# Tagged Questions

For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).

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### Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open $H:=L^2(\Omega,\mathbb R^d)$ $U$ be a separable $\mathbb R$-Hilbert space $Q:U\to H$ ...
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I'm trying to solve this task, but I'm not sure, if my solution for a) is correct. For b), i dont find a starting point. Did someone have an idea how to solve this? Thanks in advance. Be $V$ the set ...
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### Calculate the adjoint map

i'm trying to solve this task, but I don't find a starting point. Did someone have an idea how to solve this? Be V the set $\{f \in \mathbb{R}[X]| grad\,f \leq 2 \}$. This becomes to an euclidic ...
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### $(k\otimes h^\ast)^\ast=h\otimes k^\ast$?

Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $(k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this ...
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### Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g}$ we can define the adjoint representation as: $ad: \mathbb{g} \rightarrow End(\mathbb{g})$ as the map such that $ad_x(y)=[x, y]$ for all $\in \mathbb{g}$ I am ...
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### Is it true that when $X$ is a positive matrix and $Y$ is a self adjoint matrix then $XY$ is positive??

Let $X_{n \times n}$ be a positive matrix i.e $<Xy,y> \ge 0$ for all $y \in \mathbb{C^n}$ and $Y_{n \times n}$ be a self adjoint matrix. Show that $XY$ is positive or find a counterexample. So ...
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### Proving an complex operator is self-adjoint

Let V be a finite dim complex inner product space. Let T be a linear operator on V. Prove that T is self-adjoint if and only if $\langle T\alpha,\alpha\rangle$ is real $\forall \alpha \in V$. The ...
Let $T$ be a linear transformation in an inner product space $V$. Determine if the following it true or false: $$Ker (T)= Ker (T^*T)$$ Where $*$ donates the adjoint operator. Would it help proving ...