# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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### Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
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### If $Z$ is a closed subset of Hilbert Space $X$, is it true that $Z\neq X \implies Z^{\perp}\neq \{0\}$?

It is clear from Projection theorem that if $Z$ is a subspace, then since $X=Z\oplus Z^{\perp}$, $Z^{\perp}$ is not trivial (By the way, is there any reasoning that would show this without referring ...
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### How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...
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### Prove a sequence is bounded under a Hilbert space

Let $T:H\to H$ be defined by $Tx=\sum_{n=1}^\infty \lambda_n \langle x,\varphi_n \rangle \varphi_n$ where $\{\varphi_n\}_{n=1}^\infty$ is an orthogonal sequence and $\{\lambda_n\}_{n=1}^\infty$ is a ...
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### Find spectrum of integral operator

Let Af(x) = $\int_0^1 K(x,y)f(y)dy$, $A:L_2[0,1]\rightarrow L_2[0,1].$ Where $K(x,y) = \sinh(\min(x,y)\sinh(1-\max(x,y)).$ where $\sinh(x) = \frac{e^x - e^{-x}}{2}$ Find $\sigma(A), ||A||.$ I ...
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### Inversible operator in Hilbert space

Consider $\phi\in L^{\infty}[0, 2\pi]$. Let M be operator $L_2[0, 2\pi]\rightarrow L_2[0, 2\pi]$$Mf = \phi f$$ In$L_2[0, 2\pi]$we have topological basis${e^{inx}}, n\in \mathbb Z$.$L_2[0, 2\pi] =...
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### How to find out if a function belongs to $H^2$ or $H^1$

I'm beginning with Sobolev spaces and I found out, that $$H^k = W^{k,2}.$$ I've also seen the following exercise recently: $$\frac{1}{2}u'' = 1$$ And here I'm supposed to find out if $u$ ...
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### How can we compute the adjoint of the inclusion between two Hilbert spaces?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ ...
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### integral of product of three basis functions and Clebsh-Gordan coefficients

Suppose I have an orthonormal basis $\{b_i\}_{i=1}^\infty$ for an $L_2$ space (for example, the $b_i$ could be spherical harmonics on the round sphere with the Euclidean $L_2$ inner product). I want ...
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### Generalized Poincaré Inequality on H1 proof

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
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### Distance preserving function on hilbert space

It is known that an isometry on B(H) is distance preserving .I am trying to show ,conversely , that if F=R,every distance preserving function f on H( Hilbert space) has the form f(x) = f(0) +Tx for ...
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### Conjugate linear isometry of Hilbert operators

Let $H$ and $R$ be Hilbert spaces and consider an operator $T$ in $B(H,R)$. I need to show that there is a unique operator $T^*$ in $B(R,H)$ satisfying $$(Tx│y)_R = (x│ T^* y)_H,$$ $x \in H$, $y \in R$...
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### Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
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### Orthogonal sequences under A Hillbert space

I know that for two vectors $u,v\in H$ where $H$ is a Hilbert space the definition for orthogonality is $\langle u,v \rangle =0$. is thaat also corret for sequences? What is the definition for ...
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### Linear Operators on $L_2(\mathbb R)$ definfed as Integrals

Let's consider the linear operators on $L_2(\mathbb R)$ $$T_{\alpha}f(x) = \int_{-\infty}^{+\infty} \frac{e^{-|x-y|^2}}{(1+x^2)^{\alpha}}f(y)dy$$ with ${\alpha} \in [0,1]$. Find ${\alpha}$ such ...
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### If a positive operator $y$ has the same kernel as $cy$, what can we conclude about the kernel of $c$?

Let us consider the equation $x=cy$ in $B(H)$. Assume that: $y$ is a positive operator. $x$ and $y$ have the same null space. Ker($y$) is contained in Ker($c$). Can we conclude that Ker($y$)=...
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### Show that $\ker(T)=\{\varphi _n\mid\lambda_n\neq 0\}^\perp$

Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) ...