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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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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 ...
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Let $X$ be a complex manifold, $Y$ a sub-manifold, and $i \colon Y \to X$ the corresponding embedding. Then one can prove that the corresponding canonical bundles satisfy $$\omega_X \big|_Y = \... 0answers 1k views Inverse vs. adjoint operators I hope this is enough of a question (and less of a start of a discussion) to be allowed here. I am trying to get my head around the ideas of inverse and adjoint operators. To keep it simple, let's ... 0answers 41 views 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 ... 1answer 115 views Show properties of the linear operator L((a_n)_n)=\left(\frac{1}{n}*a_n\right)_n Let L\colon \ell^p \rightarrow \ell^p such that L((a_n)_n)= \left(\frac{1}{n}*a_n\right)_n. 1) Determine L'(adjoint operator), \ker(L), \ker(L'), \operatorname{rg}(L), \operatorname{rg}(... 0answers 508 views Intuition behind \ker(T)=\ker(T^*) for T a normal operator Let T : V \to V be a normal operator and V a finite-dimensional vector space. Show that \ker(T)= \ker(T^*) and \text{im}(T) = \text{im}(T^*). I know how to rigorously show this, but I'm ... 0answers 875 views Sum of operator and adjoint is self-adjoint In abstract Hodge theory there is the following lemma: Let H be a Hilbert space and A \in \mathcal{C}(H) a densely defined, closed operator (so possibly unbounded) and A^* its adjoint operator. ... 4answers 51 views Showing that is a normal operator Let H is a Hilbert space I is unit operator, T \in B(H) and \lambda \in \mathbb C T is normal operator \Rightarrow T-\lambda I is a normal operator too. I could only write : I must ... 1answer 61 views Using inner product property to determine if operator is an isomorphism. Let \varphi be an operator on a k-vector space V with an inner product \langle\cdot,\cdot\rangle. Suppose that \langle v,\varphi v\rangle = 0 for every v\in V. If we take k=\mathbb R, is ... 0answers 63 views Show that T has an adjoint, and describe T^* explicitly. Let V be an inner product space and  \beta, \gamma fixed vectors in V. Show that T \alpha = (\alpha\mid\beta) \gamma defines a linear operator on V. Show that T has an adjoint, and ... 0answers 37 views Adjoint of a polynomial in a closed linear operator. Let  H  be a Hilbert space and let  T  be a closed densely defined linear operator in  H  with domain  D(T)  and with nonempty resolvent set. We define the following polynomial in T:  P(T) :... 0answers 70 views Show that the integral-operator compact? We have the operator T on L_2[0,1] which is defined as Tf = y where y is the solution to the ODE y^{\prime \prime} + y^\prime = f with boundary conditions y(0)=0, y(1) = 1 Show that T ... 1answer 30 views Calculating T^*T when T have direct sum codomain Let \begin{equation*} \bigoplus_{ \ell_2} K_n := \{ (x_1,x_2,\cdots) \in \bigoplus_{n=1}^\infty : x_n \in K_n, \sum_{n=1}^{\infty} || x_n ||^2 <\infty \} \end{equation*} where K_n are Hilbert ... 0answers 140 views Explicit form of generators of a Lie algebra in the adjoint representation My question can be summarized as: Generators of a Lie group in the adjoint representation can be written as,$$ (T^a_\text{Ad})_{bc} = \text{i}f^{abc}, \tag{1}\label{adj} $$where f^{abc} ... 1answer 167 views Is the operator T((x_1,x_2,…,x_n))=(x_1,\frac{x_2}2,\frac{x_3}3,…) on \ell^2 self-adjoint and unitary Consider the Hilbert space$$\ell^2=\{(x_1,x_2,...,x_n),x_i\in\mathbb C\text{ for all }i\text{ and }\sum_{i=1}^\infty |x_i|^2<\infty\}$$with the inner product$$\langle(x_1,x_2,\dots,x_n)(y_1,y_2,...
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I'm studying representation theory in order to have a basis to study quantum field theory. I think the text (my professor's) i'm studying on is pretty confusing. I don't really get the difference ...
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The image of the adjoint operator in a Hilbert space

Let $H$ be a Hilbert space and $T \in L(H)$ a bounded linear operator.If $\exists c>0$ such that $Re(\langle Tx,x \rangle) \geqslant c||x||^2, \forall x \in H$, then prove that $\text{Ran}(T^*)=H$ ...
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Derivation of the adjoint poisson equation

I'm trying to follow the derivation of the adjoint Poisson equation provided in this video, but I'm getting tripped up on some of the steps that are skipped. Here is my derivation in full. Questions ...
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Consider the vector differential equations $$\mathbf{x}^{\prime}=\mathbf{A}(t)\cdot\mathbf{x}\tag{1}$$ and \mathbf{y}^{\prime}=-\mathbf{A}^{\ast}(t)\cdot\...