All Questions
3,602 questions with no upvoted or accepted answers
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214
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Compactness of multiplication operators
Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
- 6,035
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votes
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191
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Index of the Fredholm operator
I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
5
votes
0
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139
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Copies of $\ell_\infty^k$ in subspaces of the space of operators between $n$-dimensional Banach spaces
Are there a positive integer $k$ and an unbounded increasing function $d:\mathbb N\to\mathbb N$ (of growth order $\Omega(n^2)$) such that for any $n$-dimensional Banach spaces $X,Y$, the Banach space $...
- 4,535
5
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183
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Generic shadows of convex bodies
If $K \subset \mathbb{R}^m$ is a convex body (i.e., a compact convex set) and $T \mathbin\colon \mathbb{R}^m \to \mathbb{R}^n$ is a linear map then $TK \subset \mathbb{R}^n$ is a convex body as well. ...
- 2,479
5
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163
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On $\ell_1$ to $\ell_1$ operator norm of matrix with inverse Wishart distribution
Consider a random $n\times p$ matrix $X$ with $n\ll p$ and all entries of $X$ i.i.d. standard normal. For this $X$, the system of linear equations $y=Xw$ has infinitely many solutions, and the one ...
- 151
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350
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How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
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148
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Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace
Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$.
Is there a characterization of the set of projections of $f$...
- 1,245
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228
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What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
- 41.9k
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198
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Heuristic and graphic representation of BV functions and their singularities
This question is about some heuristics and graphs of BV functions.
In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are
the Heaviside function, whose ...
- 839
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438
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Green's formula and traces in weighted Sobolev spaces
Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let
\begin{equation}
\rho(x) = 1-|x| \quad \text{ for } x \in B_1,
\end{equation}
and let $\sigma >0$ be given. As per the comments, notice that $\...
- 309
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119
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Pimsner-Popa basis dealing with higher relative commutants
Let $(N \subseteq M)$ be a finite index unital inclusion of ${\rm II}_1$ factors. Let $e_1$ be the Jones' projection.
A finite subset $\{\lambda_i, i \in I\} \subset M $ is called a (right) Pimsner-...
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262
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Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
- 325
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128
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Is the Baireness a 3-space property of topological groups
It is known that the product of two Baire spaces can be meager.
On the other hand, by a recent result of Li and Zsilinszky the product of two Baire spaces is Baire if one of the spaces is countably ...
- 41.9k
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144
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Extension of elliptic complex to an exact sequence
This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.
Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial ...
- 5,824
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242
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Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
- 3,669
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110
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Clifford algebras in the context of locally convex topological vector spaces
Suppose given a locally convex Hausdorff topological vector space $V$ over $\mathbb R$ and a continuous, symmetric, bilinear map $q:V\otimes V\to \mathbb R$, where the tensor product is the completed ...
5
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231
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Which subspaces of $\ell_p^n$ are isometric?
This question is similar to the one asked here:
Extending linear isometries from subspaces of $\ell_p^n$
Let $p$ be an even integer. Let $X,Y$ be subspaces of $\ell_p^n$, and let $U : X \to Y$ be a ...
- 53
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197
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On effective constructions in the functional analysis of Volterra's integration operator
Let $V: L^2(0,1) \to L^2(0,1)$ be the Volterra integration operator: $V(f)(x) := \int_0^x f(t) \, dt$.
Is there a universal function $C(L,\varepsilon) < \infty$ such that the following uniform ...
- 13.8k
5
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answers
119
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Difference Between Eigenvalues of Schrödinger Operator with Different Boundary Conditions
Consider a Schrödinger operator
$$H=-\Delta+V$$
on a nice bounded domain $\Omega\subset\mathbb R^d$ (say, a ball or a cube), and assume for simplicity that $V$ is smooth.
Let $\lambda_D,\lambda_N$ ...
- 891
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330
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The second dual of $C(X)$ with the compact-open topology
Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
- 233
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329
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Reflexive Operator Algebra
It is known that a C*-algebra is finite-dimensional if (and only if) it is reflexive as a Banach space. What is known about the analog of this question for operator algebras? (Here, an operator ...
- 3,497
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105
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Paving property
In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property:
Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{...
5
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195
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What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
- 13.8k
5
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164
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Strong closure of nilpotent algebra
Let $A\subset \mathcal K(H)$ be a commutative algebra of compact operators on a separable Hilbert space $H$. Let $B\supset A$ be the closure of $A$ in the strong operator topology.
Assume that each $T\...
user1688
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134
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Banach space properties defined by compact operators, strictly singular operators and strictly cosingular operators
Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from $X$ into $Y$, $\mathcal{K}(X,Y)$ by the space of all the compact operators from $X$ into $Y$, $S(X,Y)$ by ...
- 3,343
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349
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Tietze extension theorem for lower semi continuous functions
On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
- 179
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725
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Stein extension Theorem in Sobolev spaces involving time
Let $n>0$ and $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$. The Stein extension Theorem states that there exists a total extension operator $E$ for $\Omega$ (see Theorem 5.24 in [1]).
As ...
- 143
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119
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Characterizing Herz-Schur multipliers using coefficient functions of uniformly bounded representations
Let $G$ be a group and let $c > 1$ be a constant. We denote by $B_c(G)$ the space of all coefficients of the representations of $G$ which are uniformly bounded by $c$; more precisely, a function $f:...
- 299
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245
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Examples of Banach lattices with positive Schur property but without Schur property
A Banach lattice $E$ has the
$(1)$ Schur property provided that any weakly null sequence in $E$ is norm null in $E$, and
$(2)$ positive Schur property provided that any weakly null sequence of ...
user114263
5
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345
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Weak to weak$^*$ continuity of the duality mapping
Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
- 799
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445
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Why are functions with vanishing normal derivative dense in smooth functions?
Question
Let $M$ be a compact Riemannian manifold with piecewise smooth boundary. Why are smooth functions with vanishing normal derivative dense in $C^\infty(M)$ in the $H^1$ norm?
Here I define $...
- 881
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103
views
Complementation problem for $\ell_p^2$
Let $n\in\mathbb{N}$ and $p,q\in(1,+\infty)$ with $p^{-1}+q^{-1}=1$. Consider isometric embedding between $\mathbb{C}$-Banach spaces
$$
\rho:\ell_p^n\to\ell_\infty(S, \ell_1^n),x\mapsto(f\cdot x)_{f\...
- 1,697
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votes
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answers
164
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Golod-Shafarevich groups and L_2- Betti numbers
Is it something known about $L^2$-Betti numbers for Golod-Shafarevich groups?
- 631
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120
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Geometric characterization of Silva distributions
There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$.
A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and ...
- 1,017
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0
answers
122
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How to solve this operator equation numerically?
I would like to know how one solves Sturm-Liouville problems on $(0,\infty)$ NUMERICALLY for the eigenvalues that are of the form
$$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$
So even if there ...
- 501
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179
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Representations of the algebra of shift-invariant operators on $\ell^\infty({\mathbb Z})$
$\newcommand{\Z}{\mathbb Z}$
By an operator on $\ell^\infty(\Z)$, I mean a bounded linear map $\ell^\infty(\Z)\to\ell^\infty(\Z)$. (Note that I am not assuming weak-star continuity.) By shift-...
- 25.8k
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254
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Examples of non-proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors
Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\...
- 7,067
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330
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Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 153
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207
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Foliations, von Neumann algebras and measurability
In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...
- 9,330
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150
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On the relation between Lipschitz free-spaces
Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point,
$Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\...
- 71
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211
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A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff
We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
- 1,270
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104
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On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
- 5,529
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answers
212
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Tensors and Nuclear/Fredholm Operators
For a locally convex Hausdorff spaces $E$, consider the canonical map
$$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$
that maps the projective tensor product to the space ...
- 9,853
5
votes
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answers
166
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Fourier basis for sub-Gaussian spaces?
Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with ...
- 519
5
votes
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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
$$f(g)...
- 1,692
5
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0
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332
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Harish-Chandra's submersive principle on closed subsets
Harish-Chandra's submersion principle says the following. Let $X,Y$ be two manifolds of dimensions $m$ and $n$ respectively. Let $\pi: X\rightarrow Y$ be a surjective smooth map which is submersive ...
- 960
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314
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C$^*$-algebras in which the spectral radius is comparable to the norm
For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:
For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \leq ...
- 3,984
5
votes
0
answers
138
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Banach spaces complemented in their ultrapowers
By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be ...
- 11.3k
5
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answers
186
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Norm of projection onto functions of mean zero
Let $X$ be a finite set and consider the space $\ell^2(X;Y)$ of functions $\zeta:X\to Y$, where $Y$ is a fixed Banach space. It decomposes into a direct sum of constant function and its complement $\...
- 51
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208
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A metric on $Homeo([0,1])$
One can define a metric on the set $Homeo([0,1])$ by setting $dist(f,g) =$ measure of support of $f^{-1}g$, that is the measure of the set of points $x$ where $f(x)\ne g(x)$. Was this metric studied ...
user6976