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Recently I found this statement -- the gradient operator is the adjoint of the minus divergence operator -- in one of my lecture notes. Knowing only a little about functional analysis, I'm looking for ...
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$T$-invariance of $U$ is equivalent to $T^{*}$-invariance of $U^{\perp}$

Is the Following argument correct? Suppose $T\in\mathcal{L}(V)$ and $U$ is a subspace of $V$. Prove that $U$ is invariant under $T$ if and only if $U^{\perp}$ is invariant under $T^*$. Proof. Given ...
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Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \frac{d^2 f}{dt^2} + f$.

Find the adjoint under the inner product $\langle f, g \rangle = \int_0^1 f(t)g(t)t \ dt$ of $\mathcal{L}(f)(t) = \dfrac{d^2 f}{dt^2} + f$ with $f(0) = 0$ and $f'(1) = 0$. Note: The weight function is ...
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Show that T is a surjective linear application

Let $X,Y$ be Banach spaces and $T:X\to Y$ a bounded linear application. Let $T^*:Y^*\to X^*$ be its adjoint (i.e. $T^*g(x)=g(Tx)$, $\forall g\in Y^*, x\in X$ ). If $||T^*g||\geq K||g||$ for some ...
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Showing that a map is open by usying lower boundedness of adjoint

Let consider the following exercise: Let $V$ and $W$ be Banach spaces and $T\colon V \longrightarrow W$ be a closed linear operator. Then, the following assumptions are equivalent: (i) R$(T)=W$; (ii) ...
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How to prove this is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to ...
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$T \in B(X,Y)$ is an isometry if and only if $T^*$ is an isometry

I would like to prove that $T \in \mathscr{B}(X,Y)$ is an isometry of $X$ onto $Y$ if and only if $T^*$ is an isometry of $Y^*$ onto $X^*$. I am not really sure what to do. I started the argument as ...
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Proving that a certain differential operator is self-adjoint

Consider the differential operator $T:u\mapsto -iu'$ for any $u\in D(T):=\{f\in AC[-\pi,\pi]~|~f'\in L^2(-\pi,\pi),f(-\pi)=f(\pi)\}$; we consider $T$ as a densely-defined operator on $L^2(-\pi,\pi)$. ...
I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator $$(Ax)(t)=x(at), x\in L^2(0,\infty), a>0.$$ My calculation is the following; I use the ...