Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
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227
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Average gap between zeros on the critical strip of the Riemann Zeta Function
Are there any approximations of the average gap between 2 successive zeros along the Riemann zeta functions critical strip up to the nth zero? If so, is it hypothesized that this average gap converges ...
- 307
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92
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Can $\ell_1(E)$ be embedd into the dual of continuous function space?
Let $E$ be a compact seperable space, for example $E=[0,1]\subset\mathbb{R}$. Denote by
$$\ell_1(E):=\{u:=(u_x\in\mathbb{C}:x\in F):F\text{ is a countable subset of }E \text{ and } \|u\|_{\ell_1(E)}:=\...
- 131
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1
answer
115
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Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?
The following inequality is trivially true
$$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
- 161
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168
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Sequence of functions tending to zero in L^2
Let us consider a sequence of functions $f_n : (0,1)\times (0,1) \to \mathbb{R}$ in $L^2((0,1)\times (0,1))$ satisfying the following condition:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 29
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144
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Optimization over the set of all bounded probability measures
Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider:
$$
\max_{\mu \in \Delta^+ (X)} \theta (\mu), \...
- 67
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152
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Predual of $BMO(\mathbb{T}^d) $
In 1971, Fefferman characterized the predual of $BMO(\mathbb{R}^d)$ as the Hardy space $H^1(\mathbb{R}^d)$.
Is there a characterization of the predual of $BMO(\mathbb{T}^d$)?
- 11
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121
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Positivity of solution for Fisher-Kolmogorov Equation
How can we prove that if $y=y(t,x)$ is the solution of the problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y(t,x)=r(x)y(t,x)-\rho(x) y^2(t,x),\ (t,x)\in (0,T)\times \Omega \\ \...
- 1,759
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302
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Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
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61
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$L^p$ estimate for perturbed heat equation
Let us consider the heat equation
$$
\begin{cases}
u_t + f(u)_x - u_{xx} = 0 & x \in (-1,1), \quad t >0\\
u(t,-1) = a(t), \\
u(t,1) = b(t), \\
u(0,x) = u_0(x)
\end{cases}
$$
where $f \in C^\...
- 131
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106
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A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
- 203
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88
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Is A an amenable $C^{*}$-algebra?
Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps ...
- 581
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44
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Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
- 11
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95
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When does a potential function with given partial derivatives exist
I am looking for the answer to the following question:
Consider an integrable function $f:X\rightarrow X$ with $X$ being a compact subset of $\mathbb{R}^n$. What are the conditions on $f$ so that a ...
- 37
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57
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Isolated eigenvalues of "bipartite" operators
Please note: This is a reformulation of a previous question of mine. The old question has been already answered to, so I prefer asking a new one. However, it looked like the old formulation did not ...
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330
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Comparison of the spectrum decomposition
In the spectral theorem, we learned that the spectrum of a linear operator $A$ is a disjoint union of: point spectrum (eigenvalues), continuous spectrum (the kernel of $zI-A$ is zero, the closure of $\...
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230
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A gap in the proof of uniqueness of functional calculus based on a spectral theorem
This question considers the proof of a fundamental theorem of functional calculus, given in the book Spectral Theory -
Basic Concepts and Applications by David Borthwick (Theorem 5.9).
Firstly we have ...
- 1,755
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62
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"Trade-off" between bound on the function and on the spectrum for functional calculus in spectral theory
Let $A$ be a self-adjoint (unbounded) operator on a separable Hilbert space $H$.
From the following form of spectral theorem, we may define a functional calculus by $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
- 1,755
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37
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Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
- 943
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52
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Functions on dense subgroups of $\mathbb{R}^n$
Let $G$ be a finitely generated dense subgroup of $\mathbb{R}^n$, and $f$ be a character on $G$.
In the situation I'm looking at $f$ is either $1$ or $-1$ at any point.
Function $f$ can be extended to ...
- 2,772
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122
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Isolated points of the spectra of self-adjoint operators on Hilbert spaces
Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$.
I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
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49
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Non-square multiplication operator matrix
Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$.
Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$
Intuitively, $$K: [L^2(0,1)]^n ...
- 617
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0
answers
113
views
Error bounds on the expansion of square root of matrix
I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using holomorphic functional calculus. However, my first ...
- 427
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57
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A function with a dense set of periods and two values, except for a countable set, is a constant a.e. with respect to Lebesgue measure
I am reading the book Vector Measures of Diestel and Uhl, especifically example 6 of Sierpinski in Chapter 2, about the construction of a function that is weak$^*$-measurable but not weakly measurable....
- 31
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105
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About the definition of lineal convexity
I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...
- 31
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168
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Creation and Annihilation operators in QFT - Part II
Following some suggestions on my previous posts, I'm trying to reformulate my question in a more specific way. This is a continuation of my original post. Since the mentioned post, I think I've ...
- 3,515
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63
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Double commutant theorem when $C^*$-subalgebra does not contain identity operator $1$
Double commutant theorem: For a unital $C^*$-subalgebra $M \subset B(H)$, one has
$$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''.$$
My question:
For a $C^*$-subalgebra $M \subset ...
- 823
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45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 5,405
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90
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Differential equation
Consider $u = u(\phi,\psi)$ where $\phi = \phi(x)$ and $\psi =\psi(x)$ are both analytic function. The following equation
$$\partial_x u - u\partial_x (\phi-\psi)=0$$
has a trivial solution $u(\phi,\...
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107
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Norm equivalences for Gaussian random functions (Cameron-Martin space)
Preliminaries
Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$.
These random functions are drawn from a Gaussian measure $\...
- 101
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107
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Open subset of compact-open topology
Let $E$ be a Banach space, $X$ a locally-compact metric space, equip $C(X,E)$ with the compact-open topology. Let $F:E \rightarrow E$ and consider the induced map $F_{\star}(f):=F \circ f$ on $C(X,E)$...
- 5,405
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113
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Viewing limit as a map
Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which
$$
f_{\infty}(x) = \lim\limits_{n \...
- 5,405
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255
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The limit of the operator norm in a Hilbert space
I am not familiar with functional analysis. Could you tell me please, how to prove the following statement (if it is true)?
$$
\lim_\limits{M \to \infty} \|T_A - T_b \| = 0,
$$
here operator norm ...
- 335
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102
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Can a quotient space of a locally convex space have finer topology that its domain?
The following is related to this post.
Let $X=X'$, as sets, and let $T:X \rightarrow X'$ be a surjective map from a countably infinite-dimensional LCS $X$ to itself and equip $X'$ with the final ...
- 5,405
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60
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The scalar convergence in $\mathcal{C}(X)$ is topologizable?
Let $(X,\|.\|)$ be a separable Banach space and $\mathcal{C}(X)$ be the collection of all nonempty, closed and convex subsets of $X$. For any $C$ in $\mathcal{C}(X)$ we set
$$
s(x^*, C) := \sup_{x\in ...
- 115
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47
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An algebraic property that makes a per-C*-algebra complete
Let $A$ be a normed *-algebra with $\|x^*x\|=\|x\|^2$. Suppose that for every subset S of A, the left annihilator
${\displaystyle \mathrm {Ann} _{L}(S)=\{a\in A\mid \forall s\in S,as=0\}\,}$
is ...
- 4,058
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161
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When linear strongly elliptic operators are invertible?
I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like:
A linear differential operator, $$A : W^{...
- 1,774
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52
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Does the following operator have a unique fixed function? Do iterations of the operator converge?
The functions considered are positive real for positive real $x$, and monotonically increasing to infinity. Given such a function $f(x)$, define the (nonlinear) operator $f\mapsto f^*$ by
$$
f^*(x) := ...
- 211
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76
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Constructing a small Radon-Nikodym derivative
Let $u:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ function. Is it possible to (explicitly construction) a function $h$ such that:
$0<h(x)$.
$\int_{x \in \mathbb{R}^n} |h(x)|<\infty$,
$\sup_{x \in ...
- 5,405
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votes
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answers
67
views
Convergence as measure vs in $H^{-2}$
Let the domain be the two dimensional torus $\mathbb{T}$, and let $f_n $ be a sequence bounded in $H^1$, such that $\sup_n |f_n|\le 1$, and $f_n \to f$ weakly in $H^1$. Let $u_n = f_n \frac{\partial ...
- 131
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42
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How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?
Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...
- 167
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68
views
Closed graph theorem for cones?
In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are ...
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0
answers
253
views
Images of measurable function
My question is as follows. Consider an $L^\infty$ function $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ such that, for almost all $y$, the function $f({\cdot}, y)$ is continuous.
$\...
- 1
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0
answers
63
views
The eigenvalue of Schrodinger opeartor
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, consider $L=\Delta+V$ where $\Delta=div\nabla$. According to section 8.12 in Gilbarg and Trudinger's book, if $V\in L^\infty(\Omega)$, then ...
- 101
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votes
0
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154
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Use of this space of very rapidly decreasing continuous functions
Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
- 5,405
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votes
0
answers
55
views
Dense stratification of a separable Hilbert space
Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
- 5,405
0
votes
0
answers
116
views
Finding a square integrable dominating function for function class
problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$
where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
- 11
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votes
0
answers
57
views
Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?
Suppose I have an equation in some reflexive separable Banach space $X$:
$$Au=f$$
for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
- 107
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votes
0
answers
254
views
The set of all functions which vanish at infinity is a subset of the set of all functions which have vanishing variation
Let $X$ be a coarse space, we define the following:
$D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$
$f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
- 313
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153
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Representations of Banach algebras
If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)...