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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8
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Gradient operator the adjoint of (minus) divergence operator?

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 ...
7
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167 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 ...
7
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0answers
170 views

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 ...
7
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0answers
117 views

Why is it called *adjunction* formula?

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 = \...
6
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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 ...
5
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1answer
130 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}(...
5
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1k 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. ...
4
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0answers
102 views

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 ...
4
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1answer
76 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 ...
3
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85 views

Diffusion-Reaction problem $u_t = \int_{-\infty}^{\infty} K(x-y) u(y) dy . u + u^3$.

I have bee stuck in this problem since more than a week. During my study I kinda understand how to find the adjoint operator for the linearization. But I have no Idea how to find the linearization to ...
3
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1answer
150 views

Does the derivative operator have an adjoint?

Determine whether the linear operator $T(f) = f'$ (taking the derivative) has an adjoint or not. Consider the inner product $\left<f,g\right> = \int_0^1 f(t)g(t)\ dt $ defined on the vector ...
3
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1answer
73 views

What's the intuition behind $T^*T$?

Let $T : V \to W$ be a linear transformation across inner product spaces over some field $\mathbb{F}$ and $T^* : W \to V$ be its adjoint. When considering $T$ composed with $T^*$, we have some unique ...
3
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0answers
66 views

adjoint matrix of an operator

considering a $2\times 2$ matrix $\bf S$, \begin{equation} {\bf S} = \begin{bmatrix} \frac{\partial}{\partial{t}} & \kappa\nabla .\\ 1/\rho \nabla . & \frac{\partial}{\partial{t}} \...
3
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0answers
90 views

Range of $𝐴^*𝐴$ and range of $𝐴^∗$ have the same closure

I have the following lemma and proof in my lecture notes ($\mathcal{R},\mathcal{N}$ denote range, kernel, respectively, and $\mathcal{X}$ and $\mathcal{Y}$ are Hilbert spaces). I was hoping there was ...
3
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168 views

What is a duality argument?

I believe this is probably a question that can have a wide variety of answers, but i believe i'm still interested. The thing is, i have been in a couple pde talks, and i saw that in both of these ...
3
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0answers
40 views

How to calculate a multiplication operator representation?

Let $H =\ \mathcal{l}^{2}(\mathbb{Z},\mathbb{C})$ and $A = R + L$, where $L$ is the left-shift operator (and $R$ is the right-shift $(Ra)_{n}=a_{n-1}$). Set $$U : \mathcal{l}^{2}(\mathbb{Z},\...
3
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34 views

Prove that for the defined $\langle .,. \rangle$ there exist $0 < a \le b$ such that $a\|x\| \le \|x\|_\ast ≤ b\|x\|$ for all $x \in H$.

Let $H$ be a Hilbert space over $\mathbb{R}$ with an inner product $(·, ·)$ and the norm $\|x\| = \sqrt {(x, x)}$. Let $A$ be a bounded strictly positive definite linear operator on $H$ with $A^\ast = ...
3
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261 views

Null space equals annihilator of range of adjoint in general spaces

(It turns out that the claim is incorrect as pointed out in the comment. Will get back soon) Let $A, B$ be normed linear spaces and let $S$ be a bounded linear map from $A$ to $B$. Define the map $S^*...
3
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63 views

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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30 views

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) ...
3
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260 views

Adjoint vs Self-adjoint operators represented by matrices

I want to see the difference between just adjoint and self-adjoint (hermitian) operator represented by matrices. If I have a matrix $$A= \begin{pmatrix} 1 & i \\ i & 1 \\ \...
3
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82 views

Check if a Differential operator is self-adjoint

Let $\Omega \in \mathbb{R}^n$ be a bounded domain with boundary of class $C^2$. Define \begin{cases} D(A) = H^2 \cap H_0^1(\Omega;\mathbb{C})\\ Au(x)=\Delta u(x)-V(x)u(x) \hspace{3mm} x \in \Omega \...
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152 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 ...
3
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128 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 $T$ ...
3
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208 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 $f^{abc}$ ...
3
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1answer
1k views

Adjoint differential equations

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\...
3
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1answer
141 views

How is the "adjoint representation" related to generic group representations?

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 ...
3
votes
1answer
206 views

Operator $f \mapsto u(f)$ solution of non-homogeneous Laplace equation is compact and self-adjoint

Let $u : L^2_0(D) \to L^2_0(D): = \lbrace f \in L^2 : \int_D f = 0 \rbrace $ be the linear operator which associates $f$ to $u(f)$ the solution of $$ \begin{cases} \Delta u = f & \text{in } D \\ \...
3
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0answers
25 views

Finding two adjoints, and showing boundedness of operators

Let $H = l_2$ and consider the following operators: $T,S:H \to H$ $Tx = (0,x_1,x_2,\ldots)$ and $Sx = (x_2,x_3,x_4,\ldots)$ Show they are bounded, and find the adjoint of both: For $T$, I have $\|Tx\|...
3
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0answers
324 views

adjoint method for computing derivatives

I am curious if anyone has heard of this problem before: Suppose that $u(x,p)$ is a function of $x$ and $p$. These arguments need not be scalars. Let $u(x,p)$ satisfy some differential equation, say:...
3
votes
1answer
331 views

Properties of adjoint matrix in a finite dimensional inner product space

let $V$ be a finite dimensional inner product space. Let $T$ be a linear operator on $V$. Prove that there exists an invertible linear operator $U$ such that $U^{-1}TT^*U = T^*T $ where $T^*$ is ...
3
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0answers
81 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = <...
3
votes
1answer
105 views

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 ...
3
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1answer
693 views

$T \in B(X,Y)$ 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 ...
3
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0answers
250 views

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 $L^2(-\pi,\pi)$. ...
3
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0answers
553 views

Find the adjoint operator

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 ...
2
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0answers
52 views

If $A^\star$ is surjective, then there is $c>0$ such that $|u| \leq c |A u|$ for all $u \in D(A)$

I'm trying to prove Theorem 2.21 in Brezis' book of Functional Analysis. The author leaves the proof as an exercise. Could you have a check on my attempt? Let $E, F$ be Banach spaces. Let $A: D(A) \...
2
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0answers
20 views

Is this a correct method for determining the domain of the adjoint?

Someone told me about this condition ("definition of the domain of the adjoint") for an operator $T$ acting on $C^\infty(L^2(\mathbb{R}))$ functions: $$f\in D(T^*) \iff (\exists M_f>0 \;s....
2
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0answers
47 views

Adjoint operator for a system of nonlinear PDEs

I have this system of nonlinear reaction diffusion equations $\begin{cases} & u_t = u_{xx} + \partial_u W(u,v),\\ & v_t = d v_{xx} - \partial_v W(u,v). \end{cases}$ I found the linearization ...
2
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0answers
256 views

Question 2.23 from Brezis' book of Functional Analysis

Question: Let $E=\ell^{1}$, so that $E^*=\ell^{\infty}$. Consider the operator $T\in \mathcal{L}(E, E)$ defined by $$Tu=\left(\frac{1}{n}u_n\right)_{n\geq 1}$$ for every $u=(u_n)_{n\geq 1}$ in $\ell^1$...
2
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0answers
65 views

Continuous linear operator on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ be an open and bounded domain and consider a positive continuous linear operator $B:L^2(\Omega)\to L^2(\Omega)$ with the property that if $f\in L^{\infty}(\Omega)\...
2
votes
1answer
63 views

On spectrum of a linear operator.

Let $H$ be a Hilbert space and $S,T$ bounded linear operators defined on $H$. I am interested in showing that: If $S,T$ are self-adjoint operators with compact $S$ and $ST=TS$ then $$\sigma(T+S)\...
2
votes
0answers
40 views

$D^*$ 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 ...
2
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0answers
32 views

Is $L^p(\mu)$ isometrically embedded into $L^q(\mu)$ when $p=\infty$ or $q=\infty$? And what can we say about the adjoint of the embedding?

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space, $1\le p,q\le\infty$ with $\frac1p+\frac1q=1$, $$\mathfrak m(f,g):=\int fg\:{\rm d}\mu\;\;\;\text{for }(f,g)\in L^p(\mu)\times L^q(\mu)$$ and $$...
2
votes
1answer
90 views

Explicit formula for codifferential

I want to show that $$\delta\eta \overset{!}{=} \nabla^*\eta = -\sum\limits_{i=1}^{n} E_i \lrcorner \nabla_{E_i}\eta$$ where $\delta:\Omega^{k+1} \to \Omega^k$ is the adjoint of the exterior ...
2
votes
0answers
32 views

Is the spectral theorem for operators or matrices?

A is a normal operator if $A A^* = A^* A$. This means for any matrix $M$ representing $A$, we have $MM^* = M^*M$. Let $M$ be one such matrix, and $P$ be an invertible but non-unitary matrix. Then $N=...
2
votes
0answers
98 views

Explicit expression of the operator $d^*d$ on 1-forms

I tried to compute the explicit expression for $d^*d\omega$ where $\omega=\omega_jdx^j$ is a 1-form on a manifold $M^n$. My hope was to find an expression that doesn't involve the Christoffel symbols. ...
2
votes
0answers
69 views

What is the adjoint operator for the Diag function?

The Diag function sends a vector x in R^n to the diagonal nxn matrix with the components of x along the diagonal. The Diag function is a linear map from R^n to S^n_+, the set of symmetric positive ...
2
votes
0answers
60 views

A sum of operators in Hilbert space

In a Hilbert space $\mathcal{H}$, let $T\in \mathcal{L}(\mathcal{H})$ be such that there exists an integer $k$ such that $T^k=\mathbb{I}$; let $\zeta := $exp$(2\pi i/k)$ and let $$P_\mu=\frac{1}{k}\...
2
votes
1answer
66 views

Justifying Interchange of Integral

I am trying to show that if $P$ is a pseudo-differential operator with symbol given by $p(x,\xi)$ i.e. the operator $$P:\mathcal S\rightarrow \mathcal S$$ defined by $$Pf(x)=\int_{\mathbb R^n}e^{ix\...

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