All Questions
3,508 questions with no upvoted or accepted answers
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144
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Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
- 1,879
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93
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Regularity of semilinear parabolic PDE in the whole space
I need regularities in Holder space of the following parabolic PDE:
$$\partial_t v = \partial_{xx} v + \partial_{yy} v + \rho \partial_{xy} v - v \partial_x v + \partial_y v + F, \forall (x, y, t) \...
- 1,399
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39
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Topology on upper semicontinuous functions and Baire one functions
Which is the standard topology on upper semicontinuous functions and on functions of Baire class one, in both cases defined on a compact real interval?
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247
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Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
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46
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Independence of variables predicted by the generator
Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...
- 449
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164
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Bound for the $\ell^3$ norm for the one-dimensional propagator
Problem: In Appendix (A.6) of Main paper is written
$$\lVert K(x; t_0, t_1, t_2, \frac{1}{2\pi}q_1,
\frac{1}{2\pi}q_2)\rVert_3 \leq \prod_{\nu=1}^{d} \lVert
p_{R^{\nu}}^{(d=1)}\rVert_3 \leq C
\...
- 101
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0
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67
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Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 617
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239
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Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
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53
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Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
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81
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What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
- 6,853
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104
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Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 61
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150
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Eigenvalues of the Laplacian and min-max formula in any space dimension
In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by
$$
\lambda_1 = \min_{u \in H^1_0(\Omega), \|...
- 99
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153
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Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$
How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
- 99
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134
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The set of bounded lipschitz functions over a compact is barrelled but not a neighborhood of zero?
I recently learned that Banach spaces are barrelled, i.e any convex, balanced, absorbing and closed subset is a neighborhood of zero (wikipedia). I'm having trouble understanding why the following ...
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231
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Definition of tensor product of dense subspaces of Hilbert spaces
Let $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ be Hilbert spaces. If $\psi_{1}\in \mathscr{H}_{1}$ and $\psi_{2}\in \mathscr{H}_{2}$, define $\psi_{1}\otimes \psi_{2}$ to be a function on $\mathscr{H}_{1}...
- 1,305
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44
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Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?
Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
- 6,853
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101
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Example of a ‘weak’ ideal which is not an ideal?
For Hilbert spaces $H$ and $K$, let $V=B(H,K)(H \neq K)$. A sub space $I$ of $V$ is called an ideal of $V$ if $$IV^*V+VV^*I \subset I$$ and $I$ is called weak ideal of $V$ If $$\text{span}\{IV^*V-VV^*...
- 1,115
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414
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Sufficient conditions for an asymptotic compactness
This question relates a theory of Mosco convergence.
Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$.
A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
- 721
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299
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Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 205
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90
views
Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\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
$$\ell_1(E):=\{u:=(u_x\in\mathbb{C}:x\in F):F\text{ is a countable subset of }E \text{ and } \|u\|_{\ell_1(E)}:=\...
- 131
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115
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Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?
The following inequality is trivially true
$$\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:
$$
\lim_{n \rightarrow \infty}\int_{1/j}^{1 - 1/j}\...
- 29
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144
views
Optimization over the set of all bounded probability measures
Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider:
$$
\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:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y(t,x)=r(x)y(t,x)-\rho(x) y^2(t,x),\ (t,x)\in (0,T)\times \Omega \\ \...
- 1,759
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302
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Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
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61
views
$L^p$ estimate for perturbed heat equation
Let us consider the heat equation
$$
\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}
$$
where $f \in C^\...
- 131
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106
views
A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
- 203
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88
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Is A an amenable $C^{*}$-algebra?
Let $A$ be $C^{*}$-algebra. Suppose that, for any $\epsilon > 0$ and finite subset $F \subset A$, there are an amenable $C^{*}$-subalgebra $B \subset A$, contractive completely positive linear maps ...
- 581
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44
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Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\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 $\...
- 1
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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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0
answers
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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0
answers
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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0
answers
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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votes
0
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90
views
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