# 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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### 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 ...

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### Unitary operators in non-separable Hilbert spaces

In Halmos's "A Hilbert space problem book" we read "If $H$ is non-separable, then it is the direct sum of separable infinite-dimensional subspaces that reduce $A$ [$A$ is any normal ...

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### 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 ...

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### 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)\...

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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|...

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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)^{...

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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:
$$...

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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 ...

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### 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 ...

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### 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
$$\...

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### 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 ...

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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 +\...

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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,...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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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):=\...

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### 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(\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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)...

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### 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 ...

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### 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 &...

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### 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 \...

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### $L^2$ 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 $\psi_h(x)$...

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### 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 ...

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### 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 ...

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### Hypo-monotone operators in Hilbert Spaces

If $H$ is a separable Hilbert space, an operator $A \subset H \times H$ is said to be $\lambda$-monotone ($\lambda \in \mathbb{R}$) if
$$ \langle v-w, x-y \rangle \ge \lambda |x-y|^2 \quad \text{ for ...

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### 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 ...

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### 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 ...

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### 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} ...

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### 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 $...

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### 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 ...

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### 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 ...

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### Operators decomposition in pseudo-Hilbert space

Let $(H,g)$ be a pseudo-Hilbert space, i.e. $H$ is an infinite dimensional vector space endowed with an indefinite symmetric product $g$. Suppose we have a linear operator $D:H\to H$ and let $D^*$ be ...

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### 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 ...

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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 ...

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### 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 $\...

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### 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 ...

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### 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 $A$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Do these properties characterize Hilbert spaces?

Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ isometric with $l_2^2$ such that $x\in E$. Does this property characterize a (...

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### Is the Sobolev space $H^1(\mathbb{R})$ contained in the domain of $(-\partial_x \alpha(x) \partial_x)^{1/2}$?

Let $\alpha(x) : \mathbb{R} \to (0,\infty)$ have bounded variation (BV) and suppose $\inf_{\mathbb{R}} \alpha > 0$. Consider the second order differential operator
$$H : =-\partial_x (\alpha(x) \...

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### How to prove the polar decomposition of unbounded operators?

Let $ T $ be a closed, densely defined operator on a Hilbert space $ H $. Then there exists a positive self-adjoint operator $ A $, $ D(A)=D(T) $ and a isometric operator $ V:R(A)\to \overline{R(T)} $ ...

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### Is $\sup\{\|A\widehat{k}_{\lambda}\|: \lambda\in\Omega\}=\sup\{\|A^*\widehat{k}_{\lambda}\|: \lambda\in\Omega\}$? where $A$ is an operator on RKHS

A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....