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### Derivative of adjoint operator-valued function

Consider an infinite dimensional complex Hilbert space $H$. I think that for a bounded operator-valued function $A: x\mapsto A(x) \in \mathcal B(H)$, where $x\in \mathbb R$, we can define the ...
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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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### Adjoint of unbounded integral operator

Consider a Borel-measurable function $a:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$, and let $T_a$ be the linear operator on $L^2(\mathbb{R})$ with domain \begin{equation} \mathcal{D}(T_a)=\left\{...
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### Find Adjoint of $L = p(x) \frac{d^2}{dx^2} + r(x) \frac{d}{dx} + q(x)$

Suppose that \begin{align*} L &= p(x) \frac{d^2}{dx^2} + r(x) \frac{d}{dx} + q(x) \\ \end{align*} Consider \begin{align*} \int_a^b vL(u) \, dx \\ \end{align*} By repeated integration ...
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Consider the vector differential equations \begin{equation} \mathbf{x}^{\prime}=\mathbf{A}(t)\cdot\mathbf{x}\tag{1} \end{equation} and \begin{equation} \mathbf{y}^{\prime}=-\mathbf{A}^{\ast}(t)\cdot\...
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### 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 ... 146 views

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### The adjoint operator of the inclusion map

Let $(H, |\cdot|)$ be a Banach space and $V$ a linear subspace of $H$. Assume that the vector space $V$ has its own norm $[ \cdot ]$ such that $(V, [\cdot])$ is a Banach space. We consider the (linear)...
### The adjoint of $T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u$
Let $a \in L^1 (\mathbb R^d)$ be a fixed function. Consider a linear operator $$T_a: L^2 (\mathbb R^d) \to L^2 (\mathbb R^d), u \mapsto a * u.$$ By Young's inequality, $\|a*u\|_2 \le \|a\|_1 \|u\|_2$...
Suppose we have a differential equation of the form $$L[f(x)] = a(x)f''(x) + b(x)f'(x) + c(x)f(x) = g(x)$$ where $a, b, c$ and $g$ satisfy the regularity conditions demanded in Sturm-Liouville ...