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

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44 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 ...
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0answers
27 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 ...
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0answers
61 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 ...
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0answers
29 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 ...
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1answer
112 views

Sequence of smooth maps converging to the identity

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
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1answer
155 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 ...
2
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1answer
70 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 ...
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143 views

Question about Lusternik-Schnirelmann Category?

I have this sets: $\Omega\subset \mathbb{R}^N, N\geq 3$ a smooth bounded domain $\Omega_{r}^+=\{x\in \mathbb{R}^N, d(x,\Omega)\leq r\}$ and ${\Omega}^-_{r}=\{x\in \Omega, d(x,\partial\Omega)\geq ...
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0answers
34 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
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53 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
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65 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 ...
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0answers
28 views

Discrete J-method of interpolation

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 ...
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0answers
54 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 ...
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0answers
57 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
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1answer
78 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}} + ...
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113 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 ...
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1answer
125 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 $$ ...
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38 views

Why Laplacian Matrix need normalization and how come the sqrt of Degree Matrix? [on hold]

I am new here. If I do any rough, please forgive me. My question: Why Laplacian Matrix need normalization and how come the sqrt-power of Degree Matrix? The ...
9
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1answer
197 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
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2answers
202 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 ...
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82 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
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2answers
172 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
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1answer
161 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
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1answer
133 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 ...
3
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2answers
250 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?
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1answer
104 views

Poisson integral

Given $g\in L(S_{2n-1}) $ we can define its Poisson integral $\mathcal{P}_{m}^{\lambda}g(z)=\int_{S_{2n-1}}\mathcal{P}_{m}^{\lambda}(z,w)g(w)dw$ my question how I can determine ...
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0answers
78 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 ...
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1answer
175 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 ...
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3answers
281 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
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1answer
203 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, ...
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1answer
125 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)? ...
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2answers
236 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 ...
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0answers
63 views

Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$max~f(x)\quad s.t.~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can show ...
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0answers
53 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 ...
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1answer
168 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 ...
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0answers
48 views

Invariant subspace of bounded self-adjoint operator [migrated]

Assume that we have a self-adjoint operator $T: H \rightarrow H$ with a representation $T(x) = \sum_{n=0}^{\infty} \lambda_n \langle x , x_n \rangle x_n,$ so we have pure point spectrum. Now, I was ...
6
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1answer
349 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 ...
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0answers
78 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 ...
4
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0answers
441 views

Can you equip every vector space with a Hilbert space structure? [migrated]

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$. Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...
0
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1answer
62 views

$\epsilon$-nearly isoclinic

Question: Two $k$-dimensional subspaes $W_1,W_2$ with associated orthogonal projections $P_1, P_2$ are isoclinic with parameter $\lambda \ge 0$ if $P_1P_2P_1=\lambda P_1$ and $P_2P_1P_2=\lambda P_2$. ...
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2answers
105 views

Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?

Should I expect the following Friedrichs/Poincare inequality to hold for $u \in C^\infty(S_n \times (0,\infty))$ with $u(x,0) = 0$: $$\int_{S_n \times (0,\infty)}|u|^2 \leq C\int_{S_n \times ...
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0answers
157 views

Estimating the kernel of Poisson semigroup $e^{-z\sqrt{-\Delta}}$ for complex $z$

Let $f(z,a)$ be an analytic function on $C^+=\{\Re z>0\}$ for each fixed $a>0$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
0
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1answer
78 views

When is a Parseval frame an orthonormal basis? [closed]

every orthonormal basis is a parseval frame. but what about the converse in the finite dimensional case? Let's say $H$ is a n-dimensional Hilbert space and $a_1,..,a_n$ a parseval frame. then, of ...
10
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1answer
207 views

Inner and extendible automorphisms of C*-algebras

If an automorphism $\alpha$ of a C*-algebra $A$ is inner then whenever $A$ is a subalgebra of another C*-algebra $B$, $\alpha$ obviously extends to $B$. Is the converse true: if an automorphism ...
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1answer
128 views

Poincare inequality on balls to arbitrary open subset of manifolds

Let $M$ be an n-dim compact Riemannian manifold with $Ric \geqslant -(n-1)$, it's well known that the following Poincare inequality holds for any function $f\in W^{1,2}(M)$ $$ \frac{1}{m(B)}\int_B ...
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1answer
161 views

Almost sure convergence and weak star convergence

Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of nonnegative measurable functions in $L_1[0,1]$. Assume that $$f_n \to f, a.e.$$ and $$\int f_n h \to \int g h, \forall h \in C[0,1].$$ Question: Do we ...
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0answers
97 views

Approximate Identity: Proof?

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Consider a left ideal $\mathcal{L}$. Denote the positive open ball by: $\mathcal{B}^+$ Then it has a right approximate identity: ...
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91 views

The category of discontinuous Banach spaces

A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
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2answers
333 views

Looking for some function

Is there a continuous function $F: R\to R$ such that $F$ is a surjection but not an injection, $F(Q)\subset Q$ and the restriction $F: Q\to Q$ is an injection, but not a surjection. Here $Q$ denotes ...
3
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0answers
98 views

Norm condition in a Banach lattice

Consider the following "condition (J)" on the norm of a (real or complex) Banach lattice $E$: whenever $x$ and $y$ are disjoint (i.e., $|x|\wedge |y|=0$) then $\|x+y\|+\|x-y\|=2\|x\|+2\|y\|$. ...