# Questions tagged [fa.functional-analysis]

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

5,349 questions

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79 views

### Can we have Levy area for N dimensional process?

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what ...

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**1**answer

123 views

### On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$

Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...

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55 views

### Connectness of $K$ and the existence of non-trivial $M$-summands on $C_0(K)$

For a locally compact Hausdorff space $K$, we denote by $C_0(K)$ the Banach space of continuous functions from $K$ to $\mathbb R$ which vanish at infinity equipped with the usual supremum norm.
For ...

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67 views

### Eigenfunctions of square root of Laplacian in an arbitrary Riemannian manifold?

Please forgive me for my inability to pose a mathematics question properly.
In one dimension the eigenfunctions of Laplacian (simply double derivative) are also eigenfunctions of its square root (...

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178 views

### Maps which are both completely positive and positive

Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...

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32 views

### Predictable Process Question (Da Prato & Zabcyzk 2014)

I've been looking at the 2014 edition of Da Prato & Zabcyzk and the sections on predictable processes. In particular, in their Proposition 3.7 (ii), they assert that if $\Phi$ is adapted and ...

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**1**answer

132 views

### Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...

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**1**answer

266 views

### Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...

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80 views

### Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.
...

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196 views

### Fundamental group and group measure space construction

Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...

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215 views

### Prescribing a gradient direction

Let $\Omega= \{(x,y) : \frac{1}{2} \leq x^2+y^2 \leq 1\}$ and $S = \{(x,y) : x^2+y^2 = 1\}$ the unit circle, and $X=w^{1.\infty}(\Omega;\mathbb{R})$ the space of Lipschitz valued functions. We denote ...

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99 views

### Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...

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69 views

### Openness of invertibility in Fréchet spaces for families parameterized by compact spaces

Consider the following setup. Let $K$ be a compact topological space, $X$ a Fréchet space and $T:K \times X \to X$ a continuous family of linear maps (i.e. $T$ is a continuous map and $T_k \equiv T(k, ...

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90 views

### $\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk

Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup_D |f|\leq1$.
Could ...

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67 views

### Convergence of self-adjoint elements in $\sigma$-weak topology

In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?

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57 views

### Separable measures on compact groups

Let us say that a (signed, finite) measure $\mu$ is separable if $L_1(|\mu|)$ is a separable Banach space.
EDIT: Suppose that $G$ is a locally compact group such that each measure on $G$ is ...

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88 views

### A sufficient condition for subdifferentiability

Corollary 9 in here (page 31) states that a proper convex function $g:Y\rightarrow \mathbb{R}\cup\{\infty\}$ (not necessarily continuous) on a locally convex space $Y$ is subdifferentiable on a point $...

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82 views

### Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding:
\begin{equation}
C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).
\end{equation}
what condition should be put ...

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57 views

### English translation of Schmidt's work on nonlinear integral equations. Part III

Does anybody know about the English translation of Erhard Schmidt's paper
Zur Theorie der linearen und nichtlinearen Integralgleichungen. III Teil. Über die Auflosing der nichtlinearen ...

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142 views

### Is this set in $\mathbb{R}^d$ closed? [closed]

Let $X$ be a convex set (in my case $X$ is infinite dimensional too) and for each $ i \in \{1,2,\dots,d \} $, let $f_{i}: X \rightarrow \mathbb{R} $ be convex functions where $C_{i} := \{ f_{i}(x) : x ...

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433 views

### Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...

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332 views

### What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales:
...the single most important fact which distinguishes locales from spaces: the ...

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58 views

### Best constant for Hölder inequality in Lorentz spaces

It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely
$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...

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153 views

### Anzai flow in noncommutative geometry

Consider Anzai flows (cf. Anzai: Ergodic Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...

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100 views

### Dual of the space of affine functions

Let $M^+(D)$ be the space of all positive measures on a closed convex subset $D$ of a locally convex topological vector space $E$. Two measure $\mu, \nu\in M^+(D)$ one can define a partial ordering $\...

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47 views

### Theta Summable operator with bounded trace

Let $D$ be an unboudned self-adjoint operator on the Hilbert space $H$. We assume that all spectrum of $D$ are eigenvalues and $D$ is theta-summable, i.e. $e^{-tD^2}$ is of trace class for all $t>...

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207 views

### Cartan subalgebra and group measure space construction

Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...

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150 views

### Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$.
Question. Is there a non-compact linear bounded operator ...

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66 views

### How does a statistical divergence change under a Lipschitz push-forward map?

Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space).
$T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...

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31 views

### Specific Bounds on Divergence operator in Sobolev setting

Given that $\vec{u}=(u_1,u_2)\in H^1(\Omega)\times H^1(\Omega)$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded with Lipschitz boundary, does there exist specific inequalities which links $\...

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150 views

### What is the trace of the integral operator $(\mathcal{L}f)(x)=\int_0^\infty (x \wedge y)f(y) \, d \pi(y)$?

Let $\pi$ denote a probability measure on $[0,+\infty)$ and let us assume that
$$m:=\int_0^\infty x \, \mathrm{d} \, \pi(x)<+\infty.$$
Let us consider the integral operator $\mathcal{L}$ on $L_2(\...

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136 views

### Density-$c_0$ in $\ell^\infty$

Let $A \subseteq \mathbb{N}$, define the upper density of $A$ as,
$$
\overline{\delta}(A) := \limsup_{N\to\infty}\frac{|A\cap\{1,2,3,\cdots,N\}|}{N}.
$$
This naturally leads to a weaker form of ...

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117 views

### The perturbation of a convex function can also be convex?

$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...

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994 views

### Intuition about L^p spaces

I have read somewhere the following very nice intuition about $L^p(\mathbb{R})$ spaces.
This graphic shows a lot of nice relations:
1) There is no inclusion between $L^p$ and $L^q$
2) $L^p$ is the ...

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55 views

### Legendre transform on signed measure space

Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...

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81 views

### eta invariant and spectral flow

We know that for a family of first-order self adjoint elliptic (Fredholm) operator $A_t$, for $t\in [0,1]$ we have the formula
$$\eta(A_1)-\eta(A_0)=spfl(A_t)_{t\in[0,1]}+\int^1_0 \omega(s)ds,$$
where ...

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56 views

### Elliptic pde L^p theory via adjoint theory

Let $ T:X \rightarrow Y$ denote some linear operator and suppose we know its one to one (here $X$ and $Y$ are Banach spaces). I believe their is results that say $Ker(T^*)= (R(T))^\perp$ (where ...

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123 views

### For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...

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169 views

### A characterisation of certain $C^*$-algebras

I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...

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153 views

### Integral with product of two infinite sums

I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...

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89 views

### Is $X = \{ B \in L^\infty(\mathbb R^n,\mathbb R^n): \nabla \cdot B \in L^\infty(\mathbb R^n,\mathbb R^n) \}$ a dense subspace?

The Sobolev space $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ is not dense in $L^\infty(\mathbb R^n,\mathbb R^n)$. In fact the functions in $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ are Lipshitz, and not ...

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109 views

### For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...

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124 views

### Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space

Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous ...

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289 views

### Is there a version of Fischer-Riesz theorem for Banach space?

$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...

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38 views

### Interpolation inequalities involving mean curvature operator

Are there any interpolation inequalities (for example, of Gagliardo-Nirenberg type) involving the mean curvature operator
$$\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)$$
(in any ...

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**1**answer

45 views

### Lyapunov-type function in a non locally-compact space and boundedness of the average

Set-up and question.
Let $\mathcal{X}$ be a complete separable metric space which is not locally-compact. Let $V: \mathcal{X} \to [0; +\infty]$ be a function and $(X_t)_{t\geq 0}$ a Markov process in $...

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383 views

### Does every positive continuous function have a non-negative interpolating polynomial of every degree?

Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...

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**1**answer

267 views

### Big list of “outstanding paper awards” [closed]

What are the prizes awarded for significant papers published in mathematical journals?
One of them is the SIAM Outstanding Paper Prize.
I'd be particularly interested in hearing about prizes ...

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**1**answer

59 views

### Examples of isomorphic W* algebra with non-homeomorphic weak topology

Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...

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119 views

### Is it true that every Banach space has at least one extreme point that is normed by some point?

Definition: Let $X$ be a Banach space and $X^*$ be its continuous dual of $X,$ that is, $X^*$ contains all bounded linear functionals on $X.$
Denote
$$B_{X^*} = \{x^*\in X^*: \|x^*\|_{X^*}\leq 1\}.$$...