Questions tagged [fa.functional-analysis]

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

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2 votes
0 answers
188 views

Best approximation of piecewise constant function by Lipschitz functions

Let $f=\sum_{n=1}^N k_n I_{E_n}$ where $E_n$ are Borel subsets of $\mathbb{R}^n$ and $k_n\in \mathbb{R}^m$ with non-negative entries, and let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. What ...
4 votes
0 answers
63 views

A standard name of a strongly extremal point of a convex set

I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
1 vote
0 answers
277 views

On measurability of certain group actions on spaces of bounded measurable functions

Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded ...
1 vote
1 answer
138 views

2-summing vs Hilbert-Schmidt norm for extended operator between Hilbert and Banach space

Let $H$ be a Hilbert space and $b$ a continuous and symmetric bilinear form on $H \times H$, such that the induced operator $T \colon H \to H^{\ast}$ (the star denoting the continuous dual) is Hilbert-...
2 votes
0 answers
86 views

How to use Fredholm alternative to check that there are only finite eigenvalues of $H$ on the imaginary axis?

On $\mathbb{R}^3$, we consider the operator \begin{equation} \mathcal{H}= \left( \begin{matrix} -\Delta +1 -2 \phi^2 & -\phi^2 \\ \phi^2 & \Delta -1 +2 \phi^2 \end{matrix} \right) , D(...
3 votes
0 answers
164 views

'Local' commutativity of self-adjoint operators

Preamble Two (unbounded) self-adjoint operators $A, B$ on a Hilbert space $\mathcal{H}$ are said to (strongly) commute if the unitary groups they generate commute or equivilantely if all the ...
4 votes
1 answer
419 views

Bochner-Minlos for moment-generating functions?

It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions? I have ...
0 votes
0 answers
135 views

Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$

How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
5 votes
1 answer
656 views

Embedding of a Banach space into a Hilbert space

Let $\mathbb H$ be a Hilbert space and let $\mathbb B$ be a Banach space continuously embedded in $\mathbb H$ and distinct from $\mathbb H$. Is it true in general that $\mathbb B$ is an $F_\sigma$ of ...
4 votes
1 answer
200 views

Mapping properties of backward and forward heat equation

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...
1 vote
0 answers
80 views

determine the spectrum of the operator [closed]

determine the spectrum of the operator $$-\Delta-5\phi^4,\phi=3^{1/4}(1+|x|^2)^{-\frac{1}{2}},x\in R^3$$
4 votes
0 answers
140 views

Continuous extension preserving modulus of continuity

Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any ...
3 votes
1 answer
163 views

Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists

The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as $$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
5 votes
1 answer
445 views

English translation of Schwartz's papers on vector-valued distributions

I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
0 votes
1 answer
70 views

Reverse Hölder type inequality for the Laplacian raised to a power

I am studying integrals of the form $\int (\Delta \rho)^{\alpha} f^{\beta}$ where $0<\alpha < 1, \beta \geq 0$ and $\rho, f \in C_c^{\infty}(\mathbb{R}^n).$ My goal is to obtain lower bound on ...
1 vote
0 answers
32 views

Condition on Functions on Overlapping patches of a Domain

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain. Let $\{p_i\}_{i=1}^m$ for some finite $m\in\mathbb{N}$ be the overlapping open sets of $\Omega$ such that $\Omega =\bigcup_{i=1}^mp_i$. Now, ...
44 votes
1 answer
2k views

Existence and uniqueness of Haar measure on compacta; a cohomological approach

I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group. I think the best way of introducing the idea I am pursuing is ...
2 votes
0 answers
50 views

Is this Beppo-Levi curl space a Banach space?

Let us define the quotient space: $$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
5 votes
2 answers
397 views

Backward heat equation and forward perturbed heat equation well posed?

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
0 votes
0 answers
89 views

Rigorous energy estimate for advection-diffusion equation

Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and $q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$ $q \in [2,4], p \in [2,\infty] \text{ if } N = 1$ and consider the ...
1 vote
0 answers
137 views

Liouville theorem for elliptic equation with advection term

How can one prove that any $L^2$ solution of $$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$ is zero if $a(x)$ is a divergence-free vector field such that $\int |\...
0 votes
0 answers
129 views

Eigenvalues of the Laplacian and min-max formula in any space dimension

In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by $$ \lambda_1 = \min_{u \in H^1_0(\Omega), \|...
1 vote
0 answers
60 views

Discrete Orlicz space estimate

We consider the discrete LlogL space of sequences $x=(x_i)$ such that $$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$ Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL ...
1 vote
1 answer
127 views

Calculation of the norm of linear combinitation of two states on a $C^*$-algebra

Let $A$ be a unital $C^*$-algebra. Suppose $\rho_1$ and $\rho_2$ are two states on $A$. If $\rho_1=\rho_2$, we have $\|\rho_1+i\rho_2\|=\sqrt{2}$. If we have $\|\rho_1+i\rho_2\|=\sqrt{2}$, can we ...
-4 votes
1 answer
357 views

Is delta function symmetric against real axis? [closed]

Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$? I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis. We can write Delta function as $$\delta(z) = \...
2 votes
0 answers
101 views

Strongly continuous semigroups on weighted $\ell^1$ space

Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$ Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
7 votes
1 answer
540 views

Is this operator bounded?

Let $T$ be an invertible positive operator and $S$ be another positive operator on a complex Hilbert space. We then study $$ \Vert (T+S)^{-1/2}T(T+S)^{-1/2}\Vert$$ I would assume that this norm is ...
2 votes
0 answers
433 views

$\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$

For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...
7 votes
0 answers
239 views

Has this Banach algebra been studied?

Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm ...
6 votes
1 answer
471 views

Almost commuting matrices, one a projection, is there a nearby projection that commutes?

Suppose that $P, A, Q \in \mathbb{M}^{n \times n}(\mathbb{R})$ (I'm still interested if it must be done over $\mathbb{C}$), (EDIT:) suppose that $A$ is given, $P$ is an orthogonal projection, and $\...
3 votes
0 answers
115 views

Clarification about extensions of Ornstein-Uhlenbeck operator

I am reading stuffs regarding the Ornstein-Uhlenbeck operator and its various extensions to $L^p(\gamma)$, with $p \in (1,+\infty)$ and with $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$. ...
0 votes
0 answers
61 views

The set of bounded lipschitz functions over a compact is barrelled but not a neighborhood of zero?

I recently learned that Banach spaces are barrelled, i.e any convex, balanced, absorbing and closed subset is a neighborhood of zero (wikipedia). I'm having trouble understanding why the following ...
1 vote
1 answer
190 views

Existence and regularity for fractional elliptic problem with gradient term: $ (-\Delta)^s u + v\cdot \nabla u = 0$ with $v \in \dot H^s$

Let us consider the problem $$ (-\Delta)^s u + v\cdot \nabla u = 0 \quad \text{ in } \mathbb R^n, $$ where $s \in (0,1)$, $(-\Delta)^s$ is the fractional Laplace operator and $v:\mathbb R^n \to \...
2 votes
0 answers
51 views

Conjugate of composition in Bochner spaces

Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
3 votes
1 answer
139 views

Translation of a paper by Salvatore Pincherle

Anyone know of an English translation of the oft-cited paper "Funktionaloperationen und Gleichungen" by Salvatore Pincherle, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ...
1 vote
2 answers
254 views

How can I prove that $(I+\lambda G)$ is invertible, where $G$ is the Green function of an elliptic operator?

How can I prove that the operator $$(I+\lambda G)$$ is invertible, where $\lambda >0$ and $G$ is the Green function of an elliptic operator $A$ in a bounded domain $\Omega$? $\Omega$ can be very ...
9 votes
4 answers
806 views

Defining the value of a distribution at a point

Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
3 votes
1 answer
1k views

Friedrichs mollifiers and Sobolev spaces

$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
5 votes
1 answer
117 views

The tensor product of two topological complexes with closed range

A Künneth formula by Grothendieck/Schwartz states the following: Let $A, B$ be chain complexes of nuclear Fréchet spaces. If the differentials $d_A, d_B$ are topological homomorphisms (meaning in ...
7 votes
0 answers
158 views

$C^*$ algebras whose nontrivial projections form a non empty compact connected set

Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set? Is there an example of this situation such that ...
0 votes
1 answer
395 views

Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure

I'm trying to figure out the connections between two contructions of Gaussian measure. Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
1 vote
1 answer
138 views

What's the size of non standard monad for weak topology?

There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space): $$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
1 vote
0 answers
58 views

Well-posedness of hyperbolic system with constant coefficients in finite domains

I'm studying the PDE $$ \frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0 $$ with $A_x, A_y, A_z$ being ...
3 votes
1 answer
112 views

The weak*-convergence of the summing basis of $c_{0}$

Suppose that $(x_{n})_{n}$ is a sequence in a Banach space $X$. We let $\textrm{clust}_{X^{**}}((x_{n})_{n})$ be collection of all the weak*-limit points of $(x_{n})_{n}$ in $X^{**}$. Let $(e_{n})_{n}$...
6 votes
1 answer
356 views

Wiener Corollary in "An introduction to harmonic analysis" by Yitzhak Katznelson

I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows: Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\...
0 votes
0 answers
212 views

Definition of tensor product of dense subspaces of Hilbert spaces

Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...
1 vote
0 answers
100 views

Fractional Sobolev embedding theorem

Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds $$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
1 vote
0 answers
63 views

Normalizing constants preserve metric entropy

Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation $$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...
0 votes
0 answers
44 views

Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?

Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
2 votes
0 answers
144 views

Are Weyl sequences polynomially bounded?

Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...

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