All Questions
Tagged with reference-request fa.functional-analysis
1,021 questions
3
votes
1
answer
220
views
Conditional expectation as square-loss minimizer over continuous functions
It is well-known that the conditional expectation of a square-integrable random variable $Y$ given another (real) random variable $X$ can be obtained by minimizing the mean square loss between $Y$ and ...
- 128k
3
votes
0
answers
74
views
Locally compact rings with reciprocals
A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
- 974
6
votes
1
answer
382
views
Sobolev embedding theorems on manifolds
I had asked the following question on math.stackexchange but did not get any response:
I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...
- 11
7
votes
1
answer
652
views
Extending Hölder functions
I originally asked this question on MathStackExchange some time ago, but it seems that MathOverflow would be more appropriate. Essentially, I would like to find references for extension theorems for (...
- 6,367
10
votes
1
answer
429
views
Direct sums of operator spaces
I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...
- 18.7k
2
votes
0
answers
124
views
Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
- 60.5k
2
votes
0
answers
126
views
Differential equations: trying to connect a nonlinear equation to a linear one
The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
- 383
1
vote
0
answers
143
views
Estimator for the conditional expectation operator with convergence rate in operator norm
Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively.
Let $L^2(\pi)$ ...
3
votes
2
answers
102
views
Reference for Wiener type measure on $C(T)$ when $T$ is open
I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
- 29.8k
5
votes
1
answer
761
views
Looking for a reference for a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces
Please I need a reference where I can find a version of the "The reverse Lebesgue dominated convergence theorem" for the Orlicz spaces analogous to Theorem 1.2.7
in Semilinear Elliptic Equations for ...
- 835
2
votes
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 2,175
1
vote
0
answers
82
views
Injective envelopes of 1-extensible spaces
Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...
- 2,605
3
votes
0
answers
140
views
Does the Kato-Ponce estimate hold on manifolds?
Recall the Kato-Ponce estimate for fractional powers of the operator $J = (1-\Delta)$,
$$ \| J^s(fg) \|_{L^r} \lesssim \| J^s f \|_{L^{p_1}} \| g \|_{L^{q_1}} + \| J^s g \|_{L^{p_2}} \| f \|_{L^{q_2}},...
- 63.9k
1
vote
0
answers
111
views
Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
- 4,461
2
votes
0
answers
325
views
Examples of RKHS that are "classical"
Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs?
It is easy to construct example of RKHSs by applying the ...
- 21
4
votes
0
answers
77
views
Reference/Help request for formula $[A,e^{-itB}]$ found in physics thread
I'm wondering if anyone has a rigorous reference or a proof of the formula (2) found in the main answer of this thread on the physics stack exchange. I want to use it but in the case where $A, B$ are ...
- 71
1
vote
0
answers
47
views
Existence for a nonlinear evolution equation with a monotone operator that is not maximal
We consider the nonlinear evolution equation
$$
\dot{u}(t) + Bu(t) = 0, \quad u(0)=0
$$
with
$$
A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
- 171
9
votes
0
answers
240
views
What is known about when $vN(G)$ is a factor, for a locally compact group $G$?
When $G$ is a discrete group, it is an elementary result in the theory of von Neumann algebras that the group von Neumann algebra $vN(G)$ is a factor if and only if $G$ is an ICC group.
What is known ...
- 146
3
votes
2
answers
147
views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 4,729
16
votes
2
answers
1k
views
Examples of Banach manifolds with function spaces as tangent spaces
I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the ...
- 6,001
2
votes
0
answers
83
views
Singular integral operators acting on Zygmund class
It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies
$$\sup_{0<R<\...
- 21
9
votes
2
answers
516
views
Why operator systems?
A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
- 42.8k
1
vote
0
answers
163
views
Reference on spectral theory of self-adjoint operators
I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self ...
- 63.9k
5
votes
2
answers
904
views
Differentiability of the Moreau envelope
I've recently come across many results discussing the differentiation of the Moreau envelope defined by
\begin{equation}
e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(h),
\end{equation}
where $f$ is ...
- 67
8
votes
1
answer
245
views
Spectral decomposition of $\Gamma\backslash X$
Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group ...
- 12.9k
7
votes
4
answers
5k
views
Intuitive functional analysis book
I want to know functional analysis book like Terence tao's real analysis and measure theory book, full of intuition. I am aware of linear algebra, real analysis, measure theory, Probability theory.
- 310
1
vote
0
answers
51
views
Error estimates for inhomogeneous semidiscrete PDE
I have the following semidiscrete problem on a meshed domain $U_h$. Let
$V_h$ be linear finite elements on $U_h$, $V_{h0}\subset V_h$ have zero trace on $\partial \Omega_h$, and
$V_{h\partial}$ be ...
- 6,367
3
votes
0
answers
151
views
Reference request: trace norm estimate
In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$
then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 869
2
votes
1
answer
166
views
Cocompact lattices in $\mathrm{Sp}(n, 1)$
This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
- 23k
2
votes
1
answer
107
views
If $b\in C^1(E, \mathbb{R})$ and $b'$ is compact, then $b$ is weakly continuous — a reference request
While reading the well known book Minimax Methods in Critical Point Theory with Applications to Differential Equations by Paul Rabinowitz, in the proof of a generalisation of the Mountain Pass Theorem ...
- 128k
0
votes
0
answers
208
views
Examples of non $w^{*}$-closed complemented subspaces of a dual Banach space that are also dual spaces
Let $Y$ be a complemented, but not $w^{*}$-closed, subspace of a Banach space $X$. It is known that certain such $Y$ are not dual spaces.
Question: What are interesting examples of subspaces of the ...
- 7,057
3
votes
1
answer
499
views
Thin-Plate-Spline understanding and solution
This is a migrated question from: Thin-Plate-Spline understanding and solution.
If I need to delete one of the questions let me know. I was suggested to post it here as well.
As I understand it a Thin-...
- 115
5
votes
2
answers
663
views
Separate continuity implies (joint) continuity
I believe that the following fact is true and I am looking for a reference.
Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V$, $W$ be Fréchet spaces.
...
CommunityBot
- 1
-1
votes
2
answers
407
views
Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
- 3,808
3
votes
1
answer
148
views
positive functional on Banach *-algebra (with appro. identity) is continuous?
Theorem (N. Th.Varopoulos): Let $\mathcal{B}$ be a Banach *-algebra
with a bounded approximate identity. Then every positive functional $T$ on $\mathcal{B}$ is continuous.
I think this theorem is ...
- 2,545
2
votes
0
answers
62
views
On a real smooth version of white noise distribution theory
In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
- 505
3
votes
1
answer
185
views
Is the weighted shift strong frequently hypercyclic?
One sided Shift
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...
- 146
5
votes
1
answer
381
views
Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$
I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...
CommunityBot
- 1
2
votes
2
answers
451
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
- 2,494
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 1,446
1
vote
0
answers
96
views
Representation formula for the continuity equation on a separable Hilbert space
The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
- 63.9k
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 188k
1
vote
0
answers
63
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 12.6k
7
votes
2
answers
398
views
Quantifying difficulty of integrals versus inverses
Recently, I have been discussing inverses with a tenth grade class and integrals with an eleventh/twelfth grade class, and this has led me to the following wonder:
Wonder. Is there a "reasonable&...
- 12.9k
1
vote
0
answers
111
views
Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
- 39k
6
votes
2
answers
457
views
$L^{p}$ isoperimetric inequalities on the Hamming cube
Let $A \subset \{-1,1\}^{n}$ be a subset of the Hamming cube with cardinality $|A|=2^{n-1}$. Define $w_{A} : \{-1,1\}^{n} \to \mathbb{N}\cup \{0\}$ so that $w_{A}(x)$ to be number of boundary edges ...
- 63.9k
1
vote
2
answers
106
views
Green function of symmetric stable process in dimension 1 and 2
Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
4
votes
1
answer
418
views
Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 52.3k
5
votes
2
answers
276
views
Dilation of bounded linear operators
Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
- 3,984
3
votes
0
answers
198
views
On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 128k