Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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2
votes
2answers
165 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
0answers
81 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
3answers
284 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
2answers
572 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 ...
6
votes
0answers
126 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
1answer
354 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
1answer
111 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 ...
1
vote
0answers
111 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
1answer
107 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
2answers
301 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
0answers
134 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
0answers
86 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
0answers
47 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
0answers
132 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
1answer
275 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
0answers
172 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
1answer
194 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
0answers
73 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
0answers
37 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
0answers
183 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
0answers
44 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
1answer
131 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
1answer
186 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
1answer
92 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
1answer
103 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
0answers
81 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
0answers
103 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
0answers
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
0answers
65 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
0answers
70 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
1answer
105 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
1answer
209 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
2answers
308 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
1answer
309 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
2answers
221 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
1answer
138 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 ...
2
votes
2answers
204 views

About a completion of a Sobolev space

Let $\Omega$ be a bounded smooth domain and define $\mathcal{C} = \Omega \times (0,\infty)$. Below, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$. The map ...
2
votes
1answer
182 views

Lusternik-Schnirelmann Theorem

In various paper i found this: But i don't find this theorem of Lusternik-Schnirelmann, have you an idea where i can find this theorem, the condition? Thank you.
3
votes
1answer
167 views

Equivalence of Gaussian measures

Let $H$ be a separable Hilbert space and $N(0, C)$ and $N(0, D)$ be Gaussian measures on it. Further, for each $v \in H$, define $R_v = \frac{\left\langle v,Cv \right\rangle}{\left\langle v,Dv ...
6
votes
2answers
308 views

Number of critical points of smooth functions on $S^1$

Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
3
votes
0answers
88 views

On a variant of Eidelheit's theorem

A theorem of Eidelheit from 1940's asserts that two Banach spaces $X$ and $Y$ are isomorphic if and only if $L(X)$ and $L(Y)$, the algebras of all bounded linear operators, are isomorphic as Banach ...
5
votes
1answer
190 views

James' theorem—going from the separable case to the general case

Consider the following famous theorem by Robert C. James (1964): Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
1
vote
3answers
293 views

Decompose the Laplacian

Is there a way to write the negative Laplacian on the 2-sphere as a decomposition of an operator $A$ and its adjoint $A^*$? I am interested in finding such a decomposition, but I could not get one by ...
6
votes
1answer
231 views

Subadditivity of the square root for matrices

For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices? If not, ...
2
votes
1answer
155 views

Survey on functional equations and inequalities

Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)? ...
1
vote
2answers
258 views

Witten index non-trivial in the context of Quantum Mechanics?

Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$. I will now consider the one-dimensional case on a compact set: So ...
0
votes
0answers
66 views

Fractional Sobolev space on compact manifold as integral

Is it possible to define the fractional Sobolev space $H^{\frac 12}(M)$ on a compact (closed) Riemannian manifold $M$ as the set of $u \in L^2(M)$ such that $$\int_M\int_M ...
1
vote
1answer
193 views

A calculus question related to the nonnegative definite functions

I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that $$ \int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge ...
7
votes
1answer
417 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
-3
votes
1answer
235 views

Hilbert space vector representation for data in a metric space. Where am i wrong in this experiment?

Consider the function space $M$ such that all its elements are of bounded variation, square integrable and of unit norm. An equivalence class is defined over this set as, $f \sim g$ iff for some ...