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### Spectrum of $H + H^*$ in Hilbert space

Consider a bounded operator $H$ on a Hilbert space $V$. Let ${\rm Re\ sp} (H)$ be the real parts of all points in the spectrum of $H$. Is it ture that ${\rm sp}( H + H^*) = 2 {\rm Re\ sp}(H)?$ Or ...
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### Finding adjoints of linear operator [closed]

I’m going through Linear Algebra Done Right, learning about singular value decomposition. I’m a bit lost when trying to calculate an adjoint of an operator. Obviously, this is simple when you are ...
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### Matrix representation of adjoint & co-adjoint orbit of $so(3)$

So I am trying to find the co-adjoint orbits of the lie algebra $so(3)^*$ from this example but I am stuck with a very trivial linear algebra property now I found the adjoint orbits and I know the ...
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### Describe the closure and adjont of operator T

Let $\mathcal{H}_2$ be a separable Hilbert space and $(T,D(T))$ be an operator in $\mathcal{H}_2$ such that there exists a Hilbert basis $(e_n)_{n\in \mathbb{N}}$ of $\mathcal{H}_2$ with $e_n \in D(T)$...
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Show that $\text{ker}(T^*T)=\text{ker}(T)$ In my solutions when showing that $x\in\text{ker}(T)\implies x\in\text{ker}(T^*T)$ the following is given: $$x\in\text{ker}(T)\implies Tx=0\implies T^*Tx=0\... • 2,445 -1 votes 1 answer 128 views ### Determine the adjoint of \tilde Q(x) for \tilde Q(x)u:=(Qu)(x) where Q:U→L^2(Ω,ℝ^d is a Hilbert-Schmidt operator and U is a Hilbert space Let d\in\mathbb N \lambda denote the Lebesgue measure on \mathbb R^d \Omega\subseteq\mathbb R^d be open H:=L^2(\Omega,\mathbb R^d) U be a separable \mathbb R-Hilbert space Q:U\to H be ... • 13.4k -1 votes 2 answers 456 views ### Isometric <=> Left Inverse Adjoint Is it true that:$$T\text{ isometric}\iff T^*\text{ left inverse}$$Obviously:$$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle...
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where From this question I concluded that L^T = L*. But when I find the transpose and then adjoint of the matrix I get as the adjoint L* of L. But it is wrong. What am I doing wrong?
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### If $\det(A) = 1$, what are possible values of $\det(\mbox{adj}(A))$?

If A is square matrix (3×3), and $det(A)=1$, is it true that $det(adj(A))=1, -1$?
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### Question on self adjoint properties

I have the follwing question and have managed to do it all apart from the last 3 parts of c) showing the various statements. Cant figure out where to start! Any help will be appreciated!
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### how to find inverse of a matrix

How to find the inverse of a 4x4 order matrix using adjoints for example A=\begin{pmatrix} 2 & -6 & -2 & -3 \\ 5 &-13 &-4 &-7 \\ -1 & 4& 1& 2 \\ 0 & 1 &...
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### Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
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