# Questions tagged [hilbert-spaces]

A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

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### Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...

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### What is lost after RKHS embedding of the L1 space?

We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...

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### Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space

$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...

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### $f\in L^2_k(\mathbb R\times \mathbb S^1)$ implies that $t\mapsto f(t) \in L^2_a(\mathbb R, L^2_{k-a}(\mathbb S^1))$?

$\newcommand{\SS}{\mathbb{S}^1}$
$\newcommand{\R}{\mathbb{R}}$
Consider a function $f:\R\times \SS\to \R$ and suppose that $f$ is in the Sobolev space $L^2_k(\R\times\SS)$ for $k>1$ so that we can ...

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### Infinite-dimensional "algebraic varieties"

This question was formerly posted on MSE but did not receive any answer or comment, so I'm re-asking it here.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its ...

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### A question about Marino–Prodi perturbation

In this paper N. Ghoussoub,
the author claims the following version of Marino–Prodi perturbation, that is :
Let $H$ a Hilbert space.
Let $f\in C^2(H, \mathbb{R}),$ $K$ is a compact subset of $K_c$ (...

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### Convergence of solutions of regularised least square problems

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces ($\mathcal{H_2}$ being finite dimensional, but I don't think that it matters). Consider $(A_{\lambda})_{\lambda>0}$ a familly of ...

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### Unicellular compact operators

An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...

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### Concentration inequality for Hilbert space valued random variables

I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,...$ is a sequence of independent $V$-...

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### Is Solèr’s theorem true in constructive mathematics?

Solèr’s theorem says that for every star division ring $R$ and every $R$-module $H$ with an orthomodular Hermitian form $\langle (-),(-) \rangle:H \times H \to R$ such that there exists an infinite ...

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### Mathematical reason for scatter states being special?

In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics.
My understanding is, if $O \in B(H)$ is a self-...

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### Extended adjoint of Volterra operator

Let $V$ be a Volterra operator on $L^2 [0,1]$.
Does there exist a nonzero operator $X $ satisfying the following system
$VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator?
$$ V(f) (x) =\...

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### When are ellipsoids in a convex hull of a sequence with prescribed growth rate?

I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like ...

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### The density of the image of a unitary irrep (a generalization of Burnside's theorem)

I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...

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### Finding the kernel of the discrete Laplace operator?

I have the discrete Laplace operator on an infinite Hilbert space H, and wanted to find if its a Fredholm operator or not. So I want to check if its kernel or cokernel are finite dimensional.
First, ...

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### How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?

I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...

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### Antilinear unbounded operator has closed graph

Let $H$ and $K$ be Hilbert spaces and $D(T)$ a vector subspace of $H$. Let $T: D(T) \to K$ be a densely defined antilinear operator. Its adjoint $T^*: D(T^*)\to K$ is defined by the relation
$$\langle ...

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### Characterize this subspace of the bounded operators on $ L^2(\mathbb{R}) $

I posted this on MSE a couple months ago and it got three upvotes but no answers or even comments so I decided to cross-post it here:
For every pair $ a,b $ of real numbers define the operator $ U_{a,...

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### Conditional Gaussians in infinite dimensions

I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the ...

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### Maximal orthonormal set with fixed distance to subspace

I'm trying to prove an upper bound on the size of a set $S$ of orthonormal vectors in a finite-dimensional Hilbert space $\mathcal{H}$ whose elements all have the same distance to the unit sphere of a ...

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### Extending an unbounded dense linear functional

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$
Let $V \subset H$ be a dense subspace of $H$
Let $f : V \to \mathbb{C}$ be a unbounded functional linear
My question is:
Is ...

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### A question on the self-adjointness of an operator

Given a Hilbert space (separable) $\mathcal{H}$ with an orthonormal basis $\{e_i\}_{i=1}^{\infty}$, define an operator $T$ with domain $\mathcal{D}(T)$ equal to the span of $\{e_i\}$ by $Te_i:=\...

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### Density of smooth function in Hilbert spaces

I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...

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### Name for certain property of equivalent norms on finite-dimensional subspaces of a Banach space

Let $X=(X,\|\cdot\|)$ be a Banach space and suppose that $F\subset X$ is a finite-dimensional subspace. There is then an equivalent norm $|\cdot|$ on $F$ such that $|\cdot|$ is induced by an inner ...

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### A convergence question in $L^2$ construction of Brownian motion

I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...

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### Is the "space of physical quantities" a field of transcendence degree $6$ or $7$ over the rationals?

Excuse my naive question and please let me explain it:
In everyday life we experience 3 spatial "dimensions" + time etc.
Usually the 3 dimensions are represented by a coordinate system and ...

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### Projection onto level set of convex functional

Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically ...

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### Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"

(This question is related to Splitting a space into positive and negative parts but different.)
Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...

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### Reverse martingale convergence theorem in Banach spaces

In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...

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### Schmidt ellipsoids to different orthonormal bases

Let $H$ be a separable, infinite dimensional Hilbert space. For an ONB $(e_n)_{n \in \mathbb{N}}$ of $H$ together with a series $(\alpha_n)_{n \in \mathbb{N}} \subset (0,\infty)$ such that $\sum\...

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### Can RKHS of Gaussian kernels over $\mathbb R^d$ have a non-zero element which is zero on a subspace $\mathbb {R^{\mathit k}\subset R}^d$ where $k>0$?

I have originally asked this question on math.sx but thought maybe here is actually a better place to ask it. Please do let me know if it is actually off-topic for mathoverflow.
I initially thought ...

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### Gaussian process and prior-posterior eigenvalue and eigenfunction

Gaussian process (GP) can be represented as an infinite sequence $\sum_{i\in I}z_i\lambda_i^{1/2}\phi_i$ where $z_i\sim \mathcal{N}(0,1)$, and $\lambda_i$ and $\phi_i$ are the eigenvalue and eigen ...

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### Measurable structures for direct integrals

I'm working with the notion of direct integrals as in Dixmier. Briefly: Given a measurable space $X$ and a family of separable Hilbert spaces $(H_x)_{x\in X}$, a measurable structure is a subspace $Y$ ...

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### eigenvalues of integral operator with centered kernel

Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that
there is a reproducing kernel Hilbert Space $\...

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### Axiomatizing projective Hilbert spaces

This question arises in connection to trying to take a different (more intrinsic) perspective on the foundations of quantum mechanics, in which projective Hilbert spaces naturally arise, e.g. see ...

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### Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...

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### Does the eigenvectors of a self-adjoint operator whose spectrum consists of simple eigenvalues construct a Hilbert basis?

I already asked this question on math.stackexchange a few days ago, but I still haven't received an answer, so I'm asking it here.
Some PDE's have what is called a Lax pair i.e. there exists two ...

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### Geometry in Hilbert spaces / spheres in high dimensions

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are ...

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### Proof: If a reproducing kernel exists for a Hilbert space, then it is unique

I really want to prove the statement in the title but I'm struggling with it. Here my current state:
Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ ...

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### Infinite dimensional topological quantum field theories?

A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. ...

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### Criterion for compactness

Let $T:H\to H$ be a continuous operator on a Hilbert space.
Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero.
Must $T$ be compact?

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### When does an RKHS contain another?

Consider a psd kernel function on the unit-sphere in $\mathbb R^d$ off the form $K(x,x') = \varphi(x^\top x')$ for some $\varphi:[-1,1] \to \mathbb R$, and let $\mathcal H_\varphi$ be the induced ...

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### Complete reducibility, in linear algebra and in topology

I thought that this is a simple question and asked it at the Mathematics StackExchange, but I now have to elevate it to MathOverflow.
Consider a representation $A(G)$ of a group $G$ in a vector space $...

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### norm estimates for Schatten class

Let $C
_p$ be the Schatten-p-classes on a separable Hilbert spaces, $p\ge 1$.
Let ${\rm Tr}$ be the standard trace.
Let $y\in C_p$ be a self-adjoint operator (or even a positive operator) and let $...

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### Drinfeld center of a tensor category

Firstly, apologies for the imprecise language, I'm a physicist trying to understand anyonic excitations from the lens of category theory.
If I have a category (say $\operatorname{Rep}(\mathbb{Z}_2)$) ...

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### How to measure perceived note similarity in music / simplicity of ratios?

I have discovered a method to measure the similarity of two successive musical notes which I wanted to share with a question:
It is known in music theory that two successive pitches $a,b$ which sound “...

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### A consequence of the Min-Max Principle for self-adjoint operators

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...

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### Link between a categorical and an algebraic characterization of (infinite-dimensional) Hilbert space

On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1 latest Arxiv version) offers a characterization of the ...

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### Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces [closed]

Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded ...

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### Complemented submodules of a Hilbert C*-module

Let $A$ be a $C^*$-algebra and $E$ be a (right) Hilbert $C^*$-module over $A$. Assume $F$ is a closed submodule of $E$ such that $F^\perp := \{x \in E: \langle x, F\rangle=0\}$ is orthogonally ...