Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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4
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1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
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3answers
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
0answers
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
0answers
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
0answers
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
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0answers
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
1answer
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
1answer
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 ...
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vote
0answers
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
3answers
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
2answers
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
2answers
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
1answer
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
1answer
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
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0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
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0answers
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
2answers
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
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0answers
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
3answers
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
0answers
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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 ...
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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
1answer
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
1answer
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 ...
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vote
0answers
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
1answer
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 ...