# Questions tagged [adjoint-operators]

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

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### Finding adjoint operator $T:L^2((0,1))\rightarrow \mathbb{R}^2$

Find the adjoint operator of $T:L^2((0,1))\rightarrow \mathbb{R}^2$ $x\mapsto (\int^1_0sx(s)ds, \int^1_0s^2x(s)ds)$ My attempt was to look at $\langle Tf,g\rangle = \langle f, T^{\ast}g\rangle$ \...
1answer
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### Adjoint operator $T^{\ast}$ in space $l^1$

Find the adjoint operator of $T:l^1 \rightarrow l^1$, $(x_k)_{k\in \mathbb{N}}\mapsto (\sum^{\infty}_{k=1}x_k,0,0,...)$ In our lecture we defined the adjoint operator as \begin{align*} T^{\ast}(y^{\...
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### Adjoint orbits of the Lie algebra $sl(2)$

So I have been trying to figure out the ad-joint and co-adjoint orbits of the lie algebra $sl(2)$, I found online that they are supposed to be hyperboloids but I can't seem to get that using my matrix ...
0answers
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### $D^*$ invertible iff $D$ invertible but with unexpected inner products.

I have a (densely defined) operator of the form $$D:=\frac d {dt} + L + h:L^2(\mathbb{R}\times Y)\dashrightarrow L^2(\mathbb{R}\times Y)$$ where $L:C^\infty(Y)\to L^2(Y)$ is self-adjoint, elliptic ...
0answers
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### For an unbounded operator A, when is $D(A^{-1}) = D(A)'$? Is there a canonical isomorphism?

I am reading a text at the moment and I am not sure what the authors mean by $D(A^{-1})$. In this situation $A: D(A) \subset H \longrightarrow H$ is an unbounded operator on a hilbert space $H$ which ...
1answer
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### If $V=W$ (vector spaces), then $\det A^{\dagger}=\det A$, where $A$ is a map and $A^{\dagger}$ is its adjoint

Let $(V,g)$ and $(W,h)$ be scalar product spaces, and let $A:V\to W$ be linear. We define $A^{\dagger}:W\to V$ by $$A^{\dagger}=S\circ A^*\circ\mathcal{F}$$ where $S$ is the sharp map corresponding to ...
1answer
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### Exponential form of a unitary operator

I have to solve the following exercise: Let $U(t)$ be a unitary operator ($t$ is a real parameter) such that $U(0)=\mathbb 1$ (identity). Show that $$U(t) = \exp(itH)$$ to a first order approximation....
0answers
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### Finding the adjoint operation of a black-box differential operator

I have a black-box code (automatic differentiation) that computes $Du$, where $D \in \mathbb{R}^{n \times n}$ and $u \in \mathbb{R}^n$. Note that I do not have the matrix $D$ explicitly and neither do ...
2answers
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### A question regarding normal and adjoint operators

Let $(X,A,\mu)$ be a measurable space. For all $f\in L^{\infty}(X,A,\mu)$ define $M_f\in B(L^2(X,A,\mu))$ by: $M_fg:=fg$ for all $g\in L^2(X,A,\mu)$. So, $M_f^*=M_{\overline{f}}$ and then $M_f$ is ...
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### Inner product and adjoint mapping in matrices

Let $\mathcal{A}_1:S^n \to S^n$ be defined by $\mathcal{A}_1(P)=-(A^*P+PA)$. How can I find $\mathcal{A}_1^{adj} (Z)$, where $\mathcal{A}_1^{adj}:S^n \to S^n$?
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### Is there any reason to restrict the Hermitian adjoint to linear operators?

The Wikipedia page on the Hermitian adjoint is inconsistent about whether that operation is only defined for linear operators on a single Hilbert space, or more generally for arbitrary linear maps ...
1answer
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