**1**

vote

**0**answers

44 views

### Linear independence of an odd set of measurable functions

Let $g(t)$ be a convex positive function. I'm trying to show that the set $\{ |t|^ne^{\frac{g(t)}{t^n}}\}_{n\in \mathbb{N}}$ is linearly independent in the space of measurable functions on $\mathbb{R}...

**1**

vote

**1**answer

97 views

### Bounded operators $T: B(H)\to H$ whose Kernel is a Lie algebra

Assume that $H$ is an infinite dimensional Hilbert space.The space of all bounded operators on $H$ is denoted by $B(H)$.We consider the Lie algebra structure $[T,S]=TS-ST$ on $B(H)$.
Is there a ...

**1**

vote

**1**answer

96 views

### Some integrals with respect to a Gaussian measure on a Hilbert space

Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...

**0**

votes

**1**answer

83 views

### Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...

**0**

votes

**1**answer

133 views

### Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...

**3**

votes

**1**answer

65 views

### $C(X)$-compact operators and families of compact operators

In this question Operators on Hilbert $C^*$-module and families of Fredholm operators I asked about the relation between being a family of compact operators $F:X \to K(H)$ on Hilbert space $H=\ell^2$ ...

**0**

votes

**2**answers

121 views

### Strictly convex norm on an infinite-dimensional Hilbert space

Question: Consider the Hilbert space $ H=\ell^2(\mathbb{Z})$. Let ${\rm L}(H)$ be the set of all linear
operators on $H$ onto itself. Find a norm $N$ and a domain $DN\subset {\rm L}(H)$ for $N$ ...

**4**

votes

**0**answers

147 views

### How can a sequence of functions be dense in L^2

Assume $\Omega$ is a bounded domain in $\mathbb R^d$ with sufficiently smooth boundary. Let $\{\lambda_n,\varphi_n\}_{n=1}^\infty$ be an orthonormal eigensystem of the Laplacian opertor $-\Delta$, ...

**2**

votes

**0**answers

23 views

### Multivariable Weighted shift and subnormality

I have asked this question in mathstackexchange but didn't get any answer. I hope, I will get my answer here.
Let $\mathbb B^m$ denote the Euclidean ball in $\mathbb C^m.$ Does there exist a ...

**0**

votes

**0**answers

145 views

### Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...

**0**

votes

**1**answer

71 views

### Finding a vector representation for a data where we only know the inner products

I am an engineer working on speech signal processing and I have a problem that I have encountered while trying to model speech signals. The mathematical formulation is not entirely pure and I try to ...

**0**

votes

**0**answers

40 views

### Prokhorov convergence of Gaussian measures

Consider a Hilbert space $\mathcal{H}$ and a sequence of centered Gaussian measures $\mu_n$ on it. The covariance operators of $\mu_n$ are defined via their eigenpair(eigenbasis and eigenvalue)) as ...

**2**

votes

**1**answer

172 views

### Selfadjointness of hamiltonian with 1/x potential

Let us consider the Hilbert space $L^2([0,\infty))$ and operator $H=-\frac{d^2}{dx^2} + \frac{1}{x}$ on the domain of $C^{\infty}_0((0,\infty))$ (smooth functions with compact support away from $0$).
...

**3**

votes

**2**answers

131 views

### The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant

In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on:
If $ \mathbb{H} $ is an RKHS and we denote the ...

**6**

votes

**1**answer

188 views

### Definitions of Hilbert Bundles

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...

**4**

votes

**2**answers

222 views

### Error estimate in the spectral theorem of compact operators on a Hilbert space

Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...

**0**

votes

**0**answers

57 views

### What equals $\ker[(A-\lambda I)^+]$ for a negative unbounded operator $A$?

We have the following result:
$\{ E_{\lambda}; \, -\infty <\lambda < + \infty\}$ is a spectral family, where $E_{\lambda}$ is the projection of $H$ onto the null space $\mathscr N \left(A_\...

**6**

votes

**1**answer

333 views

### Current status of computable spectral theorem and interpretation of quantum mechanics

The spectral theorem states if $A$ is a Hermitian operator acting on an $n-$dimensional Hilbert space space $H$, and $\lambda_1, ... \lambda_m$ are $m \leq n$ distinct eigenvalues of $A$, then
$$ H=\...

**0**

votes

**1**answer

43 views

### Simplify the expression of $ T^+$ for an unbounded operator $T$?

For a negative unbounded operator $T$, what equals the operator
$$ T^+ = \left[\frac{1}{2}(|T| + T) \right]^{**},$$
where $|T|= (T^2)^{1/2}$ and $A^{**} $ is the minimal closed extension of an ...

**1**

vote

**3**answers

129 views

### Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...

**3**

votes

**0**answers

140 views

### A strongly open set which is not measurable in the weak operator topology

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...

**5**

votes

**2**answers

493 views

### Holomorphy of a function with values in a Hilbert space

I asked the same question on MathStackExchange. EDIT: This question has now a open bounty worth +50 reputation on MSE.
Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq ...

**1**

vote

**0**answers

42 views

### Complex conjugate and unitary complex conjugate

Definition: Let V be complex finite dimensional inner product space
Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex ...

**4**

votes

**1**answer

110 views

### Every self-adjoint trace class operator on $L^2$ has integral kernel

I have asked this question on MSE but did not receive an answer. I thought I could try it here.
Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be ...

**2**

votes

**1**answer

120 views

### Visualizing ANOVA Decomposition [closed]

Let $f \in L^2[0,1]^d$ be a measurable function where $d \in \mathbb{N}$. For a given subset $u \subseteq D := \{1,2,\ldots,d\}$ consider the projections $P_u : L^2[0,1]^d \to L^2[0,1]^{|u|}$ given ...

**10**

votes

**1**answer

240 views

### Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...

**7**

votes

**1**answer

173 views

### Orthonormal bases on Reproducing Kernel Hilbert Spaces

Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional $...

**0**

votes

**0**answers

49 views

### Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function $f=f(...

**7**

votes

**1**answer

270 views

### A Hilbert space characterization via retractions--a conjecture

Given a Banach space $X$ and a functional $f:X\rightarrow \mathbb R$, let
$$ X_f := \{x\in X : f(x)\ge 0\} $$
("functional" means "non-zero linear functional"). Also, given a topological space $E$ ...

**5**

votes

**0**answers

112 views

### Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of
quantum mechanics. On the positive side it demonstrates how the probabilistic
structure of quantum theory follows from its logical ...

**1**

vote

**0**answers

21 views

### Efficient packing in Gaussian measures on Hilbert spaces

Let $B(0,1)$ be the unit ball in a separable Hilbert space $H$ with a Gaussian measure $\mu$ on it. For a small $r > 0$, can we have $x_i \in B(0,1)$, $0 < r_i \leq r$ and a constant $K > 0$ ...

**6**

votes

**0**answers

94 views

### orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...

**5**

votes

**1**answer

110 views

### Discrete Wavelets

I am looking for research that has been done in Discrete wavelets. Let me be specific as Google doesn't give me what I want when I say "discrete wavelets". I don't want countable basis for $ L^2(\...

**0**

votes

**0**answers

42 views

### Hilbert Manifold structure on global sections of a sheaf

Let $<U_i,\phi_i>$ be an atlas on a Hilbert manifold $M$. If $F(M,U_i)$ is a Hilbert space for every $U_i$ then does this imply that $F(M,M)$ can be made into a Hilbert manifold also with ...

**5**

votes

**2**answers

185 views

### Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general:
Let $X$ be a separable Banach space and $X^*$ the dual space. The mean
value $...

**4**

votes

**1**answer

149 views

### Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
$$P_{\{\lambda_1,...\lambda_n\}}=\frac{...

**0**

votes

**1**answer

69 views

### Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?

**37**

votes

**5**answers

5k views

### Are dagger categories truly evil?

Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...

**0**

votes

**1**answer

75 views

### Norm of derivative of rank one projector

I asked this question on math.stack but I got no answer, so I try here.
Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation}
i\partial_t\phi(t)=-\Delta\phi(t)+(V*|\phi|^...

**3**

votes

**0**answers

138 views

### Deformation and Representations

Let $\widetilde{U_q(sl_n)}$ denote a deformation of the algebra $U_q(sl_n)$. In particular, $\widetilde{U_q(sl_n)}$ is defined by the same generators and relations and $*$-operations as $U_q(sl_n)$ ...

**1**

vote

**0**answers

56 views

### Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions:
Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...

**4**

votes

**2**answers

63 views

### Distribution of the RKHS norm of the posterior of a Gaussian process

In a classical noisy regression setting, let $\big(f(x)\big)_{x\in\cal X}$ be a centered Gaussian process of covariance $k$ on a compact $\cal X$, and $\mathcal{F}_n$ be the filtration generated by ...

**2**

votes

**0**answers

43 views

### Equivariant exponential map on Hilbert manifolds

Let $M$ be a Hilbert Riemannian manifold and $G$ a finite-dimensional Lie group acting on $M$. It is well-known that when $M$, too, is finite-dimensional
$$\exp_p: U \subset T_pM \rightarrow \exp_p(U) ...

**0**

votes

**1**answer

156 views

### On the Riesz representation theorem II

I have a follow-up question to On the Riesz representation theorem .
Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the ...

**4**

votes

**1**answer

386 views

### On the Riesz representation theorem

Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument - not necessarily a Hilbert space. Let $\Phi$ be an antilinear functional on $V$.
What are the precise (...

**6**

votes

**1**answer

513 views

### An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...

**1**

vote

**0**answers

101 views

### Algorithm for finding eigenfunctions

I have an $ L^2(\mathbb{R}) $ operator that looks like
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |,
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...

**-2**

votes

**1**answer

181 views

### No Hilbert space can have countable Hamel basis without using Baire's Category theorem [closed]

I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...

**2**

votes

**1**answer

187 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{...

**10**

votes

**2**answers

620 views

### Schur's Lemma for Hilbert spaces

Let $H$ be a complex Hilbert space and let a group $G$ act on $H$ such that there are no invariant closed subspaces besides $H$ and $(0)$. Let $D$ be the ring of bounded operators which commute with ...