Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,248
questions with no upvoted or accepted answers
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Adjoint of sum of two operators. Kato-Rellich
Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to ...
- 41
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292
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When is the sum of a weak-$*$ closed convex cone and a subspace also weak-$*$ closed?
Let $X$ be a Banach space. Suppose $C \subset X^*$ is a convex cone and $V \subset X^*$ is a subspace, and suppose both $C$ and $V$ are closed in the weak-$*$ topology. Are there any general ...
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693
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On the projective tensor product of $c_{0}$ by $c_{0}$
Let $E$ be the projective tensor product of $c_{0}$ by $c_{0}$. Does it follow that $E$ is isomorphic to no subspace of $C(K)$, where $K$ is countable compact metric space?
When $C(K)$ is isomorphic ...
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632
views
Define the space of distributions with algebraic decay?
A tempered distribution $u\in \mathcal{S}'(\mathbb{R})$ is said to be rapidly decreasing if for every $f \in \mathcal{S}(\mathbb{R})$, $u*f \in \mathcal{S}(\mathbb{R})$.
One rough way to motivate ...
- 2,174
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372
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Hilbert Schmidt Operators and the Conditional Expectation Operator
Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
- 289
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975
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Predual of a von Neumann algebra in terms of trace class operators
For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ where $\mathcal{B(H)}$ is the space of all bounded linear operators on the Hilbert space $\mathcal{H}$, there is a Banach space $ \...
- 71
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301
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Convergence of Schwartz Kernels
I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...
- 9,591
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130
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The proximality of low rank function approximation
The paper "Best $n$-Dimensional approximation to sets of functions" by A. L. Brown in 1964 gave a negative answer to the following question:
Q1: Is there for a given integer $n$ always a best ...
- 3,177
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550
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$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?
Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra
$$A(\mathbb R):= \{f\in L^{1}(...
- 1,051
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307
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Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,267
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429
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Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
- 403
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285
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Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
user24451
4
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Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets
I was involved into this subject when I answered
this
question from MSE. Trying to generalize my answer, I am thinking about a following
Question. Let $X$ and $Y$ be metric spaces. When each ...
- 4,102
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269
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Exterior powers and singular values on Hilbert spaces
I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer ...
- 6,186
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answers
2k
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metric entropy for Lipschitz functions
Suppose $(X,d)$ is a metric space of unit diameer and let $F$ be the collection of all $1$-Lipschitz functions mapping $X$ to $[-1,1]$, equipped with the sup-norm $||\cdot||_\infty$.
I am interested ...
- 6,061
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221
views
Any references on infinite-dimensional Fourier-Plancherel theory?
Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
- 3,970
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111
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status of Invariant subspace problem on Krein Space
What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
- 2,076
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278
views
Markov operators and existence of ergodic measures
My question refers to the yesterday's question (see here)
of John Learner and goes as follows:
Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
- 141
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answers
89
views
$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology
This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
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284
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trace-class embeddings
There is a classical theorem of Riesz-Kolmogorov that characterizes compact embedding in $L^p$-spaces of some subspace of them. A generalization to arbitrary metric spaces has been recently obtained ...
- 3,328
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109
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How fast is discrete-time diffusion on a continuous set?
This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
- 7,573
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94
views
Algebraic conditions of separability
Let $X$ be a real vector space (without any norm), and $Y$ be a convex subset of $X$, $0\notin Y$. The goal is to find a hyperplane $L$ passing through 0 such that $Y$ lies in a closed halfspace ...
- 103k
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239
views
When separation in $L^1$ is possible?
Let $A$, $B$ be disjoint convex closed subsets of the Banach space $L^1[0,1]$. Assume additionally that $A$ is bounded and $A$, $B$ are closed under convergence in measure. Then there exists a closed ...
- 103k
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1k
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Applications of Riesz's lemma for the unit ball
I should give a talk on something I'm working on, and I'd like to have a list, as complete as possible, of applications, in and out of functional analysis, of the following classical result by F. ...
- 9,636
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296
views
Which orbits of a separable representation of the infinite unitary group are closed?
Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...
- 965
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436
views
Binomial Expectation of Convex Function
Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate
$$\frac{\partial}{\partial \alpha} \...
- 41
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157
views
Does this construction yield an injective hull ?
Let $K$ be an object of $\mathbf{CHaus}$, the category of compact Hausdorff spaces, and $K \xrightarrow{\ \ \sigma \ \ } K$ be an involutory morphism without fixed points. Define $C^{\sigma}(K)$ as ...
- 7,199
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137
views
When is $A^*A$ invertible for Banach space?
Let's consider a linear functional $A$ from smooth objects to smooth ones. It is first order operator in the sense that it extends to be a map from $W^{k+1,p}$ to $W^{k,p}$. Assume that we have $L^2$ ...
- 527
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1k
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Resonance of Schrödinger operator
Consider the dispersive estimates for the Schrödinger flow
$$
e^{itH}P_{c},\quad H=-\Delta+V \quad \text{on}\quad \mathbb{R}^n,n\ge 1
$$
where $P_{c}$ is the projection onto the continuous spectrum ...
- 1,614
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1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509
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165
views
Relationship between sequential compactness of a convex set and its extremal points
Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a ...
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484
views
Laplace Transform: Are there theorems similar to the Bernstein Theorem?
Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b)
Are ...
- 93
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450
views
Quantum sheaves
Are the following definitions known?
Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:
(a) {0} and H lie in Sigma
(...
- 1,368
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votes
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answers
102
views
quasinilpotence and finite spectrum II
Let A be a quasinilpotent operator on a Hilbert space and let every
operator of the algebra generated by $A$ and $A^{*}$ have finite
spectrum. Does then follow, that A is nilpotent ?
See also ...
- 2,768
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answers
234
views
dimension of induced comodule
Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...
- 41
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answers
185
views
Boundedness criterion for operators on mixed Lebesgue spaces
Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences
${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that
...
- 969
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633
views
Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
- 706
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256
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A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.5k
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483
views
Convolutions and Toeplitz Operators
Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...
- 2,044
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votes
1
answer
2k
views
Norm of convolution operator
By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
- 2,710
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votes
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45
views
About the J-method of interpolation
In the classic text of J.Bergh and J.Lofstrom, Interpolation Spaces, the $J$-method of real interpolation defines a functor in the following way: For $0<\theta<1$ and $1\leq q\leq \infty$ we ...
3
votes
0
answers
93
views
Image of trace operator on $W^{2,1}(\mathbb{R}^2)$
It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.
For ...
- 929
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votes
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193
views
Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
- 1,641
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111
views
Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
- 31
3
votes
0
answers
134
views
Interchange limit and supremum of functionals over a bounded convex set
Let $(H, \langle\cdot,\cdot\rangle)$ be a separable real Hilbert space and $B\subset H$ be (nonempty) convex and bounded, and suppose that $(\alpha_k)\subset H$ is a sequence for which the limit $\...
- 451
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votes
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59
views
About the naturality of Krasnoselskii genus on Variational Methods
I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
- 131
3
votes
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answers
93
views
Finitely generated Banach lattice $C(K)$ and partitions of unity
Let $E$ be a Banach lattice. A Banach sublattice $L$ of $E$ is called finitely generated if there exists a finite subset $F \subseteq E$ such that
$$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \...
- 419
3
votes
0
answers
99
views
Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
- 2,749
3
votes
1
answer
99
views
Can gradient zero implies that a function is constant with Hörmander vector fields
Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
- 561
3
votes
0
answers
126
views
Takesaki's duality in representation theory of $C^*$-algebras
In M.Takesaki's 1967 article titled A Duality in the Representation Theory of C-Algebras*, admissible operator fields are defined in order to generalize Gelfand transform to a non-abelian setting.
...
- 139