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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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 ...
Ali's user avatar
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23 votes
4 answers
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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 ...
MathMath's user avatar
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2 votes
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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}...
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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 ...
mathoverflowUser's user avatar
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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 $n$...
mathoverflowUser's user avatar
7 votes
3 answers
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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 ...
potionowner's user avatar
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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}=...
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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 ...
JP McCarthy's user avatar
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1 answer
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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 ...
Black's user avatar
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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 ...
apyshkin's user avatar
3 votes
2 answers
363 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-...
Simon Henry's user avatar
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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 ...
katago's user avatar
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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 ...
Ivan Feshchenko's user avatar
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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 $...
ABB's user avatar
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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. ...
Artemy's user avatar
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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 ...
Maurizio Moreschi's user avatar
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1 answer
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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 ...
Maurizio Moreschi's user avatar
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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 ...
Artemy's user avatar
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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 ...
groupoid's user avatar
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4 votes
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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 ...
Peg Leg Jonathan's user avatar
5 votes
1 answer
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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? ...
Peg Leg Jonathan's user avatar
4 votes
1 answer
267 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 ...
groupoid's user avatar
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7 votes
4 answers
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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 $F\cup F^\perp$?
G. Panel's user avatar
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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 ...
Maurizio Moreschi's user avatar
1 vote
0 answers
81 views

Geometry for an odd perfect number? (Second question)

Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{R})$. Let $h(n) = J_2(n)$ be the second Jordan totient function. Define: $$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
mathoverflowUser's user avatar
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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 ...
Peter's user avatar
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2 votes
1 answer
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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)}...
mathoverflowUser's user avatar
1 vote
1 answer
711 views

A bounded operator $T$ is compact if and only if $\sigma_{\mathrm{ess}}(T)=\{0\}$ [closed]

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 ...
Andrés Felipe's user avatar
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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 ...
NewB's user avatar
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3 votes
1 answer
162 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 ...
user160747's user avatar
3 votes
1 answer
423 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 ...
zeraoulia rafik's user avatar
5 votes
1 answer
156 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 ...
Mathmo's user avatar
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2 votes
0 answers
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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 $G$-...
user27182's user avatar
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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 $\...
Mathmo's user avatar
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2 votes
1 answer
88 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,...
Andrés Felipe's user avatar
1 vote
0 answers
65 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}^...
ABIM's user avatar
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1 vote
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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\|...
ABIM's user avatar
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6 votes
3 answers
659 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 ...
MathMath's user avatar
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4 votes
1 answer
303 views

Boolean ring of unitary divisors / Structure of unitary divisors?

I hope this question is appropriate for MO: Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors. We can make $U_n$ to a boolean ring: $$a \...
user avatar
1 vote
1 answer
405 views

Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following. Let $H$ be ...
ABIM's user avatar
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3 votes
1 answer
401 views

Is the maximum of derivatives of a function in (s,2)-Sobolev space (an RKHS) bounded by their norms?

Let $f(x) \in W^{s,2}(\Omega) \equiv H^s$, where $\Omega \subseteq \mathbb{R}^d$, $s > d/2$ and $W^{s,2}$ is a $(s,2)$-Sobolev space. Clearly, $W^{s,2}$ is an Reproducing Kernel Hilbert Space (RKHS)...
qwer1304's user avatar
1 vote
0 answers
24 views

On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators

Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book: Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert Space. CRC press above the statement of ...
Manuel Norman's user avatar
4 votes
1 answer
192 views

A kind of holomorphicity of maps on Hilbert space

Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?: 1)For every open set $U\subset H$ and every ...
Ali Taghavi's user avatar
0 votes
0 answers
86 views

How to introduce not-orthonormal base on Reproducing Kernel Hilbert Spaces?

I read some tutorial papers and slide,and find that the bases on Reproducing Kernel Hilbert Spaces always be orthonormal. For examples,you can refer to this link for the content about Reproducing ...
Max's user avatar
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0 answers
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Norm equivalences for Gaussian random functions (Cameron-Martin space)

Preliminaries Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$. These random functions are drawn from a Gaussian measure $\...
ares's user avatar
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0 answers
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A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?

In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
UserA's user avatar
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5 votes
1 answer
176 views

An extension of Lomonosov Theorem

Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result: Theorem (Lomonosov): Every nonscalar $T \in B(H)$ which commutes ...
Manuel Norman's user avatar
3 votes
1 answer
787 views

RKHS norm of Lipschitz functions

Given a set $\mathcal{X}$ and RKHS $\mathcal{H}$ of functions on $\mathcal{X}$, we can recover a (pseudo)metric on $\mathcal{X}$ by $d(x,y)=||\phi_x-\phi_y||_{\mathcal{H}}$, where $\phi_x=k(x,\cdot)$. ...
Tyler6's user avatar
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7 votes
0 answers
223 views

Freys elliptic curves and Hilbert spaces?

Consider the Frey-Hellegouarch curve given $a,b$ positive rational numbers: $$y^2= x\left(x-\frac{a}{\gcd(a,b)}\right)\left(x+\frac{b}{\gcd(a,b)}\right)$$ The j-invariant is given by: $$j(a,b) = \frac{...
user avatar
4 votes
1 answer
251 views

On the automorphisms of the unitary group in the strong operator topology

Let $H$ be an infinite dimensional complex (or real) Hilbert space, and let $U(H)$ be the unitary (or orthogonal) group. We equip $U(H)$ with the strong topology. Now, suppose that $\phi: U(H) \...
Peter's user avatar
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