**1**

vote

**0**answers

94 views

### Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...

**2**

votes

**3**answers

465 views

### Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...

**6**

votes

**1**answer

188 views

### Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...

**6**

votes

**1**answer

433 views

### Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?

It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...

**1**

vote

**1**answer

99 views

### Approximation Property: Decomposition

This thread originated from MSE: Approximation Property: Decomposition
Given a Banach space $E$.
Consider a finite rank operator $F\in\mathcal{F}(X,E)$.
Introduce a basis on the finite dimensional ...

**2**

votes

**0**answers

345 views

### Homemorphism between $X$ and $\mathcal{P}(\mathcal{P}(X))$

Let $X$ be a topological space, $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$. Endow the latter with the weak* topology. I was wondering whether there exists a (nontrivial) ...

**0**

votes

**0**answers

54 views

### On Properties Of Lusternik-Schnirelmann Category

I have this part of proof from "Analysis and Topology in Nonlinear Differential Equations" book page 292:
I don't see how we find that $cat(\Omega)\leq cat( N_{\varepsilon}\cap ...

**0**

votes

**0**answers

66 views

### Approximation Property: Characterization

Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...

**0**

votes

**0**answers

67 views

### Compact Approximation

This thread originated from MSE: Compact Approximation
This is meant as lemma for: Approximation Property
Given a Banach space $E$.
Denote compact domains by $\mathcal{C}$.
Denote compact ...

**2**

votes

**0**answers

72 views

### Convergence in distribution of random measures

Let $M$ denote the space of real Radon measures on $\mathbb{R}$ as the topological dual of $C_c(\mathbb{R})$ equipped with the inductive limit topology (for possibly unbounded Radon measures) or ...

**1**

vote

**0**answers

110 views

### Harmonic extension of $L^\infty$ function is in $L^\infty$?

Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = ...

**4**

votes

**0**answers

80 views

### Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...

**5**

votes

**1**answer

257 views

### Non-reflexive Banach space s.t. X,X*,X**,… are separable

Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?

**3**

votes

**1**answer

177 views

### Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...

**2**

votes

**2**answers

142 views

### Quantum Field theory - integral notation

I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists.
I'm a student of mathematics, and I understand the notion of a ...

**1**

vote

**0**answers

67 views

### Spectrum of convolution operator

This question was asked already on Stack Exchange under http://math.stackexchange.com/q/1114095 . It might be not on a research level, but as it could not be answered on Stack Exchange, I hope for ...

**8**

votes

**3**answers

277 views

### Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D ...

**7**

votes

**2**answers

559 views

### exp(S) exp(T) = exp(S+T) for commuting operators

The standard way to prove the exponential law for two bounded commuting operators $S, T$
$$
\exp(S)\exp(T) = \exp(S+T)
$$
is to pass by the binomial formula and the power series of $\exp(.)$. I wonder ...

**5**

votes

**0**answers

124 views

### Proof that the second Borel cohomology group of $(\mathbb R, +)$ is trivial

Does anyone have a reference for a fairly direct proof that the second Borel cohomology group for $(\mathbb R, +)$ (with the trivial action on the circle group) is trivial? The motivation is to show ...

**11**

votes

**1**answer

341 views

### Schemes over topological rings

I have recently been interested in studying an extension of 'usual' algebraic geometry to take into account the topology of $R$ in the definition of the affine scheme $\mathrm{Spec}\, (R)$ when the ...

**0**

votes

**1**answer

103 views

### Does Trudinger inequality implies this critical Sobolev embedding?

Let $1<p<\infty$, $\mathrm{L}^p_s(\mathbb{R}^n)=J_s(\mathrm{L}^p(\mathbb{R}^n))$, where $J_s=(I-\Delta)^{-\frac{s}{2}}$, or $\mathscr{F}(J_sf)(\xi)=(1+|\xi|^2)^{-\frac{s}{2}}\hat{f}(\xi)$. And ...

**0**

votes

**0**answers

98 views

### Perturbation of Laplacian via Kato-Rellich theorem

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian
$$-\Delta+V(x)$$
is self-adjoint on $H^2(\mathbb{R}^3)$.
My idea is to use Kato-Rellich theorem; ...

**2**

votes

**1**answer

98 views

### Is $ {C_{c}}(G) $ a meager subset of $ {L^{2}}(G) $ for a second-countable locally compact Hausdorff group $ G $?

The following problem is a stumbling block in a research project that I am working on:
Problem. Let $ G $ be a second-countable locally compact Hausdorff group with a fixed Haar measure. Is it ...

**8**

votes

**2**answers

294 views

### Compact operators on Lebesgue spaces

Let $K:{\rm L}^p({\bf R}^d)\to {\rm L}^p({\bf R}^d)$ be a bounded linear operator for every $p\in(1,\infty)$.
Assume that for some $r\in(2, \infty)$ it holds that $K$ is compact on ${\rm L}^q({\bf ...

**1**

vote

**0**answers

130 views

### Solution to a PDE with constant data - what is the fault in my proof? [closed]

Let $C=\Omega \times (0,\infty)$. We want to find a solution $v \in H^1(C)$ such that given $u \in H^{\frac 12}(\Omega)$,
$$\int_0^\infty\int_\Omega \nabla v \nabla \varphi + v_y\varphi_y = ...

**0**

votes

**0**answers

80 views

### Can we conclude that $\varphi(L_xf_0)\neq0$ for every $x\in G$?

Let $H$ be a compact subgroup of locally compact topological group $G$ and $A=\{f\in L^1(G); R_hf=f(a,e)\forall h\in H\}$ as a subalgebra of $L^1(G)$ by convolution of $L^1(G)$. If $\varphi \in ...

**0**

votes

**0**answers

46 views

### Completeness of spaces $\Lambda(\varphi, p)$

Definition 1. Let $f$ be measurable function on a mesurable subset $E\subset \mathbb R^n$. Non-increasing rearrangement of $f$ is a function $f^\ast(x)=\inf\{s>0: \operatorname{mes} E[|f| > s] ...

**1**

vote

**0**answers

110 views

### Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...

**4**

votes

**1**answer

229 views

### Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...

**1**

vote

**0**answers

162 views

### What is the spectrum of $L^1(G:H)$?

Let $H$ be a compact subgroup of a locally compact topological group $G$ and
$$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in ...

**5**

votes

**1**answer

186 views

### Can we extend a multiplicative linear functional of a closed left ideal on whole of the algebra?

Let B be a closed left ideal of a Banach algebra A. Also, B has a right approximate identity (in B).
If g is a nonzero multiplicative linear functional on B, can we always extend g to a ...

**0**

votes

**0**answers

66 views

### Existence of a complementary closed subspace extending a given subspace

Let $H$ be an infinite dimensional (separable) Hilbert space (or any infinite dimensional Banach space in which every closed subspace has a complementary subspace). Suppose that $X$ and $Y$ are closed ...

**0**

votes

**0**answers

35 views

### Dependence of weak solution of an equation on a parameter

For each $p \in [a,b]$, let $X_p$ be a Hilbert space with $Y_p \subset X_p$ a subspace and we are given a bilinear form $a_p(\cdot,\cdot):X_p \times X_p \to \mathbb{R}$.
Given $u_p$ with $p \mapsto ...

**3**

votes

**0**answers

174 views

### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective?
The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...

**1**

vote

**0**answers

41 views

### Measurability of eigenelements $s \mapsto (\varphi_k(s), \lambda_k(s))$ of Laplace-Beltrami on $M_s$

For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary.
Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k ...

**1**

vote

**1**answer

130 views

### Sequence of smooth maps converging to the identity [closed]

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...

**3**

votes

**1**answer

178 views

### adjoint of this closed (?) operator

I am currently dealing with an unbounded operator
$T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow ...

**3**

votes

**1**answer

85 views

### If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz?

Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.
Suppose $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of ...

**2**

votes

**0**answers

71 views

### Nuclearity properties of Integral operators with Schwartz type kernels

Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth ...

**3**

votes

**0**answers

74 views

### Stationarity of Brownian motion with drift

Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...

**7**

votes

**0**answers

93 views

### Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...

**1**

vote

**0**answers

43 views

### Discrete J-method of interpolation [closed]

The discrete $J$ method is, given Banach spaces $A_0$ and $A_1$:
The interpolationn space $[A_0, A_1]_\theta$ is defined by: $a \in [A_0, A_1]_\theta$ if and only if $a$ can be written as ...

**0**

votes

**0**answers

58 views

### Restriction of derivations on $C^\infty(X)$

In 'Kriegl, Michor - A convenient setting for global infintite-dimensional analysis', they say that for an element $x$ in a convenient (i.e. Mackey-complete locally convex) space $X$, a bounded ...

**1**

vote

**0**answers

66 views

### A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space

Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule
...

**2**

votes

**1**answer

98 views

### Proper domain for operators

in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + ...

**2**

votes

**1**answer

198 views

### Integration in C^* algebra

Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that
$$
\int d s \, f(s)\, \alpha_s(A)
$$
is well defined as a ...

**1**

vote

**2**answers

252 views

### A question on the Lebesgue differentiation theorem

In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit
$$
...

**11**

votes

**1**answer

290 views

### Krein Milman theorem without the axiom of choice

The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...

**3**

votes

**2**answers

216 views

### Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...

**0**

votes

**0**answers

89 views

### Functional Calculus and Fredholm index

Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...