All Questions
Tagged with fa.functional-analysis reference-request
1,021 questions
0
votes
3
answers
291
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Smallest norms on crossed product $C^*$-algebras
Let $A$ be a commutative $C^*$-algebra with a discrete group $G$ acting on it. The reduced crossed product is the completion of the algebraic crossed product $C_c(G,A)$ in the reduced norm $\Vert \...
- 2,263
5
votes
1
answer
242
views
Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I ...
- 743
6
votes
2
answers
304
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Block version of Maurey Pisier theorem
The Maurey-Pisier theorem states that if $p_X$ is the supremum of those $p$ such that the Banach space $X$ has Rademacher type $p$, then $\ell_{p_X}$ is finitely representable in $X$.
For $1\leq p&...
- 2,429
7
votes
1
answer
606
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Weak* continuity of positive parts, again
Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\...
CommunityBot
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2
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0
answers
77
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When do finite dimensional approximations approximate the spectral absicssa of a linear operator?
I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.
I would like to compute the spectrum of a known non-normal, ...
- 121
5
votes
1
answer
227
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On the 'usefulness' of the 'original' definition of viscosity solution
In [CEL84, Theorem 1.1, p.489], Crandall, Evans, and Lions give three equivalent definitions of viscosity solution.
As the authors note, the first two are "more appealing in some respects and more ...
- 331
2
votes
1
answer
367
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Equivalence of alternative definitions of 'viscosity solution'
Consider the first-order Hamilton-Jacobi equation (HJ):
$$H(x,u,\nabla u) = 0 \quad \text{ on } \ \Omega,$$ where $\Omega$ is an open set of $\mathbb{R}^n$, $u:\Omega \to \mathbb{R}$, and $H:\Omega \...
- 331
4
votes
1
answer
2k
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Homogeneous fractional Sobolev spaces
Given $s\in (0,1)$ and a measurable function $u:\mathbb{R^n}\to\mathbb{C}$, let us define $$\|u\|_{\dot H^s(\mathbb{R}^n)}^2:=\iint\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$
and let $\dot H^s(\...
- 3,085
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0
answers
272
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Fixed-point iteration depending on a parameter
Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration
\begin{align}
x_{k+1} = f(x_k,\...
- 2,712
1
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0
answers
74
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Reference for a text book on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$
We need basic reference on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$. Usually text books cover much more subtle case of $T:l^{\infty}(\mathbb Z_+,...
- 109k
4
votes
1
answer
221
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If $K$ is a countable compact metric space is the set of extreme point of $Ba(C(K))$ countable?
The question is the title. The set $Ba(C(K))$ is the unit ball of $C(K)$. This has to be known, but I can't find the answer explicitly in the literature. There is some literature about polyhedral ...
- 109k
2
votes
1
answer
401
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Reference on Probability theory on functional spaces (in special Hilbert spaces)
Currently, I am working on some sort of stochastic optimization problems defined over function spaces.
I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...
CommunityBot
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5
votes
0
answers
215
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Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$
Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)...
- 1,692
3
votes
2
answers
397
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The space $L^p(\partial\Omega)$ in cited references
The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
- 215
5
votes
1
answer
328
views
Is a space with p-norm a Finsler manifold?
Suppose $\mathbb{R}^n$ is equipped with the p-norm $\left\Vert x \right\Vert_p$. Let $x\in \mathbb{R}^n$ and let $y$ be in a neighborhood of $x$. The distance between $x$ and $y$ can be defined as $\...
- 26.3k
4
votes
1
answer
1k
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Reference on the countable product of Banach spaces
If we are given countably many Banach spaces $X_1, X_2, \ldots, X_k, \ldots$, then we can norm their Cartesian product by
$$
\left\| (x_k)_{k \in \mathbb N} \right\|_p := \left( \sum_k \| x_k \|_k^p \...
CommunityBot
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2
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0
answers
346
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When is the sum of complemented subspaces complemented?
Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
- 45.7k
2
votes
3
answers
978
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"Must read" papers in functional analysis and PDE (à la Trefethen) [closed]
The following question has been inspired by "Must read" papers in numerical analysis.
In 1993, Prof. Trefethen published a NA-net posting with a list of thirteen papers that he had used in ...
CommunityBot
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1
vote
0
answers
135
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infinite dimensional funtional ito calculus
I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
- 5,405
32
votes
19
answers
23k
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Good books on theory of distributions
Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
- 101
1
vote
1
answer
1k
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Introductory text to Sobolev spaces and PDE's [closed]
I'm looking for a good introductory to Sobolev, preferably with an emphasis to their relationship to PDE's analysis.
I have only seen thus far Giovanni Leoni's "First Course in Sobolev Spaces" which ...
- 579
2
votes
0
answers
106
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Type-cotype inequalities for arbitrary orthonormal systems
Let $X$ be a B-convex Banach space and let $v^1 = (v^1_1,…,v^1_n), …, v^n = (v^n_1,…,v^n_n)$ be an orthonormal basis of $\mathbb{R}^n$. My question is what one can say about $\left( \sum_i \Vert \...
- 1,163
3
votes
1
answer
145
views
Reference for explicit quasicentral BAI in K(H) as ideal in B(H)?
As observed by Arveson and Akemann+Pedersen, if $J$ is an ideal in a ${\rm C}^\ast$-algebra $B$, then one can always find a contractive approximate identity for $J$, call it $(e_\lambda)_{\lambda\in\...
- 42.8k
10
votes
1
answer
802
views
Restriction of irreducible unitary representation to normal subgroup of finite index
Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
- 11.2k
1
vote
1
answer
237
views
Could I affirm that $f$ is not identically 0?
Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the
space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric:
$
d(x,y)=\sum_{i\geq 1}\frac{|...
- 42.8k
4
votes
1
answer
364
views
$H^1$-continuity of Laplace's equation with respect to boundary data
Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
- 13k
6
votes
0
answers
272
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Extension operators for topological vector space-valued smooth functions on closed sets
There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...
- 35.5k
0
votes
1
answer
317
views
Some questions related to the unitary operators
A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...
- 5,529
5
votes
1
answer
365
views
Does the Nash inequality hold on manifolds with Lipschitz boundary?
Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...
- 687
5
votes
0
answers
341
views
Real interpolation of weighted Sobolev spaces with different weights
Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
2
votes
2
answers
339
views
List of tensor product spaces with uniform crossnorms
Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products
$H:=\bigotimes_{j=1}^n H^{(j)}$ and $G:=\...
CommunityBot
- 1
3
votes
0
answers
95
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Strengthening of the local smoothing estimates for the free Laplacian
The classical local-smoothing estimates for the free Laplacian asserts that:
$$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$
where $B\subset\mathbb{...
CommunityBot
- 1
9
votes
1
answer
777
views
Abstract result on partitions of unity?
A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...
- 7,817
3
votes
0
answers
73
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On the principal eigenvector of an elliptic operator
Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...
CommunityBot
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4
votes
0
answers
172
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Donnelly-Fefferman growth of eigenfunctions
Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
- 41
2
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0
answers
282
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Reference request: functional analysis results used in Taubes paper (1980)
I am studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I am looking for a reference of three following theorems:
Let $f(x)$ be a convex funtional ...
- 6,249
3
votes
2
answers
435
views
A possible norm on a subspace of $C^\infty([0,1])$?
I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...
CommunityBot
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6
votes
1
answer
357
views
Travelling waves for nonlinear Schrödinger equation
Consider the following nonlinear Schrödinger equation:
$$
-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,
$$
where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
CommunityBot
- 1
4
votes
0
answers
508
views
Good reference for noncommutative $L^p$ spaces
I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
- 3,574
3
votes
1
answer
305
views
The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold
Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
- 7,426
5
votes
1
answer
256
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Injectivity/Surjectivity of $F_A :=\frac{d}{dt} +A(t) $ for a hyperbolic path of matrices $A(t)$ on $H^1 $
I am looking for a reference to the following problem:
Given a hyperbolic (no purely imaginary eigenvalues), continuous path of matrices $A(t)$ in $\mathbb{R}$ with hyperolic limits at $\pm \infty $.
...
- 211
5
votes
2
answers
310
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Error estimate in the spectral theorem of compact operators on a Hilbert space
Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
- 3,574
4
votes
1
answer
355
views
(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation
If we have nonnegative $V \in L^1_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for ...
- 3,574
5
votes
1
answer
137
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Operator space structures on CB(H,K) where H and K are Hilbertian operator spaces?
(I'd be grateful if anyone thinking of putting MathJax in the question title refrains from doing so.)
By consulting various standard sources (Effros-Ruan's book, Pisier's book, the lexicon of ...
- 9,486
8
votes
2
answers
367
views
$l^1$ versus $l^2$
Is there an elementary proof of this Banach space fact?
If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...
- 52.3k
1
vote
1
answer
66
views
Extremal of an L^1 continuous functional on a compact bounded set
Please, I need a small help with a reference.
Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...
- 3,574
9
votes
3
answers
2k
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Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
- 52.3k
1
vote
0
answers
469
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Multilinear Interpolation
Suppose I have a multilinear map $\Gamma(u,v)$ satisfying
\begin{align}
\big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\
\big\| \Gamma(u,v)\big\|_{L^\infty} &\...
- 516
6
votes
1
answer
227
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Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
CommunityBot
- 1
11
votes
2
answers
1k
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Do non-stable Banach spaces exist?
Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:
Is every infinite ...
- 11.3k