Questions tagged [adjoint-operators]
For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).
300
questions with no upvoted or accepted answers
10
votes
1
answer
285
views
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 ...
- 374
10
votes
0
answers
4k
views
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 ...
- 1,182
8
votes
0
answers
154
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 = \...
- 9,728
7
votes
0
answers
238
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 ...
- 5,802
7
votes
0
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207
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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 ...
- 4,470
6
votes
0
answers
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 ...
- 532
5
votes
1
answer
140
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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}(...
- 411
5
votes
0
answers
1k
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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. ...
- 1,134
4
votes
0
answers
111
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Norm of the adjoint operator of an operator of the form $\int_{0}^{1} k(t,s)f(s)ds$
I am asked to compute the adjoint of the operator $A: L^2[0,1] \longrightarrow L^2[0,1] $ defined by
$$
Af(t)=\int_{t}^{1} t (s-1/2) f(s) ds
$$
and then to compute its norm, that is $\lVert A^* \rVert$...
- 474
4
votes
0
answers
110
views
If the boundary is adjoint to the differential, what is the "coboundary" adjoint to the codifferential in the continuum?
For a smooth manifold, Stoke's theorem says that the differential/exterior derivative $\mathrm{d}$ is adjoint to the boundary operator $\partial$, i.e.
$$\int_{\partial U} \omega = \int_{U} \mathrm{d}\...
- 1,151
4
votes
0
answers
143
views
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\{...
- 349
4
votes
0
answers
127
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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 ...
- 2,609
4
votes
1
answer
2k
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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\...
- 791
4
votes
1
answer
83
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 ...
user155861
3
votes
1
answer
146
views
Existence and uniqueness of formal adjoint operator on manifolds
Let $(\mathcal{M},g)$ be a (pseudo)-Riemannian manifold and $E$ a real vector bundle over $\mathcal{M}$, which we equip with a non-degenerate metric $\langle\cdot,\cdot\rangle_{E}\in\Gamma^{\infty}(E^{...
- 2,196
3
votes
0
answers
96
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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 ...
- 1,372
3
votes
1
answer
437
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 ...
- 700
3
votes
0
answers
102
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
votes
0
answers
109
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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 ...
- 713
3
votes
0
answers
315
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 ...
- 469
3
votes
0
answers
50
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
$$U : \mathcal{l}^{2}(\mathbb{Z},\...
- 31
3
votes
0
answers
38
views
Prove that for the defined $\langle .,. \rangle$ there exist $0 < a \le b$ such that $a\|x\| \le \|x\|_\ast ≤ b\|x\|$ for all $x \in H$.
Let $H$ be a Hilbert space over $\mathbb{R}$ with an inner product $(·, ·)$ and the norm $\|x\| = \sqrt
{(x, x)}$. Let $A$ be a bounded strictly positive definite linear operator on $H$ with $A^\ast = ...
- 1,361
3
votes
0
answers
449
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Null space equals annihilator of range of adjoint in general spaces
(It turns out that the claim is incorrect as pointed out in the comment. Will get back soon)
Let $A, B$ be normed linear spaces and let $S$ be a bounded linear map from $A$ to $B$. Define the map $S^*...
- 103
3
votes
0
answers
76
views
Show that T is a surjective linear application
Let $X,Y$ be Banach spaces and $T:X\to Y$ a bounded linear application.
Let $T^*:Y^*\to X^*$ be its adjoint (i.e. $T^*g(x)=g(Tx)$, $\forall g\in Y^*, x\in X$ ).
If $||T^*g||\geq K||g||$ for some ...
- 728
3
votes
0
answers
36
views
Showing that a map is open by usying lower boundedness of adjoint
Let consider the following exercise:
Let $V$ and $W$ be Banach spaces and $T\colon V \longrightarrow W$ be a closed linear operator. Then, the following assumptions are equivalent:
(i) R$(T)=W$;
(ii) ...
- 649
3
votes
0
answers
301
views
Adjoint vs Self-adjoint operators represented by matrices
I want to see the difference between just adjoint and self-adjoint (hermitian) operator represented by matrices.
If I have a matrix $$A=
\begin{pmatrix}
1 & i \\
i & 1 \\
\...
- 1,463
3
votes
0
answers
142
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 \...
- 1,884
3
votes
0
answers
195
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
...
- 2,817
3
votes
0
answers
82
views
adjoint operators in the vector space of real polynomials
This problem is about the space $V$ of real polynomials in the variables $x$ and $y$. If $f$ is
a polynomial, $d_f$ will denote the operator $f(d/dx,d/dy)$ , and $d_f(g)$ will denote the result of
...
- 507
3
votes
0
answers
146
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 $T$ ...
- 171
3
votes
0
answers
277
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 $f^{abc}$ ...
- 296
3
votes
1
answer
161
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 ...
- 131
3
votes
1
answer
236
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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
$$
\begin{cases}
\Delta u = f & \text{in } D \\
\...
- 31
3
votes
0
answers
27
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Finding two adjoints, and showing boundedness of operators
Let $H = l_2$ and consider the following operators: $T,S:H \to H$ $Tx = (0,x_1,x_2,\ldots)$ and $Sx = (x_2,x_3,x_4,\ldots)$ Show they are bounded, and find the adjoint of both:
For $T$, I have $\|Tx\|...
- 31
3
votes
0
answers
334
views
adjoint method for computing derivatives
I am curious if anyone has heard of this problem before:
Suppose that $u(x,p)$ is a function of $x$ and $p$. These arguments need not be scalars.
Let $u(x,p)$ satisfy some differential equation, say:...
- 3,913
3
votes
1
answer
401
views
Properties of adjoint matrix in a finite dimensional inner product space
let $V$ be a finite dimensional inner product space. Let $T$ be a linear operator on $V$.
Prove that there exists an invertible linear operator $U$ such that
$U^{-1}TT^*U = T^*T
$ where $T^*$ is ...
- 1,089
3
votes
0
answers
91
views
Do I have the correct mental map for adjoint operators for inner product spaces?
Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = <...
- 11.1k
3
votes
0
answers
1k
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Linear map is diagonalizable iff its adjoint is diagonalizable
Problem
Let $V$ be a finite inner product space and let $T:V \to V$ be a linear transformation. Prove that $T$ is diagonalizable if and only if the adjoint transformation $T^{*}$ is diagonalizable.
...
- 2,591
3
votes
1
answer
107
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How to prove this is a self-adjoint operator?
I have this operator from $H^1_0$ to $H^1_0$ defined by: $$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}$$
I want to ...
- 5,183
3
votes
1
answer
754
views
$T \in B(X,Y)$ is an isometry if and only if $T^*$ is an isometry
I would like to prove that $T \in \mathscr{B}(X,Y)$ is an isometry of $X$ onto $Y$ if and only if $T^*$ is an isometry of $Y^*$ onto $X^*$. I am not really sure what to do. I started the argument as ...
- 491
3
votes
0
answers
275
views
Proving that a certain differential operator is self-adjoint
Consider the differential operator $T:u\mapsto -iu'$ for any $u\in D(T):=\{f\in AC[-\pi,\pi]~|~f'\in L^2(-\pi,\pi),f(-\pi)=f(\pi)\}$; we consider $T$ as a densely-defined operator on $L^2(-\pi,\pi)$. ...
- 1,372
3
votes
0
answers
643
views
Find the adjoint operator
I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator
$$
(Ax)(t)=x(at), x\in L^2(0,\infty), a>0.
$$
My calculation is the following; I use the ...
user34632
2
votes
1
answer
75
views
Domain of the adjoint operator of a bounded operator (on a Hilbert space). (Experimental physicist)
Let $A$ be a bounded operator on a Hilbert space $H$ and $D_A$ its domain. We can define the following functional on $D_A$: $$f_\eta(\xi)=(\eta,A\xi).$$
Then we have that: $$||f_\eta(\xi)||=||(\eta,A\...
- 21
2
votes
0
answers
23
views
Brezis' exercise 6.24.1: prove that $T$ is positive IFF $\sigma (T) \subset [0, \infty)$
Let $E$ be a real Banach space. Let $\mathcal L(E)$ be the space of bounded linear operators on $E$ and $\mathcal K(E)$ its subspace consisting of compact operators. Let $\| \cdot\|$ be the operator ...
- 16.2k
2
votes
0
answers
44
views
On an operator inequality
Let $A$ and $B$ be two positive, self-adjoint and traceclass operators mapping from some Hilbert space $H$ to it self. Note that we have for any $a, b > 0$, $p>1$
$$(a+b)^p\leq 2^{p-1}(a^p+b^p)$$...
- 73
2
votes
0
answers
31
views
Assume $T\in \mathcal L(E, F)$. Then $T\in \mathcal K(E, F)$ IFF $T^* \in \mathcal K(F^*, E^*)$
Let $E, F$ be Banch spaces. Let $\mathcal L(E, F)$ be the space of bounded linear operators from $E$ to $F$. We endow $\mathcal L(E, F)$ with the operator norm. Let $\mathcal K(E, F)$ be the subset of ...
- 16.2k
2
votes
0
answers
40
views
Discrete spectrum under a finite rank perturbation
Let $A$ be a bounded self-adjoint operator on a Hilbert space. Assume the discrete spectrum of $A$ is empty. Let $B$ be a finite rank operator.
I have learned that the essential spectrum of $A+B$ and $...
- 597
2
votes
0
answers
74
views
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)...
- 16.2k
2
votes
0
answers
53
views
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, $\|a*u\|_2 \le \|a\|_1 \|u\|_2$...
- 5,128
2
votes
0
answers
62
views
Sturm-Liouville theory and adjoint operator
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 ...
- 533