Questions tagged [hilbert-spaces]
For questions involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.
8,710 questions
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Under what conditions is a subset of the Span($X$) (where $X$ is a totally bounded subset of a Banach space) relatively weakly compact?
More specifically, I'm considering the following problem: Let $v: \Omega \to \mathcal H$ be a Pettis-integrable Hilbert space-valued function defined on a perfect probability space $(\Omega, \mathcal ...
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Do non-GNS representations of C*-algebras exist?
Consider a representation $(\mathcal{H}, \pi)$ of a C* algebra $\mathcal{A}$. Does there always exist a state $\omega$ of $\mathcal{A}$ such that the corresponding GNS representation $(\mathcal{H}_\...
- 4,019
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What are the minimal assumptions for the projection onto a vector subspace to be well-defined?
I have been teaching (pre-)hilbert space theory recently, and started asking myself a few questions regarding the minimal sets of assumptions guaranteeing that the metric projection onto a set is well-...
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Representing a radially weighted integral of the radial derivative of a holomorphic function
Let $\mathbb{B}_d$ denote the unit ball in $\mathbb{C}^d$ and $\mathrm{Hol}(\mathbb{B}_d)$ the space of holomorphic functions on it. For $f \in \mathrm{Hol}(\mathbb{B}_d)$ we define the radial ...
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Question on controlling norm of output of a semidefinite operator by inner product
I'm asked to prove the following proposition:
Prop. If $H$ is a complex Hilbert space, $A\in L(H)$ satisfies that
$$
(Ax|x)_H\ge0,\ \forall x\in H,
$$
then
$$
\|Ax\|_H^2\le\|A\|_{H\rightarrow H}(Ax|x)...
- 395
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Orthogonal of the kernel [duplicate]
Let $H$ a Hilbert space and $T \in L(H)$ a linear self-adjoint operator, I cannot find the theorem on the internet that tells me that $\ker(T)^{\bot} = \overline{\operatorname{Ran}(T^*)}$.
I just want ...
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Representations of a separable Hilbert space as a Banach lattice
An infinite dimensional separable Hilbert space $H$ has at least
two non-equivalent representations as a Banach lattice: $\ell_2$
and $L_2(0,1)$.
Are there other non-equivalent representations of $H$ ...
- 276
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Compact operators in Hilbert space
I have tried to solve this exercise:
Let $H$ be a Hilbert space and let $T = T^* \in B_{\infty}(H)$ a compact operator. Given $\psi_0 \in H$ consider the equations:
\begin{align*}
(1)\quad &T\psi =...
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Confusion when projecting a function onto a function basis
Let $\phi_i(x)$ denote the Fourier basis vectors on $[-1, 1]$. Such that if $i$ is even it's a cosine and if it is odd it is a sine.
$\phi_0(x)$ is just the constant 1.
So its standard norm is 2 on ...
- 3,847
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Function in Hilbert space with bounded derivatives
We have a function $f \in H^2(\mathbb{R}^2)$ and $f \in C^{2,\alpha}(K)$ for any compact set $K \subset \mathbb{R}^2$ and for any $\alpha \in (0,1).$ Moreover, we have that $\|\Delta f\|_{L^{\infty}(\...
- 1,395
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Bichteler-Dellacherie Theorem for Hilbert Spaces
The Bichteler-Dellacherie theorem characterizes good integrators as semimartingales. More accurately, for a cadlag, adapted stochastic process $(Z_t)_t$, these are equivalent
$\{\int_{0}^{t} \xi \...
- 11
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What is a "symmetric" elliptic operator?
I'm reading chapter 6 of Evans "Partial Differential Equations" (2nd Ed) and I am reading section 6.5.2, "Eigenvalues of nonsymmetric elliptic operators".
Here we are studying ...
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Enveloping von Neumann algebra of multiplication operators
I'm trying to verify something I think should be true that is proving to be a bit tedious just due to the level of generality. Let $G$ be a locally compact group with Haar measure $\mu$ and $A \subset ...
- 2,004
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Is this true in Hilbert spaces? $P_{T(U)}=TP_UT^+$
Let $\mathcal{H}$ and $\mathcal{K}$ be a real Hilbert space, $U\subseteq \mathcal{H}$ a closed linear subspace and $T\in\mathcal{B}(\mathcal{H},\mathcal{K})$ a continuous linear operator. I am ...
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Are the $\sqrt n$ prefactors "natural" in creation/annihilation operator definitions?
On the symmetrized (bosonic) Fock space $\mathcal F_{\mathcal B}$, the standard creation and annhilation operators are defined by
\begin{align*}
A^{\dagger}(e_k) |\, n_1,n_2,...,n_k,... \rangle
& ...
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