# 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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### Entire functions and Bergman spaces

Given an open set $D \subset \mathbb{C}$ with compact closure, let us consider the Bergman space $A^2(D)$ of all holomorphic functions on $D$ that are square integrable on $D$ with respect to the ...

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### Domain of a Jacobi operator with unbounded coefficients

Is it possible to describe the domain of a Jacobi operator explicitly?
Let $J$ be the linear operator acting on a real sequence $(u_{n})_{n\in\mathbb{N}}$ by
$$
J(u_{n}) = a_{n+1} u_{n+1} + a_{n} u_{n-...

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### Dependence and $L^2$ projections of functions

tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...

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### Time regularity vs space regularity for parabolic PDE

Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...

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### What is the quotient of the unit sphere by the bilateral shift on infinite-dimensional separable real Hilbert space?

Let H denote the real Hilbert space 𝓁2(ℤ) with its usual inner product.
If {en | n ∈ ℤ} denotes its standard orthonormal basis, define the unitary mapping W : H → H via W(en) = en+1, extended ...

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### Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?

Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow,
$ \forall n \geq 1 $,
$$ f_n (z) = \dfrac{1}{n^{z}} $$
I would like to ask you if it is possible to construct a ( non-...

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### Compiling a List of Schauder Bases for Function Spaces

Schauder bases are fundamental tools in functional analysis, allowing us to represent functions within a space as infinite sums of basis elements. These bases play a crucial role in studying the ...

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### Book on Hilbert spaces, including non-separable

I am looking for a book that develops the theory of Hilbert spaces, including the spectral theorems and unitary representations, but includes non-separable Hilbert spaces in the main exposition. Any ...

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### Orthogonality in Hilbert algebras and congruence

Consider a finite-dimensional Hilbert space $V$ (say, over $\mathbb{C}$) and a finite-dimensional Hilbert algebra $A$ (i.e., Hilbert space with a compatible associative unitary algebra structure). ...

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### Subspaces of $C_0$ on which $p$-norm are equivalent?

I have a question concerning the generalization of the following fact.
Let $E = C^0([0,1],\mathbb{R})$ endowed with the $\|.\|_\infty$ norm. One can show that if $F$ is a subspace of $E$ for which ...

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### Analogue of $\ell^2(X)$ over an arbitrary Banach ring

Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...

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### Minimal norm problem with linear combination of translation operator to be estimated

Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...

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### About normal states in abstract von Neumann algebras

In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16
but this was state only for concrete von Neumann algebras (because ...

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### $L^2$ space of Hilbert-Schmidt operator valued functions

Let $\mathscr{S}$ denote the space of all Hilbert-Schmidt operators on $L^2(\mathbb R)$. Consider the Hilbert space $L^2(\mathbb R, \mathscr S)$ of square-integrable $\mathscr S$-valued function, that ...

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### A question on projective unitary representation of a Lie group

$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...

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### Is such a Banach space $X$ isometrically isomorphic to a Hilbert space or not?

Let $X$ be a real or complex Banach space. $X$ satisfies:There exists real or complex series $\{a_k\}_{k=1}^n,\{b_k\}_{k=1}^n$ (which satisfies that:$\begin{cases}
a_k,b_k\in \mathbb{R}\ \forall k=1,\...

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### Derivative of Moreau envelope in Hilbert space with respect to regularization parameter $\lambda$ using Hopf-Lax formula?

Let $\mathcal H$ be a Hilbert space, $f \colon \mathcal H \to (- \infty, \infty]$ a proper, convex, lower semi-continuous function and $\lambda > 0$.
The $\lambda$-Moreau envelope of $f$ is
$$
f_{\...

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### What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows:
Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...

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### Proof that generalized Laplacian is essentially self-adjoint (Heat Kernels and Dirac Operators)

According to Proposition 2.33 in Heat Kernels and Dirac Operators each symmetric generalized Laplacian $H$ is essentially self-adjoint. This is an immediate consequence of the fact that
\begin{...

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### Taylor expansion theorem for Gateaux differentiable functions

I am having a hard time studying Gateaux derivatives (see https://en.wikipedia.org/wiki/Gateaux_derivative), it seems that every author mentions the concept but only as a cliffhanger to study Fréchet ...

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### Question about the ergodic mean

This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...

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### Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?

How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...

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### Are the paths of the Brownian motion contained in a suitable RKHS?

Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...

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### Is a bounded convex function $g$ that is non-negative on this particular convex set also non-negative on the its closure?

Setup :
Let $S$ be the simplex on $\mathbb N$, i.e. the set of probability distribution on the natural numbers. Suppose we have $p\in S$ such that, for all $n\geq 1$, $p_n > 0$. For any $\emptyset\...

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### Properties of the relatively bounded probability distributions on the simplex over the natural numbers

Setup :
Let $S$ be the simplex over $\mathbb N$, i.e. the set of probability distribution over $\mathbb N$. Let $p\in S$ be such that $0 < p_n$ for all $n\in \mathbb N$. Let $H$ be the Hilbert ...

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### Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?

(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...

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### Kirszbraun-like extension of periodic functions

Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...

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### Self-adjoint operator with pure point spectrum

Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true:
A has pure point spectrum (i.e., the ...

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### Non-linear transforms of RKHS question

I was reading the paper Norm Inequalities in Nonlinear Transforms (referenced in this question) but ran into difficulties, so I was wondering if anyone could help?
I think I follow the paper until I ...

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### What are the current open problems in dilation theory?

I just started doing my PhD in mathematics. My topic is unitary dilations of operators. I've been reading a lot of papers on that subject so far (especially about the dilation of $n \ge 3$ commuting ...

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### Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...

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### On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...

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### Operator identity

Let $T:\mathcal{D}(A)\to\mathcal{H}$ be a unbounded, self-adjoint, operator with positive spectrum $\sigma(T)\subset [\varepsilon,\infty)$ for $\varepsilon>0$. Hence $T$ is bijective with bounded ...

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### How to show that every Von Neumann algebra is unital?

I was reading the book on operator algebra by Kehe Zhu. The proof of theorem 17.7 (page 107) goes like this :
He first considered the set of all non-empty finite subsets of the set of all projections ...

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### If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?

Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $\|\cdot \|$ its induced norm. Let $a: H \times H \to \mathbb R$ be a bilinear form. We say that
$a$ is coercive IFF there is $C>...

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### A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessarily measurable?

Let $H:= L^2(S^1)$, where $S^1$ is the circle, and $\tau_a : H \to H$ be the translation operator for each $a \in S^1$:
\begin{equation}
(\tau_a f)(x):= f(x+a)
\end{equation}
Then, it is clear that ...

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### Wold decomposition of toral endomorphisms

Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^...

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### Is this a well known space? Perhaps homogeneous Sobolev-like space?

The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...

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### Can Rep(G) tell us whether G is discrete?

Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...

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### Equivalent characterization of weak derivative in Bochner space

Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...

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### finite dimensionality of a subspace of a Banach space

Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...

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### Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...

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### A compact embedding claim

Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...

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### Orthogonal vectors translation using standard vectors

When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$
$$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$
$$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$
It is ...

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### Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?

I am trying to prove that the function:
$$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$
is a positive definite function over the natural numbers. What has sometimes ...

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### Is $L^2(I,\mathbb Z)$ homeomorphic to the Hilbert space?

I am somehow puzzled by the subset $G:=L^2(I,\mathbb Z)$ of $H:=L^2(I,\mathbb R)$ of all integer valued functions on $I=[0,1]$ (in fact I mentioned as an example in this old MO question).
Some simple ...

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### Qualitative values between two electrons in an atom or how to interpret these values?

This question is a little bit trying to understand physics through geometry of simplex:
Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...

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### Perturbation of matrices

Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$.
Question. Does there exist a Lebesgue measurable ...

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### If a Hilbert space-valued mapping is norm-decreasing, can we make sense of the limit of sums consisting of projected-values?

Let $H$ be some separable Hilbert space with a given orthonormal basis $\{ e_n \}$. Write the projection onto the subspace spanned by first $N$ basis elements to be $P_N$.\
Now,let $g(t) : [0,1] \to H$...

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### A RKHS interpretation of the Rydberg formula for hydrogen and an application for physics?

I was thinking if it is possible to define an inner product between two small physical objects with a positive definite kernel and was led to look at the Rydberg formula:
The Rydberg formula for ...