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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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 ...
Gabriel Palau's user avatar
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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 ...
Ribhu's user avatar
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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{...
Mahtab's user avatar
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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,\...
anyon's user avatar
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1 answer
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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_{\...
ViktorStein's user avatar
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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 ...
S-F's user avatar
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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{...
Filippo's user avatar
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2 answers
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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 ...
Antonio Martins Alves Veloso d's user avatar
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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 ...
Vakos's user avatar
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3 votes
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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 ...
Mueller's user avatar
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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\...
P. Quinton's user avatar
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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 ...
P. Quinton's user avatar
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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. ...
jetSett's user avatar
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124 views

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 ...
user3476591's user avatar
2 votes
1 answer
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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 ...
Mat's user avatar
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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 ...
S-F's user avatar
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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 ...
Daniel Goc's user avatar
2 votes
1 answer
169 views

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 ...
B.Hueber's user avatar
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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 ...
UtsabrajSarkar's user avatar
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165 views

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>...
Akira's user avatar
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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 ...
Isaac's user avatar
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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}^...
an_ordinary_mathematician's user avatar
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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_{\...
Dan1618's user avatar
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18 votes
0 answers
353 views

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 ...
André Henriques's user avatar
2 votes
1 answer
106 views

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) \, ...
Mandelbrot's user avatar
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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 \...
Ali's user avatar
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2 votes
0 answers
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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 ...
B.Hueber's user avatar
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2 votes
1 answer
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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 ...
Ali's user avatar
  • 4,045
1 vote
1 answer
117 views

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 ...
ABB's user avatar
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2 votes
1 answer
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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 ...
mathoverflowUser's user avatar
6 votes
0 answers
174 views

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 ...
Pietro Majer's user avatar
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1 vote
1 answer
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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 ...
mathoverflowUser's user avatar
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1 answer
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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 ...
Ali's user avatar
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1 vote
1 answer
54 views

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 $g(t) : [0,1] \to H$...
Isaac's user avatar
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0 votes
1 answer
588 views

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 ...
mathoverflowUser's user avatar
2 votes
1 answer
139 views

$(S\otimes T)^{it}= S^{it}\otimes T^{it}$ for unbounded operators

Let $S,T$ be unbounded, closed operators in Hilbert spaces $H,K$. In that case, we can form the tensor product operator $S\otimes T$ on the Hilbert space $H\otimes K$ which is the closure of the ...
Andromeda's user avatar
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0 answers
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Question about a step in the proof of the min-max principle

I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
MathMath's user avatar
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2 votes
1 answer
139 views

Variation of concept of a Lusin space

Citing from Wikipedia, A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space. Is there a (previously studied) analogous concept of a Hausdorff (...
iolo's user avatar
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3 votes
0 answers
104 views

Unitary operators with the same inner product as vectors

Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
user3826143's user avatar
1 vote
1 answer
107 views

An inner product and projection property in RKHS

I'm reading an article regarding the condition for the existence of a solution to a constrained Pick interpolation problem, and the author wrote the following equality - for background, we have the ...
GBA's user avatar
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2 votes
1 answer
129 views

Orthonormal bases in RKHSs via interpolating sequences

Definitions and setting Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
ABIM's user avatar
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2 votes
0 answers
229 views

Examples of RKHS that are "classical"

Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs? It is easy to construct example of RKHSs by applying the ...
lost_analyst's user avatar
0 votes
1 answer
147 views

Spectrum of a product of a symmetric positive definite matrix and a positive definite operator

Let $\mathbf H$ be an infinite dimensional Hilbert space. I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
SAKLY's user avatar
  • 63
1 vote
0 answers
103 views

Formula for the kernel of an operator

Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...
dennis's user avatar
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1 vote
1 answer
325 views

Hilbert–Pólya conjecture with Grommer inequalities?

The Grommer inequalities are equivalent to RH and formulated on page 20 of Conrey - Riemann's hypothesis: Let $$\Xi(t) := \xi(1/2+it).$$ Then RH is equivalent to : All zeros of $\Xi(t)$ are real. The ...
mathoverflowUser's user avatar
1 vote
0 answers
71 views

Commutator of self-adjoint operators and $C^1$-type formula

Let $\mathcal{H}$ be a (complex) Hilbert space. Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
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