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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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 ...
DerGalaxy's user avatar
4 votes
2 answers
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When are finite-dimensional representations on Hilbert spaces completely reducible?

Let $G$ be a group and $\pi$ be a finite-dimensional (not necessarily unitary) representation of $G$ on a complex Hilbert space $H$. We shall say that $\pi$ is completely reducible if there exists a ...
Nanoputian's user avatar
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1 answer
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Intersection of Hilbert spaces with Schauder basis

Let $H$ be a infinite dimensional, separable, complex Hilbert space, $\{v_{1_n}\}_{n \in \mathbb{N}}$ be a sequence in $H$, $V_1=\operatorname{span}\{v_{1_n}\}_{n \in \mathbb{N}}$ $U_1=\overline{V_1}$...
Matey Math's user avatar
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Is a RKHS defined using a feature map over another RKHS bigger than the latter RKHS?

I am interested in learning more about what happens when 'composing' two reproducing kernel Hilbert spaces (RKHS). Let $\phi\in C(\mathbb{R})$ and $X=[-1,1]^d$. Suppose we have two RKHSs with the ...
ChocolateRain's user avatar
1 vote
0 answers
114 views

Logic which depends on the point of view / perspective? (Semantic space of logic)

I am not sure if this question is appropriate for MO, since I have only a basic understanding of Boolean logic, and am maybe not qualified to ask questions beyond that, but still I will try to write ...
mathoverflowUser's user avatar
2 votes
2 answers
153 views

LF or LB space that happens to be finite dimensional

Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further ...
Isaac's user avatar
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2 votes
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Orthogonal complement of arbitrary intersection of Hilbert subspaces

Let $H$ a Hilbert space, and $\mathcal C$ an arbitrary set of closed subspaces of $H$. Is it true that $$\left( \bigcap_{Z\in \mathcal C}Z\right)^\perp = \overline{\sum_{Z\in \mathcal C} Z^\perp}$$ ...
Nathaël's user avatar
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Inequality for normed power n, m

Let $ B (H) $ indicate the set of all bounded linear operators on a complex separable Hilbert space $ H $. Let $ A \in B(H) $, where $ A $ is a positive semi-definite operator in $ H $ (i.e. $ \langle ...
Anas Abbas H.'s user avatar
3 votes
1 answer
57 views

Why does the normalization term disappear when computing the MLE of decomposed Gaussians

Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
hdeplaen's user avatar
6 votes
1 answer
288 views

Log-convexity of determinant

Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
António Borges Santos's user avatar
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Compact embedding of homogeneous weighted Sobolev spaces

Let $n\geq 2$ and let $\Omega$ be the open unit ball with the origin removed. For each $\delta>0$ and each $u\in C^{\infty}(\Omega)$ let us define $$ \|u\|^2_{L'^1_\delta(\Omega)}= \int_{\Omega} |x|...
Ali's user avatar
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4 votes
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A question on products of linear combinations of complex matrices

Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds $$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
user493645's user avatar
1 vote
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Nonlocal elliptic problem - what is its associated energy?

It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem: $$...
Bogdan's user avatar
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Mach's principle, Newton's law and Hilbert sphere?

(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.) I wanted to share with you something I stumbled upon ...
mathoverflowUser's user avatar
-4 votes
2 answers
467 views

Inverse square-law as a positive definite kernel?

Newtons law for gravity states that: $$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$ The function : $$k(x,y):=\exp(-| x-y|^2)$$ is known to be a positive definite function, called the RBF-kernel. It ...
mathoverflowUser's user avatar
2 votes
0 answers
124 views

A question about Gauss-Green formula - a weaker assumption

The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place $$\...
Bogdan's user avatar
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0 votes
1 answer
90 views

A property of the canonical dual frame in a Hilbert space

Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as \begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation} is a Hilbert space ...
an_ordinary_mathematician's user avatar
0 votes
1 answer
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Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?

Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that $${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$ $${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...
user92646's user avatar
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1 vote
0 answers
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Wave equation on $[0,1]$ with mixed boundary conditions

Consider the wave equation $u_{xx}-u_{tt}=0$ on the unit interval $x\in[0,1]$. Take mixed boundary conditions ($\alpha_{1,2}^2+\beta_{1,2}^2 \neq 0$) \begin{align*} \alpha_1 u(0,t) + \beta_1u_x(0,...
J_P's user avatar
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Norm of the Linear Operator

Let $G$ be a compact group, and $\pi : G \rightarrow \mathcal{U}(H)$ be a continuous unitary representation. Let $f \in L^{1}(G)$ be arbitrary. By Riesz Representation Theorem we can find a bounded ...
Peg Leg Jonathan's user avatar
3 votes
1 answer
156 views

Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?

This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
tparker's user avatar
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192 views

Convergence of inverse operator with projections

Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...
John's user avatar
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1 vote
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75 views

Integration over a finite-dimensional subspace of Hilbert space

Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
John's user avatar
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2 votes
1 answer
209 views

If $A$ is a closed operator, is $A^k$ closed?

Let $A$ be a closed (densely defined) operator on a Hilbert space $H$. We define for a natural number $k$, the operator $A^k$ with its natural domain. Is $A^k$ closed?
Andromeda's user avatar
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Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$

Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
DeltaEpsilon's user avatar
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1 answer
79 views

Derivative in Sobolev space extended by zero

Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero. How to find $J'(u)$ for $$ J(u)= \int\limits_0^l |u(x)|^2\operatorname{d\!}x\;?? $$ In $L_2$ it's easy: $$ J'(u) = \left(\...
anon.for's user avatar
2 votes
1 answer
228 views

Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book. Suppose ...
Andromeda's user avatar
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4 votes
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118 views

Maximally fine topologies on $B(H)$ making the unit ball compact

Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
Aareyan Manzoor's user avatar
3 votes
1 answer
321 views

Takesaki II Lemma 1.13: stuck in proof

Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"): Here, we associate with an ...
Andromeda's user avatar
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3 votes
2 answers
257 views

Unbounded positive self-adjoint without $0$ in its spectrum: can we construct its inverse using functional calculus?

Let $P$ be a positive, self-adjoint (unbounded) operator in a Hilbert space $H$ with $0\notin \sigma(P)$. Consider its spectral decomposition $$P = \int_{\sigma(P)} t dE(t).$$ Since $0 \notin \sigma(P)...
Andromeda's user avatar
  • 189
5 votes
2 answers
247 views

Dilation of bounded linear operators

Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
SKNEE's user avatar
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0 answers
35 views

Uniqueness of solution to abstract wave equation with unsigned energy

Let $H$ be a self-adjoint operator on a Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle$). Suppose the spectrum of $H$ in $(-\infty, 0)$ consists of only finitely many eigenvalues $\mu^2_k &...
JZS's user avatar
  • 469
3 votes
0 answers
190 views

On a paper of von Neumann

Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality $$ \lVert p(T)\rVert \leq \sup \...
HaSa's user avatar
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1 vote
0 answers
54 views

$L^2$ norm of a kernel with a variable width

Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
Caroline Wormell's user avatar
2 votes
0 answers
65 views

RKHS lying in another RKHS

Suppose $H_1$ and $H_2$ are reproducing kernel Hilbert spaces such that $H_1 \subset H_2$. For $f \in H_1$, when can I bound $\|f \|_1$ with $C\|f\|_2$ (for some $C$)? Is there a relationship between ...
Athere's user avatar
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1 vote
1 answer
141 views

The space of linear operators between Hilbert spaces has martingale type 2

I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded ...
Shashi's user avatar
  • 113
2 votes
1 answer
209 views

When is Euclidean distortion finitely determined?

The Euclidean distortion of a metric space $X$, denoted $c_2(X)$, is the infimum of $c$ for which there exists a map $f\colon X\to\ell^2$ such that $$d_X(x,y) \leq \|f(x)-f(y)\|_{\ell^2} \leq c\cdot ...
Dustin G. Mixon's user avatar
2 votes
0 answers
132 views

How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]

I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
bernard's user avatar
  • 205
1 vote
1 answer
80 views

Show $A_0 + S$ is invertible, where $S$ is the Riesz projection of bounded $A_0$

Let $A_0$ be a bounded linear operator on a Hilbert space $H$. Suppose $0$ is an isolated point of the spectrum of $A_0$. Let $S$ be the corresponding Riesz projection, namely, $$S = -\frac{1}{2\pi i} ...
JZS's user avatar
  • 469
6 votes
0 answers
170 views

Measurability of eigenvalues-eigenvectors of a positive compact operator

Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$. By the spectral theorem, given $a \in A$, there are scalars $...
user127022's user avatar
1 vote
1 answer
223 views

Isometries of Hilbert space

It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two ...
Markus's user avatar
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1 vote
0 answers
84 views

Invariance signature in infinite dimension

Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that: $g_0$ is positive-definite $g_t$ is non-degenerate for ...
John117's user avatar
  • 395
2 votes
0 answers
125 views

What are examples of infinite-dimensional Banach spaces that are also measure spaces?

I am interested in examples of infinite-dimensional vector spaces that are Banach spaces or even Hilbert spaces are measure spaces Instead of the full vector space, subsets with measure structure ...
shuhalo's user avatar
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1 vote
1 answer
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

This is a follow-up to this previous question, but under stronger assumptions. Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real scalar field). Let $\tilde ...
Jochen Glueck's user avatar
7 votes
2 answers
379 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
Jochen Glueck's user avatar
4 votes
1 answer
781 views

Left and right eigenvectors are not orthogonal

Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
Guido Li's user avatar
0 votes
0 answers
190 views

Eigenvalue multiplicity of tensor product of positive operator with itself

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
Dasherman's user avatar
  • 203
3 votes
0 answers
119 views

Domain of operator which is used in operator monotone function

We are studying the paper Rupert L. Frank, Leander Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, Journal für die reine und angewandte Mathematik (Crelles ...
Houa's user avatar
  • 561
7 votes
1 answer
220 views

A characterization of Hilbert spaces by norm one projections

Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
Markus's user avatar
  • 1,361
4 votes
1 answer
351 views

Motivation for Heisenberg's modeling of observables

What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...
MrPajeet's user avatar
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