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.
525
questions
6
votes
1answer
172 views
Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?
Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
Question. Is the function $k(g,h) = \...
0
votes
0answers
77 views
Is this set a basis of $L^2(0,T;X)$? [closed]
The space $L^2((0,a)\times(0,b))$ admits an orthogonal basis of the form $(\phi_n^1,\phi_m^2)_{n,m\geq1}$ where $\phi_n^1(x)=\sin(\frac{n\pi x}{a})$, $\phi_m^1(x)=\sin(\frac{m\pi x}{b})$. My question ...
0
votes
0answers
97 views
Extension of a Hilbert basis
The picture below is taken from this paper: http://real.mtak.hu/22877/.
The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
5
votes
1answer
176 views
Initial conditions in the Klein-Gordon equation
I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$)
\begin{equation}\label{kg}
\left\lbrace
\begin{array}{ll}
(\square+m^2)F(x)=0\\
...
3
votes
1answer
128 views
Dense subspace of $\operatorname{Ind}_{H_1 \times H_2}^{G_1 \times G_2} \chi$
Let $H = H_1 \times H_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G_1 \times G_2$, with $H_i \leq G_i$. Let $\chi = \chi_1 \otimes \chi_2$ be a unitary ...
1
vote
1answer
86 views
lifting theorem for n case
I am aware of the following statement of the lifting theorem. For $i\in \{1,2\}$ let $B_i$ be a contraction on a Hilbert space $H_i$ and let $A_i$, acting on the Hilbert space $K_i$, be the minimal ...
0
votes
0answers
21 views
Empirical covering number bounds of functions in rkhs ball
In this paper https://arxiv.org/pdf/1905.11882.pdf, the authors rely on showing the optimal "potential" functions lie in a set with a bounded Holder norm. They're able to show specific ...
0
votes
1answer
99 views
Are there any function spaces with bounded gradients?
Are there any known function spaces where the gradients are uniformly bounded? For a problem I’m working on, I’ve been able to show my functions are bounded in a ball within an RKHS (reproducing ...
0
votes
0answers
35 views
Dual of union of Reproducing Kernel Hilbert Spaces
I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...
0
votes
0answers
33 views
Reproducing kernel Hilbert space of matrix montone function
It is well known that a function $f:(a,b) \to \mathbb{R}$ is matrix-monotone (i.e. $f(A) \leq f(B)$ for any hermitian $A,B \in \mathbb{C}^{n \times n}$ with $A \leq B$ and $\sigma(A),\sigma(B) \...
8
votes
1answer
154 views
Reference for “Every compact quasinilpotent operator is the limit of nilpotent ones”
It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
1
vote
0answers
106 views
$\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$
For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...
1
vote
1answer
102 views
What's the size of non standard monad for weak topology?
There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space):
$$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
1
vote
1answer
73 views
A linear algebra question for semi-Euclidean norm
Let us consider Minkowski inner product on $\mathbb R^{1+n}$, defined by
$$ \langle v,w \rangle = -v_0w_0+\sum_{j=1}^n v_j w_j\quad \,\forall\, v,w \in \mathbb R^{1+n}.$$
We say that a vector $v$ is ...
20
votes
4answers
1k views
Rigged Hilbert spaces and the spectral theory in quantum mechanics
I'm trying to learn some quantum mechanics by myself, and because of my mathematics background, I'm trying to understand it in a rigorous way. Since then, I've been intrigued by the use of rigged ...
2
votes
0answers
91 views
Interpolation of embedded Hilbert spaces and intersection
I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...
4
votes
1answer
225 views
Can the Lagarias inequality be written as a “kernel inequality”?
The Lagarias inequality, which is equivalent to the Riemann hypothesis, is:
$$\sigma(n) \le H_n + \exp(H_n) \log(H_n) =:L(n)$$
for all natural numbers $n$, where $\sigma=$ sum of divisors, $H_n=n$-th ...
0
votes
0answers
154 views
The d(abc)-theorem, the abc-conjecture and positive definite kernels over the natural numbers?
I noticed in the work of Hector Pasten, Th.1.11 the $d(abc)$ theorem and have a question, after doing some experiments with sagemath.
Let $s_k(n) = \sum_{d|n}{ d^k }$ be the sum of divisiors $d$ of $n$...
6
votes
3answers
286 views
Essential spectrum of multiplication operator
Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its essential spectrum. Is there any way to explicitly compute its ...
1
vote
1answer
40 views
Is there any quantitative relationship between the two terms of a Helmholtz decomposition?
Let $\Omega \subset \mathbb R^3$ denote an open, bounded and simply connected set with smooth boundary. The Helmholtz decomposition
$$ L^2(\Omega) = \nabla H^1_0(\Omega) \oplus L^2(\operatorname{div}=...
4
votes
1answer
298 views
Reference request for Deterministic $\subset$ Random $\subset$ Quantum
I hope this post is on topic as a reference request.
I have seen somewhere the idea of (and saw it written just like this):
$$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$
I am ...
2
votes
1answer
167 views
Is it possible to classify non-closed subspaces of Hilbert's space?
Let $H$ be Hilbert's space.
Motivated by my previous question about wildly discontinuous linear functionals, which may be interpreted as an attempt to classify dense hyperplanes in $H$, let me now go ...
2
votes
1answer
125 views
When the adjoint of an unbounded operator on a Hilbert space coincides with the formal adjoint on its natural domain?
This is almost a copy of https://math.stackexchange.com/questions/3931318/when-the-adjoint-of-an-unbounded-operator-on-a-hilbert-space-coincides-with-the
I am trying to work with infinite matrices in ...
2
votes
2answers
234 views
Spin-statistic for free quantum fields
Short version of the question:
Can someone explains what physicist's 'Spin-statistic theorem' says rigorously in the context of Free Quantum fields when they are describe as the Fock space of some '1-...
3
votes
1answer
248 views
Explicit example $f_k \to f$ converging strongly in $L^6(R^3)$, but only weakly in $H^1(R^3)$
Can we find an explicit example of a sequence of functions $f_k \in H^1({\mathbf R}^3)$ such that, $f_k \rightharpoonup f$ weakly converges in $H^1({\mathbf R}^3)$ and $f_k \to f$ strongly converges ...
19
votes
2answers
1k views
Can we take a supremum over all Hilbert spaces?
In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$,
$n\geqslant 2$, by
$$
f_n(c)=\sup\{\|P_n\dotsm ...
0
votes
0answers
139 views
A commuting pair of isometries
Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$.
The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
1
vote
0answers
43 views
Continuity of a linear functional for sequences of projections
Let $T_+$ be the set of a positive trace-class operators over some separable Hilbert space and
$A: T_+ \to \mathbb{R}\cup \{\infty\}$ some linear functional.
In general, $A$ will not be continuous. ...
0
votes
0answers
46 views
Isolated eigenvalues of “bipartite” operators
Please note: This is a reformulation of a previous question of mine. The old question has been already answered to, so I prefer asking a new one. However, it looked like the old formulation did not ...
0
votes
1answer
112 views
Detecting isolated eigenvalues from local spectral measures
Please note: This question has been edited after it became clear from Christian Remling's answer that the original formulation was far from what I really meant to ask.
Let $T\ne 0$ be a self-adjoint ...
1
vote
1answer
129 views
Linearity of the directional derivative of a convex functional at the minimum
Let $H$ be a Hilbert space, $T_+(H)$ the set of positive self-adjoint trace-class operators on $H$, and $f : T_+(H) \to [0,m]$ a non-negative, bounded, convex functional. I don't necessarily know that ...
4
votes
1answer
108 views
Canonical multiplication representation of self-adjoint operator in quantum chemistry and coding theory research
In my applied math research group, we are studying and going over functional analysis results from papers and theses from our institution to generalize their results and apply them in our discrete ...
4
votes
0answers
133 views
Solution without using any k-theory tools
Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...
4
votes
1answer
107 views
hereditary C*-subalgebra of a non-elementary simple C*-algebra
A is said to be elementary if A is isomorphic to some $K(H)$ or $M_n$.
A C*-subalgebra $B$ is said to be hereditary if for every $0≤a≤b∈B$ we have $a∈B$.
I wanted to know that is this statement true?
...
4
votes
1answer
136 views
Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus
In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim ...
7
votes
3answers
164 views
Existence of a non-null and non-negative vector in $F\cup F^\perp$
In $\mathbb{R}^n$ ($n\ge 1$) endowed with the usual dot product, for any linear subspace $F$, does there exist a non-null vector with non-negative coordinates in $F\cup F^\perp$?
0
votes
0answers
73 views
Isolated points of the spectra of self-adjoint operators on Hilbert spaces
Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$.
I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
4
votes
1answer
80 views
If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once?
Let $F$ a Fréchet space.
This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p_{n} \}_{n \in \mathbb{N}}$.
Let's ...
2
votes
1answer
259 views
A geometric approach to the odd perfect number problem?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function.
Define:
$$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
2
votes
1answer
204 views
A bounded operator $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$
Theorem: Let $T$ be a bounded self-adjoint operator on a complex infinite-dimensional Hilbert space $H$. Then $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$.
Proof: If $T$ is compact ...
1
vote
1answer
117 views
Example of linear functionals on $B(H)$
Let $H$ be a Hilbert space and $B(H)$ denotes the space of all continuous linear operators on $H$. I am looking for a class/example of bounded linear functionals $B(H)\to \mathbb C$ which cannot be ...
3
votes
1answer
74 views
Tangent space of smooth Hilbert submanifolds
Let $X, Y$ be Hilbert spaces and $F:X \rightarrow Y$ smooth. Assume that $M := F^{-1}(0) \subset X$ is a smooth submanifold. Is it true that for any $x\in M$, the tangent space $T_xM$ is a Hilbert ...
3
votes
1answer
356 views
What is Bouziani space and what are its applications in mathematics?
I have accrossed a new topological space seems were derived from Hilbert Space and it used to solve some boundary value problem for PDE and ODE , Inspired by this paper (page 4, Definition 3.1) , The ...
5
votes
1answer
144 views
For self-adjoint $A$ and $B$, when is $(A+iB)^*$ the closure of $A-iB$?
Suppose that I have two self-adjoint operators $A$ and $B$ such that $\mathcal{D}(A)\cap\mathcal{D}(B)$ is dense and $B$ positive. Then $A\pm iB$ (with domains $\mathcal{D}(A)\cap\mathcal{D}(B)$) are ...
2
votes
0answers
90 views
invariant theory for non-polynomial functions (eg Hilbert spaces)
I am looking for references regarding the study of group invariant functions that are not polynomials. In particular, I have a (nice) group $G$ and I am interested in what can be said about the $G$-...
2
votes
0answers
120 views
Does a spectral theorem exist for linear operator pencils?
I was wondering if a version of the spectral theorem (the projection valued measure case) holds for linear pencils of the form
$$
A-\lambda B
$$
where $A,B$ are self-adjoint on some Hilbert space $\...
2
votes
1answer
60 views
On the dimension of the range of the resolution of the identity
I want to prove the following: Let $A,B$ be bounded self-adjoint operators in a complex-Hilbert space and $E_A(\lambda)$, $E_B(\lambda)$ its corresponding spectral resolutions, i.e.,
$$A=\int_{[m_A,...
1
vote
0answers
48 views
Multivarate “RKHS” Examples
I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
1
vote
0answers
32 views
Various definitions of coercivity
In this post one says that a functional $F:H\rightarrow [0,\infty]$ on an infinite-dimensional Hilbert space $H$ is (strongly) coercive if there exists a constant $k>0$ such that
$$
F(x)\geq k\|x\|...
6
votes
3answers
237 views
Representation theorem for quadratic form on Hilbert space
I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things ...
