All Questions
Tagged with fa.functional-analysis reference-request
1,021 questions
12
votes
2
answers
2k
views
Reference for invariance of essential spectrum under relatively compact perturbations
I'm looking for a proof of the following statement:
Let $X$ be a Banach space and $T$ be a closed map on $X$. For any relatively compact map $A$ the essential spectrum of $T$ and $T+A$ are the same.
...
- 4,461
2
votes
0
answers
231
views
A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - \...
- 183
4
votes
1
answer
1k
views
definition of accretive operator
A relation T with domain and range in a Hilbert space is said to be accretive if the
transformation $ (T − \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for ...
CommunityBot
- 1
1
vote
1
answer
345
views
L logL space and compactness
I think that if a sequence of L^1 functions have the integral
$$
\int f_n \log (f_n)dx
$$
uniformly bounded, then there is a subsequence that converges strongly in $L^1$.
The questions are:
1) Is ...
- 17k
6
votes
1
answer
322
views
Self-adjoint operator
Assume that $B$ is a self-adjoint operator and $\alpha\in(0,1)$. I need a reference for the following equality $$B^{-\alpha}=\frac{\sin\alpha \pi}{\pi}\int_0^\infty \lambda^{-\alpha}(\lambda+B)^{-1}\...
- 53.3k
2
votes
1
answer
288
views
Reflexive Besov spaces Bs,p,q
I don't know whether the Besov space $B^s_{p,q}$ with $1<p,q<\infty$ is reflexive or not? Can someone help me please?
- 16.4k
4
votes
0
answers
2k
views
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 ...
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4
votes
0
answers
277
views
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 ...
- 6,206
8
votes
0
answers
1k
views
On the classification of injective Banach spaces
A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a norm-...
- 4,461
0
votes
2
answers
180
views
A basic question about JL Lions' transformation of a Stefan problem
In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...
- 3,388
18
votes
1
answer
1k
views
Who introduced the notion of "stability" in numerical analysis?
I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...
- 4,729
2
votes
1
answer
240
views
Support-preserving pseudodifferential operators
Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the ...
- 16.2k
0
votes
2
answers
663
views
Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
- 190
5
votes
1
answer
226
views
A question about extensions of Markov semigroups
I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. ...
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4
votes
0
answers
226
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 ...
- 4,010
1
vote
1
answer
193
views
If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$?
Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain.
Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$
where $w'$ means the weak ...
- 5,380
14
votes
0
answers
3k
views
Tanh version of a Fourier Transform?
I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
- 2,834
34
votes
1
answer
4k
views
Theme of Isbell duality
Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
- 457
2
votes
1
answer
130
views
Linear combination of i.i.d. $Z_i$ distributed as $Z_1$
A classical property of the Gaussian distribution is that, if $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z_i \sim N(0,1)$) and $S = \sum_{i=1}^n a_i Z_i$ where $...
- 11.4k
2
votes
0
answers
117
views
Maximum Principle with Banach Control Space
This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...
- 197
3
votes
1
answer
245
views
Orthonormal basis in $\ell^n_p$
Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of
$$
\sum_{i=1}^k \|a_i\|_{\ell^p}^2
$$
for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis.
...
- 4,432
9
votes
2
answers
706
views
Measures whose projections are absolutely continuous
Since my question was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
CommunityBot
- 1
2
votes
0
answers
266
views
Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$
Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of $L^2(0,...
- 171
1
vote
1
answer
400
views
Nonlinear parabolic PDEs existence with Galerkin method?
Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...
- 4,729
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 19.3k
1
vote
0
answers
258
views
Is an exact operator, unitary equivalent to a banded operator?
Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis.
$T \in B(H)$ is ...
CommunityBot
- 1
1
vote
1
answer
527
views
Reference request: Spectral analysis of advection diffusion PDE
As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the ...
- 2,243
4
votes
1
answer
370
views
Norms for complex measures
I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
CommunityBot
- 1
2
votes
2
answers
403
views
Resource on Infinite Systems of Difference Equations
I have asked this question previously at Math.stackexchange, but it seems to receive little attention there.
In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer ...
- 352
8
votes
1
answer
394
views
Origin of the term Riesz Basis
The term Riesz basis is in wide use for the image of an orthonormal basis in a Hilbert space under a bounded invertible map (there are lots of equivalent definitions). But I've been unable to find ...
- 21.5k
4
votes
0
answers
90
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 ...
7
votes
2
answers
1k
views
For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?
(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
CommunityBot
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1
vote
1
answer
172
views
If a function is defined in terms of a solution to an initial value problem, is it also solution to an initial value problem?
Say $f:\mathbb R^{n+1}\to \mathbb R^p$ is a solution to an initial value problem, and $g:\mathbb R^{n+1}\to \mathbb R^q$, so that the components of $g$ can be expressed as polynomials in $f$, $f'$, ...
- 21.5k
4
votes
0
answers
109
views
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 ...
CommunityBot
- 1
1
vote
1
answer
184
views
Special kind of operators
Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$
where $(...
- 25.3k
4
votes
0
answers
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 ...
- 109k
4
votes
0
answers
1k
views
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. ...
- 10.5k
0
votes
1
answer
305
views
Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
CommunityBot
- 1
1
vote
0
answers
618
views
A question on a variant of Hardy's inequality.
I would like to know a proof of a variant of Hardy's inequality below. Could anyone introduce me a reference or give me a proof? Thank you very much for your assistance.
Set $0<r<\frac{2(n-s)...
- 11
2
votes
1
answer
891
views
Riesz representation theorem for vector-valued fields
Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual ...
- 2,409
1
vote
1
answer
2k
views
PhD in operator algebras and non-commutative geometry [closed]
I do not know whether it is a good place to ask this question or not.
I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...
- 170
2
votes
1
answer
347
views
The point of view of semicats in functional analysis
I'm completing a paper about (Mitchell's) semicats (well, not exactly, but let's say so for simplicity), and as a motivational example I'd like to mention at some point that the monic/epic morphisms ...
- 25.3k
1
vote
0
answers
61
views
Multi-completely monotone functions
Consider a $C^{\infty}$ nonnegative function $f(x,y,z)$, $x,y,z>0$ and let $\lambda f(\lambda x, \lambda y,\lambda z) \equiv f(z,y,z)$ for any $\lambda > 0$ (positive homogenity). Define
$$
g_{...
- 1,329
0
votes
1
answer
813
views
a problem in functional analysis that erdos solved in 2 lines
https://math.stackexchange.com/questions/261685/paul-erdoss-two-line-functional-analysis-proof .
does anyone know about what the problem was and what was his solution.
[Edit by quid:] please follow ...
CommunityBot
- 1
1
vote
1
answer
650
views
Riesz representation for an infinite-dimensional space
Suppose $X$ is an infinite-dimensional Banach algebra (hence not locally compact).
Does there exist any sort of Riesz representation theorem that says something about elements of $C(X)^*$?
- 4,010
6
votes
2
answers
460
views
Terminology: Banach spaces equipped with continuous associative product?
This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach algebra/...
- 42.8k
9
votes
2
answers
1k
views
Hilbert transforms of measures
Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
CommunityBot
- 1
6
votes
4
answers
1k
views
Reference for integral of functions taking values in a topological vector space.
(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", "...
- 41.1k
0
votes
0
answers
218
views
Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
- 42.8k
0
votes
0
answers
80
views
relationship between different function classes
I was wondering if there is a survey of relationship between several different well-studied function classes ?
ps - The question may be vague but I am looking for something along the lines of - http:/...
- 1