All Questions
10,448 questions
0
votes
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193
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Dissipative operator
Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative.
is it true that : if $\|y\|\leq \|z\|$ then $\|Ay\|\leq \|Az\|$?
Thank ...
- 11
4
votes
1
answer
243
views
When is Prim(A) of an infinite discrete group hausdorff ?
Does anyone know, if the following result has been proved ?
Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology.
The result is :
...
- 41
2
votes
0
answers
564
views
Young inequality in weighted spaces
Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...
- 1,205
3
votes
1
answer
842
views
An elementary introduction of Colombeau's generalized function theory
Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a ...
- 1,649
0
votes
1
answer
454
views
Is this set of functions compact?
Let $\mathcal{F}$ be the set of continuous functions $\varphi$ from $\mathbb{C}$ to $[0,1]$ that satisfy $\begin{align}\varphi(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(z+e^{i\theta})d\theta\end{align}$ ...
- 1
1
vote
0
answers
127
views
Mixed Tsirelson Norm
A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...
- 125
2
votes
0
answers
110
views
Is there any weighted sobolev embedding with non-decaying weight
Is there any weighted sobolev embedding like
$$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$?
Here $s>0$, and for some appropriate $p, q$.
- 31
0
votes
1
answer
223
views
Relation between the wave front set and the semiclassical frequency set
I need to prove that the wave front set of a distribution (as defined in Hormander's "The analysis of linear partial differential operators I") is equal to the semiclassical frequency set of an h-...
2
votes
1
answer
224
views
Subalgebras of $B(E)$
Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...
- 11.3k
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 $(...
- 9,853
2
votes
0
answers
132
views
Birkhoff orthogonal of a Banach space in its bidual
Let $X$ be a Banach space embedded in $X^{**}$ in the usual way.
We consider the set
$$
O_X := \{ x^{**}\in X^{**} : \|x^{**}-x\|\geq \|x\| \textrm{ for all }x\in X\}.
$$
I think this is the ...
- 4,461
1
vote
0
answers
101
views
On existence of right units with control of their norms
Does, there exist a Banach algebra with a family of right units with norms converging to 1, but without right unit of norm 1?
- 1,697
0
votes
1
answer
168
views
local moments of measures whose Fourier transform vanish in an interval
Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form
\begin{equation*}
\int_{-\delta}^{+\delta}|h|(dt)\le C(\...
- 859
1
vote
1
answer
59
views
Partial Constraint of Low Rank Matrix
Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...
- 773
0
votes
1
answer
156
views
Calculation of L2-dimension
For a group $G$, can we calculate $dim^{(2)}_{\mathcal{N}G}(\ell^2 G)$, where $\mathcal{N}G$ is the von Neumann algebra of $G$ and $\ell^2 G$ is the Hilbert space on $G$? I want to see whether this is ...
- 537
5
votes
1
answer
419
views
positive hermitian elements in $M_n(\mathbb{C})$
Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers
have some special properties:
(i) they are closed under sum,
(ii) they are closed under multiplication by positive ...
- 179
3
votes
0
answers
125
views
Copositivity in matrix pencils
Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
- 6,962
0
votes
0
answers
54
views
Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$
Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...
- 1
3
votes
1
answer
1k
views
Adjoint/transpose of wavelet transform
I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...
- 170
1
vote
1
answer
172
views
If a function is defined in terms of a solution to an initial value problem, is it also solution to an initial value problem?
Say $f:\mathbb R^{n+1}\to \mathbb R^p$ is a solution to an initial value problem, and $g:\mathbb R^{n+1}\to \mathbb R^q$, so that the components of $g$ can be expressed as polynomials in $f$, $f'$, ...
- 4,312
4
votes
1
answer
1k
views
When can a partial isometry $u$ in $\mathcal B(H \otimes K)$ be extended to a unitary in $1 \otimes \mathcal B(K)$?
Let $H$ and $K$ be Hilbert spaces, and let $u$ be a partial isometry in $\mathcal{B}(H \otimes K)$ between projections $p_0 = u^\ast u$ and $p_1 = u u^\ast$ such that $p_0, p_1 \leq 1 \otimes (1-q)$ ...
- 1,199
5
votes
1
answer
577
views
Does generator of continuous time random walk map heat kernel from L^2 to L^2?
Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi_{xy}$, where, given $e=\{x,y\}\in E$, we write $\...
- 269
2
votes
1
answer
362
views
Quasinilpotent example [duplicate]
Possible Duplicate:
Quasinilpotent operator
Do you know any example of a quasinilpotent operator such that every its power is non-compact?
Of course direct sum of nilpotent operators(or Volterra ...
- 21
3
votes
1
answer
556
views
Convergence of mountain pass solutions of $-\Delta u+u=u|u|^{p-2}$
Consider the following equation in $\mathbb{R}^N, N \ge 3$:
$$
(E) \quad -\Delta u +u=|u|^{p-2}u,
$$
where $2 < p < 2^{*} =2N/(N-2)$.
Denote by $J: H^1(\mathbb{R}^N) \to \mathbb{R}$ the ...
- 397
5
votes
0
answers
221
views
Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
3
votes
1
answer
1k
views
Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
- 1,338
11
votes
1
answer
603
views
Reference for a particular Radon transform on non-positively curved spaces
Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for ...
- 14.4k
0
votes
1
answer
92
views
Finite dimension implies regularity
Let $\mathscr{D}'(\mathbb R)$ be the set of distributions on $\mathbb R$ and $X$ be a linear subspace of $\mathscr{D}'(\mathbb R)$, which is closed under translations, i.e., if $\varphi\in X$ and $h\...
- 2,933
7
votes
0
answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
8
votes
1
answer
431
views
Injectivity for bimodules and Hochschild cohomology
Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...
user2412
0
votes
1
answer
643
views
Is this (interpolation) inequality right?
Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.
Question: Can we say that $F$ ...
- 61
1
vote
0
answers
465
views
Reference book for a stronger version of Goldstine theorem
Let me quote the proof of Goldstine theorem from wikipedia ( http://en.wikipedia.org/wiki/Goldstine_theorem ):
Given an $x^{**} \in B_{X^{**}}$, a tuple $(\phi_1, \dots, \phi_n)$ of linearly ...
- 311
4
votes
1
answer
783
views
Dependence of norm of extension map on Sobolev spaces and $(\epsilon,\delta)$ domains
Let $D\subset \mathbb{R}^n$ be a bounded domain.
An extension map is $E_D: W^{p,k}(D)\to W^{p,k}(\mathbb{R}^n)$ satisfying:
...
- 175
0
votes
0
answers
141
views
Property of quasipositive matrices
I saw this theorem stated in a paper without proof and I have difficulty proving it.
If $A$ is an $n\times n$ matrix with non-negative off-diagonal entries, let $s(A)$ be the real eigenvalue such ...
- 143
-1
votes
1
answer
152
views
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
2
votes
0
answers
223
views
optimal regularity for Laplace equation with inhomogeneous L^p Robin boundary condition
Consider the problem
$$-\Delta u = 0 \mbox{ in }\Omega,\qquad \partial_\nu u+\tau u=g\mbox{ on }\partial\Omega,$$
where $\Omega\subset R^n$ is a bounded $C^2$-domain, $\tau>0$ is a constant, and $...
2
votes
0
answers
52
views
About norm on $H^{\frac 12}(M \times \{0,1\})$
Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary.
Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space ...
- 171
6
votes
1
answer
453
views
The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
- 8,512
1
vote
1
answer
188
views
bounding the sum of the entries of the inverse of a 0-1 matrix away from 1?
Let $A\in\mathbb{R}^{n\times n}$ be an invertible 0-1 matrix. Is it possible that the sum $a:=\sum_{i,j}(A^{-1})_{ij}$ of entries of $A^{-1}$ is not equal to 1, but exponentially close (w.r.t. $n$) to ...
- 14k
1
vote
0
answers
91
views
Representing a Pullback as an Infinite Matrix
Let $M$ and $N$ be manifolds and let $T: M \to N$ be a bijective map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) be the space of all functions from $M$ (resp. $N$) to $\...
- 61
2
votes
0
answers
136
views
equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)
Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$,
$$
\...
- 1,835
3
votes
0
answers
456
views
Morphism of von Neumann Algebras
Hello,
Is there a counterexample to the following statement:
let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ?
( a $W^* $-...
- 663
3
votes
0
answers
396
views
Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$
Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That ...
- 5,327
1
vote
0
answers
443
views
Diagonalizing matrix with a special conjugate transpose property
Hi all,
I'm looking for the minimum criterion on $A\in M_{3x3}(\mathbb{C})$ (a $3x3$ complex matrix) such that:
1) $A$ is diagonalizable by a matrix $T\in M_{3x3}(\mathbb{C})$
2) $T$ is such that $...
- 111
2
votes
1
answer
276
views
Asymptotic Behavior of Non-Analytic Function of the Eigenvalues
Hello,
Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$.
If $A_n$ were a sequence of Hermitian ...
- 43
4
votes
1
answer
167
views
A Laplacian semi-group estimation
Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by
$$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
- 43
1
vote
0
answers
112
views
Existence of orthogonal projections generating Von Neumann algebras
Let $V$ and $V'$ be Abelian Von Neumann algebras of projections on some Hilbert space $H$, and let $V_{1}$ and $V_{2}$ be minimal sub-algebras of $V$ and $V'$ generated by projections $P_{1} \in V$ ...
- 631
3
votes
0
answers
185
views
spectrum of a polygon and zeta function
Let $\Delta(x,y) = 1,0$ according to whether $(x,y)$ is in some polygon (symmetric with respect to the diagonal axis).
E.g. The convex hull of three points (taken from a paper on dominoes)
$$ \...
- 22.8k
2
votes
0
answers
161
views
When is it possible to split a non-linear operator into a composition of a linear and local one?
Let $A: L^2(R^n)\to L^2(R^n)$ be a non-linear operator. Is it known when it's possible to split $A$ into a composition of a linear operator $B: L^2(R^n)\to (L^2(R^n))^k$ and a local operator $C: (L^2(...
- 2,205
1
vote
0
answers
181
views
Infinite dimensional quotients of L_1 by isomorphic subspaces
Let $M$ be a subspace of $L_1(0,1)$. If the subspace $M$ is isomorphic to $L_1(0,1)$ and complemented, then the quotient $L_1(0,1)/M$ is clearly non-reflexive if it is infinite dimensional. So as we ...
