transform of $f$ with ...

**9**

votes

**1**answer

246 views

### The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, ...

**6**

votes

**0**answers

197 views

### Commutation preserving operators

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...

**3**

votes

**0**answers

104 views

### Eigenvalues and eigenvectors of the q-Bernstein operator

The Bernstein operator maps $f\in C[0,1]$ to its Bernstein
polynomial $B_n f.$ The eigenvalues and eigenfunctions of the
Bernstein operator on $C[0,1]$ have been described in [1]. Similar description ...

**1**

vote

**0**answers

79 views

### A version of the Weak Regularity Lemma

Definitions: Given a graph $G$ and $S$, $T \subseteq V(G)$, let $e_G(S, T)$ denote the number of edges of $G$ with one endpoint in $S$ and the other in $T$ and let
$$d_G(S, T) := \frac{e_G(S, ...

**4**

votes

**1**answer

127 views

### Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure

I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question.
Recently I was reading a book "Operator Function and system" ...

**0**

votes

**1**answer

143 views

### Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of ...

**1**

vote

**0**answers

71 views

### One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...

**0**

votes

**1**answer

113 views

### Nonstable $K$-theory question

Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$.
Question A: Is ...