Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,435 questions with no upvoted or accepted answers
1
vote
0
answers
125
views
Probabilistic interpretation of von Neumann's approach to quantum mechanics
One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...
- 1,305
1
vote
0
answers
126
views
Non-surjective isometries of $l_p$
It is well known that all surjective isometries of $l_p$ for $p\neq 2$ are the signed permutations of the unit vector basis $(e_n)$. Is there a characterization for the linear non-surjective ...
- 1,361
1
vote
0
answers
89
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
- 111
1
vote
0
answers
96
views
Representation formula for the continuity equation on a separable Hilbert space
The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
- 111
1
vote
0
answers
140
views
Does a Borel transform uniquely determine a Borel measure?
It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question.
I came across the concept of a Borel transform of a Borel ...
- 3,535
1
vote
0
answers
63
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 11
1
vote
0
answers
152
views
Uniqueness of Borel functional calculus for unbounded self-adjoint operators
I was reading these short notes on the Borel functional calculus where the author discusses the uniqueness property of this calculus for both bounded and unbounded self-adjoint operators.
When it ...
- 1,305
1
vote
0
answers
87
views
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$?
what is the relationship betwen $L(s,sym^mf\times sym^mg)$ symmetric L function of $f$ and $g$ and $\lambda_{f}(n^m)$, $\lambda_{g}(n^m)$ ?
- 11
1
vote
0
answers
89
views
Reference for an application of J. Simon "Compact sets in the space $L^p(0,T;B)$"
Theorem 5 p.84 of J. Simons paper "Compact sets in the space $L^p(0,T;B)$" states a generalization of the Aubin-Lions lemma which relaxes the required regularity in time to the existence of ...
- 469
1
vote
0
answers
273
views
Characterization of the Hamiltonian's spectrum in quantum mechanics
This is basically the same question I made on physics stack exchange The spectrum of the Hamiltonian in quantum mechanics, but I got no answers so far and decided to move it to mathoverflow with some ...
- 1,305
1
vote
0
answers
77
views
Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
- 503
1
vote
0
answers
111
views
Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
- 131
1
vote
0
answers
89
views
Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
- 111
1
vote
0
answers
98
views
Representations of the dual Banach algebra pair $(\ell_1,c_0)$
Let $\displaystyle E_p=(\bigoplus_{n\in\mathbb{N}} \ell^1_n)_{\ell^p}$ for some $1<p<\infty$ and $\ell^1 = \ell^1(\mathbb{N})$ be equipped with the convolution. Then, there exists an isometric &...
- 2,605
1
vote
0
answers
93
views
About vector valued measure algebras
Let $G$ be a locally compact group and $A$ be a Banach algebra. By $L^1(G,A)$ and $M(G,A)$ we denote the $A$-valued group, and measure algebra.
Is $M(G,A)$ a Banach algebra (with convolution as the ...
- 2,118
1
vote
0
answers
70
views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 39
1
vote
0
answers
120
views
Smooth vectors of unitary representations and reducibility
A representation $(\pi,V)$ of a locally compact group $G$ is unitary if $V$ is a Hilbert space endowed with a $G-$invariant bilinear form. In general, a unitary representation can't be expressed as a ...
- 103
1
vote
0
answers
120
views
Is a discrete harmonic function bounded below on a large portion of $\mathbb{Z}^2$ constant?
In the paper https://doi.org/10.1215/00127094-2021-0037, the main result is if we partition the plane $\mathbb{R}^2$ into unit squares (cells) so that the centers of squares have integer coordinates ...
- 543
1
vote
0
answers
64
views
Sequential Hölder-norm for functions in $H_{\alpha}([0,1]^{d})$?
I have come across a nice result attributed to Ciesielski (Ciesielski, Z. (1960). On the isomorphisms of the spaces $H_{\alpha}$ and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217–222.), even ...
- 149
1
vote
0
answers
385
views
Densely defined and closed operator
Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be ...
- 85
1
vote
0
answers
63
views
$L^2$ norm of a kernel with a variable width
Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
1
vote
0
answers
166
views
About a weak$^*$ convergent net
Let $G$ be a locally compact abelian group and $A$ be semisimple commutative Banach algebra such that $A^{**}$ has Radon-Nikodym property. Denote by $\Gamma$ and $M(G)$ the dual group and the measure ...
- 2,118
1
vote
0
answers
106
views
Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 286
1
vote
0
answers
85
views
Spectral projection with height less than $\lambda$
Let $x\geq 0$ be a positive element in a von Neumann algebra $\mathcal M.$ Then b y functional calculus the projection $e_\lambda=1_{[0.\lambda)}(x)$ has the property that $e_\lambda$ commutes with $x$...
- 1,879
1
vote
0
answers
116
views
A convergence problem in the space of tempered distributions
Let $K(x):=|x|^{-\alpha}$ be a function on $\mathbb{R}^{n}\setminus\{0\}$ with $0<\alpha<n$. Suppose $\phi$ is a $C^{\infty}_{c}(\mathbb{R}^n)$ function such that
$$\text{(i)}\quad \text{supp}\...
- 852
1
vote
0
answers
136
views
Banach spaces in which every DP-set is a limited set
Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset.
$A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \lim_{n\to\...
- 2,605
1
vote
0
answers
153
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 185
1
vote
0
answers
151
views
Weak convergence using tensor product
I haven't got to see this argument used in the PhD thesis of [R. Ryan]: Applications of topological tensor products to infinite dimensional holomorphy, doctoral thesis, Trinity College, Dublin (1980), ...
- 203
1
vote
0
answers
96
views
Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures
TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures.
Let $(X, d)$ ...
- 67
1
vote
0
answers
212
views
Relationship between Beilinson’s resolution of the diagonal and functional analysis
I have been wondering for long enough to embarrass myself on here by asking: is there a reason why Beilinson’s resolution of the diagonal “Coherent Sheaves on Pn and Some Problems of Linear Algebra” ...
- 511
1
vote
0
answers
220
views
Reference for Schwartz kernel theorem on vector bundles
In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
1
vote
0
answers
161
views
The norm of Sobolev space involving the time
Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help.
Let's assume we have a function
$$
\mathbf{u} (\mathbf{x}; t) = \...
1
vote
0
answers
142
views
Could we characterize elements in the second dual by the character space?
Let $A$ and $B$ be two semisimple commutative Banach algebras. Assume that $A\mathbin{\tilde\otimes} B$ is a Banach algebra obtained by completing $A\otimes B$ with respect to a cross norm not ...
- 2,118
1
vote
0
answers
165
views
What pre knowledge does Mumford's Tata collections on theta need?
I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...
- 11
1
vote
0
answers
128
views
Surjectivity of perturbed linear operators
Consider two Banach spaces $X$ and $Y$ and two linear bounded operators $A,B:Y\rightarrow Y$.
Suppose the following:
(1) Y is reflexive (or even uniformly convex);
(2) $X\cap Y$ is dense in $X$ and $Y$...
- 11
1
vote
0
answers
78
views
Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space
Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...
- 519
1
vote
0
answers
82
views
Name for natural norm on functions to non-linear targets
Is there a standard term for the quasi-norm
$$\|f\|_{[k]}=\sum_{i=1}^k(\sup\|f^{(i)}\|)^{1/i}$$
?
It is useful due to the fact that it is reasonably compatible with post-composition by smooth ...
- 18.7k
1
vote
0
answers
52
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
- 287
1
vote
0
answers
277
views
Fundamental Solution to Biharmonic Equation in 3D
(This is a repost of a question posed in StackExchange that didn't get any replies.)
Is anything known about the fundamental solution to the equation:
$$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$
for ...
- 431
1
vote
0
answers
71
views
Lipschitz isomorphisms of $C(\omega^\omega+)$
Let $C(\omega^\omega+)$ denote the Banach space of continuous, scalar-valued functions defined on $\omega^\omega+=[0,\omega^\omega]$. Suppose that $X$ is a Banach space and $U:C(\omega^\omega+)\to X$ ...
- 81
1
vote
0
answers
190
views
Is the strong topology the strongest?
Let $X$ be a topological vector space. We know that the weak topology $\sigma(X,X^*)$ is the weakest locally convex topology in $X$ that make every $f \in X^*$ continuous. Consequently, if $\tau$ is ...
- 203
1
vote
0
answers
180
views
Generalizations of Fubini-Tonelli's theoem
Fubini-Tonelli's theorem: If $(X, A, \mu)$ and $(Y, B, \nu)$ are $\sigma$-finite measure spaces and $f: X\times Y \to [0,\infty]$ is a measurable function, then
$$ \int _{X}\left(\int _{Y}f(x,y)\,{\...
- 11
1
vote
0
answers
119
views
On some integral equation
Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows:
$$
1-t= \int\limits_0^\infty p(x)\Phi\left(\frac{...
- 1,399
1
vote
0
answers
229
views
How to prove a concentration isoperimetric inequality for a non-Lipschitz function
Definition $1$. A probability measure $\mu$ on $\mathbb{R}^{d}$ satisfies c-isoperimetry if for any bounded L-Lipschitz $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$, and any $t \geq 0$,
\begin{align}
\...
- 79
1
vote
0
answers
199
views
Integration with respect to Haar measures normalised over a subspace
Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...
- 111
1
vote
0
answers
74
views
"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$
Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
- 839
1
vote
0
answers
158
views
When can we characterize Sobolev space $W^{2k,p}(\Omega)$ only via the Laplacian-like terms
One of the characterizations of the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$ uses the Fourier transform $\mathcal{F}$:
$f \in W^{s,p}(\mathbb{R}^n)$ iff $f$ is a tempered distribution such ...
- 820
1
vote
0
answers
155
views
Proving a sign rule for $f_{2n}$
If $t_{1},...,t_{n}$ are real numbers, consider the set of indexed linear operators $T(t_{1}),...,T(t_{n})$ on a Hilbert space $\mathcal{H}$ and define its ordering by:
$$\pi[T(t_{1})\cdots T(t_{n})] :...
1
vote
0
answers
52
views
Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel
Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional
Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
- 705
1
vote
0
answers
78
views
A statement on completeness of complex exponentials
I'm currently reading a paper by Olevskii on almost integer translates: https://www.sciencedirect.com/science/article/pii/S0764444297878731
In this paper the author considers for a given sequence $\{ \...
- 437