Questions tagged [adjoint-operators]
For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).
910
questions
0
votes
1answer
21 views
Finding adjoint operator $T:L^2((0,1))\rightarrow \mathbb{R}^2$
Find the adjoint operator of $T:L^2((0,1))\rightarrow \mathbb{R}^2$ $x\mapsto (\int^1_0sx(s)ds, \int^1_0s^2x(s)ds)$
My attempt was to look at $\langle Tf,g\rangle = \langle f, T^{\ast}g\rangle$
\...
2
votes
1answer
34 views
Adjoint operator $T^{\ast}$ in space $l^1$
Find the adjoint operator of $T:l^1 \rightarrow l^1$, $(x_k)_{k\in \mathbb{N}}\mapsto (\sum^{\infty}_{k=1}x_k,0,0,...)$
In our lecture we defined the adjoint operator as
\begin{align*}
T^{\ast}(y^{\...
0
votes
0answers
17 views
Adjoint orbits of the Lie algebra $sl(2)$
So I have been trying to figure out the ad-joint and co-adjoint orbits of the lie algebra $sl(2)$, I found online that they are supposed to be hyperboloids but I can't seem to get that using my matrix ...
2
votes
0answers
37 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 ...
1
vote
0answers
25 views
For an unbounded operator A, when is $D(A^{-1}) = D(A)'$? Is there a canonical isomorphism?
I am reading a text at the moment and I am not sure what the authors mean by $D(A^{-1})$.
In this situation $A: D(A) \subset H \longrightarrow H$ is an unbounded operator on a hilbert space $H$ which ...
0
votes
1answer
18 views
Adjoint of the operator $(Mf)(x, \omega)=e^{3x/2}f(e^x\omega)$
Let $r>1$ and $R=\ln{r}$ and $M: L^{2}(B(0, r)\setminus B(0, 1))\to L^{2}((0,R)\times S^{2})$ be the operator $(Mf)(x, \omega)=e^{3x/2}f(e^x\omega)$. What is its adjoint?
My attempt: Let $f\in L^{...
3
votes
0answers
44 views
Poisson bracket and co-adjoint orbits for $sl(2)$
So I am trying to do this problem from Peter Olver's book Application of Lie groups to differential equations and I am wondering if somebody could check my work because I am not really sure about it ...
-1
votes
1answer
42 views
Matrix representation of adjoint & co-adjoint orbit of $so(3)$
So I am trying to find the co-adjoint orbits of the lie algebra $so(3)^*$ from this example but I am stuck with a very trivial linear algebra property
now I found the adjoint orbits and I know the ...
0
votes
1answer
42 views
Show $M$ is complemented.
For a closed subspace $M \subset X$, show the following statements:
If $\dim M < \infty$, $M$ is complemented.
If $\operatorname{codim} M (= \dim X/M) < \infty$, M is
complemented.
I know that ...
0
votes
0answers
18 views
Differential operator hermitian conjugate
I'm trying to prove that over the function space $V=\{f:\Bbb{R} \rightarrow \Bbb{C}\}$ with the inner product $\langle f,g\rangle =\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}$, the ...
0
votes
1answer
46 views
Consider the boundary value problem $-u''=x, u(0)=u(1)=0$. Find solution, write down a minimization problem and the value of the quadratic functional
I am working on problem 9.3.1 from Introduction to Differential Equations bij Peter J. Olver. I have just been introduced to self-adjointness and positive (semi) definiteness but I find it quite hard ...
1
vote
1answer
63 views
Extending a function T from a subspace X of a Hilbert space H to H
$\phi:N to N$, $H$ is Hilbert space with O.N.B $\{e-i\}-i=1 to \infty$and $X=span\{e-1,e-2,...\}$ and $T:X to H$ so that $ T(e-i)=e-\phi(i)$. Now we want to characterize (by finding necessary and ...
-2
votes
0answers
18 views
Bounded operators and adjoint operator
consider the operator $V:L_2(0,1) \to L_2(0,1)$ defined as
$$
(Vf)(x) = \int_0^x f(x)\,dt.
$$
Demonstrate that V is bounded and find the adjoint operator $V^*$
0
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0answers
24 views
How to prove eigenvalues are real when given self-adjoint and positive operators?
Let's say I have some T, self-adjoint operator, as well as S, which is a positive operator on a complex finite-dimensional inner product space. How could I go about proving that all eigenvalues of ST ...
0
votes
1answer
20 views
U is T-invariant if and only if orthogonal complement of U is T*-Invariant
as told on the topic i need to prove
if U is a subspace of V and V is a vector space. $T\in L(V)$ which L is the set of all operators.
I need to prove the both side of this statement below:
U is T ...
0
votes
1answer
13 views
Norm of $\hat{A}\psi$ in terms of norm of $\psi$ - confused in derivation
In part of a problem I'm asked to obtain an expression for the norm of $\hat{A}\psi$ in terms of the norm of $\psi$, i.e. I would like to simplify
$\int_{-\infty}^\infty (\hat{A}\psi)^*(\hat{A}\psi)dx$...
0
votes
0answers
29 views
Eigenvalues of a linear operator $A$, knowing all the traces of $A^n$.
If I know the trace of all the $n$-th powers of a linear operator $A$ in an analytic form $f(n)= {\rm Tr} A^n$, is there a way to compute all the eigenvalues of the operator $A$?
In particular, I am ...
1
vote
0answers
37 views
Why if T is normal linear map, then image of T if the same as image of T*?
If $T$ is a normal linear map from vector space $V$ to itself, then image of $T$ is the same as image of $T^*$.
Can anyone help me to prove this?
1
vote
1answer
41 views
What's the adjoint of a translation operator on a Hilbert space? [closed]
Let $H$ be a $\mathbb R$-Hilbert space, $x\in H$ and $$\tau_x:H\to H\;,\;\;\;y\mapsto y+x.$$
What's the adjoint operator of $\tau_x$?
This should be an easy question, but I'm not able to deduce the ...
0
votes
0answers
13 views
Proof of relations involving function operators [duplicate]
I need proof this property of operators, where should I start?,
A and B are operators, and
$$\hat{e}Be^{-A} = B+[A,B]+ \frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+... $$
Thx
0
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0answers
40 views
Doubt in computation of inner product space.
I need to show that if $$(Ta|a)=(T^*(a) |a) $$
Then $$T=T^*$$
Some hints would be helpful instead of a complete answer.
4
votes
1answer
62 views
Some Geometric intuition behind self-adjoint operators
In my Functional Analysis class, we have been studying self-adjoint compact operators for the past week or so (more specifically their spectrums). I have a geometric idea of what it means for an ...
3
votes
1answer
101 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 ...
0
votes
1answer
19 views
Exercise with normal operators.
Let $T \in Hom(ℂ^3)$, $T$ a normal operator and $T(1,1,1)=(2,2,2)$. If $(x,y,z) \in ker(T)$, then $x+y+z=0$.
I was trying to solve this question, one of my attempts was using the fact that if $T$ is a ...
0
votes
1answer
40 views
For $T\colon V\to V$ every anti self-adjoint $T$ is orthogonally diagonalizable.
I know how to prove that every self-adjoint operator is orthogonally diagonalizable (using spectral the and Gram Schmidt process). But I am not sure how to apply to an anti self-adjoint operator. Can ...
0
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0answers
37 views
The transpose of the position operator
I understand that $\hat{\mathbf{x}} = (\hat{x}, \hat{y}, \hat{z})^{T}$, and I think that $\hat{\mathbf{p}}=-i\hslash \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\...
2
votes
1answer
50 views
If $V=W$ (vector spaces), then $\det A^{\dagger}=\det A$, where $A$ is a map and $A^{\dagger}$ is its adjoint
Let $(V,g)$ and $(W,h)$ be scalar product spaces, and let $A:V\to W$ be linear. We define $A^{\dagger}:W\to V$ by $$A^{\dagger}=S\circ A^*\circ\mathcal{F}$$ where $S$ is the sharp map corresponding to ...
-2
votes
1answer
31 views
Prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n =0 $ iff $a_{n} \in \mathbb{R}$ [closed]
How to prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n=0 $ iff $a_{n} \in \mathbb{R}$ and $[u_n]$ is an orthonormal sequence?
Edit: does it have something to do with the equality: $\...
0
votes
0answers
29 views
adjoint operator for gradient descent optimization
In any inverse retrieval problem, the forward model can be stated in matrix form as:
$$Ax = y.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)$$
Here, $A$ is the operator acting on the variables $...
0
votes
0answers
29 views
Find the adjoint of the following linear operator.
I have the boundary value problem $$xy''=ky$$ $$y(0)=y'(1)=0$$ and want to find the adjoint, so I have
$L=x\frac{d^2}{dx^2}-k I$
$$(Lu,v)=(x\frac{d^2u}{dx^2}-k u,v)=\int_0^1 v(xu''-k u)dx=$$
$$\int_0^...
0
votes
1answer
29 views
Relation between adjoint and dual space
I'm learning tensors recently. And I found that the notation for dual spaces $V^*$, the star, is the same as the notation of adjoint operators. My definition of adjoint is the adjoint of $T \in L(V,W)...
1
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0answers
19 views
Question on operator and its adjoint in Hilbert space: $A^{*}A = \text{id}_H$
Let $\{e_n\}_{n \in \mathbb{N}}$ be a orthonormal sequence in a Hilbert space $H$. Further, let $A : H \to \ell^2$ be a linear operator, and $A^{*} : \ell^2 \to H$ its adjoint, where
$$Ax = \left(\...
1
vote
1answer
44 views
Trace norm and Bures distance
I am given that the Bures distance between two density matrices (i.e. semi positive-definite self-adjoint maps which have trace equal to unity) $\rho_1,\rho_2$ is: $D_B(\rho_1,\rho_2) = 2(1-\|\rho_1^{...
0
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0answers
30 views
Inclusion operator on half-integer weight modular forms and its adjoint
We have an inclusion $\iota: S_{k+1/2}(8N) \hookrightarrow S_{k+1/2}(16N)$, whose adjoint with respect to the Petersson scalar product is apparently given by $$\iota^{*} = Tr: \begin{pmatrix} 1 & ...
2
votes
0answers
22 views
Is $L^p(\mu)$ isometrically embedded into $L^q(\mu)$ when $p=\infty$ or $q=\infty$? And what can we say about the adjoint of the embedding?
Let $(\Omega,\mathcal A,\mu)$ be a finite measure space, $1\le p,q\le\infty$ with $\frac1p+\frac1q=1$, $$\mathfrak m(f,g):=\int fg\:{\rm d}\mu\;\;\;\text{for }(f,g)\in L^p(\mu)\times L^q(\mu)$$ and $$...
0
votes
1answer
52 views
Let $H$ a Hilbert space and $T\in \mathcal{B}(H)$. $T$ is a isometry $\Leftrightarrow$ $T\circ T^{*}=I$
$(\Leftarrow)$ If $T\circ T^{*}=I$ we had $$\langle x,x \rangle=\langle (T\circ T^{*}) (x),x \rangle=\langle T(x),T(x) \rangle, \forall x\in H.$$
Hence, $\Vert T(x)\Vert=\Vert x \Vert$, $\forall x\in ...
1
vote
1answer
26 views
Exponential form of a unitary operator
I have to solve the following exercise:
Let $U(t)$ be a unitary operator ($t$ is a real parameter) such that $U(0)=\mathbb 1$ (identity). Show that
$$U(t) = \exp(itH)$$
to a first order approximation....
1
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0answers
19 views
Finding the adjoint operation of a black-box differential operator
I have a black-box code (automatic differentiation) that computes $Du$, where $D \in \mathbb{R}^{n \times n}$ and $u \in \mathbb{R}^n$. Note that I do not have the matrix $D$ explicitly and neither do ...
0
votes
2answers
42 views
A question regarding normal and adjoint operators
Let $(X,A,\mu)$ be a measurable space.
For all $f\in L^{\infty}(X,A,\mu)$ define $M_f\in B(L^2(X,A,\mu))$ by:
$M_fg:=fg$ for all $g\in L^2(X,A,\mu)$.
So, $M_f^*=M_{\overline{f}}$ and then $M_f$ is ...
0
votes
0answers
21 views
Inner product and adjoint mapping in matrices
Let $\mathcal{A}_1:S^n \to S^n$ be defined by $\mathcal{A}_1(P)=-(A^*P+PA)$. How can I find $\mathcal{A}_1^{adj} (Z)$, where $\mathcal{A}_1^{adj}:S^n \to S^n$?
1
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0answers
29 views
Is there any reason to restrict the Hermitian adjoint to linear operators?
The Wikipedia page on the Hermitian adjoint is inconsistent about whether that operation is only defined for linear operators on a single Hilbert space, or more generally for arbitrary linear maps ...
0
votes
1answer
32 views
On the adjoint of a densely defined unbounded operator
Let $H,K$ be Hilbert spaces, $D(T)\subset H$ be a dense vector subspace.
Suppose $T: D(T) \rightarrow K$ and $D(T^*)=K$. What can you say about $T$?
By definition, we have $D(T^*)= \{k \in K: \exists ...
2
votes
1answer
61 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 ...
1
vote
1answer
88 views
Are the following two inner products on differential forms equal?
There are two inner product on differential forms:
$\langle \alpha,\beta\rangle$ induced from Riemannian metric $g$ by defining on 1-forms as dual of vector fields then extending to all differential ...
1
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0answers
44 views
Product of Hermitian operators
I am having trouble to proof that the product of two hermitian operators is hermitian iff they commute. Given the definition of hermitianity:
D is hermitian if it satisfy $$\int f^*(x)Dg(x)dx=\int g(x)...
4
votes
1answer
67 views
Comparing Hamiltonians - Quantum harmonic oscillator
For standard 1D quantum harmonic oscillator we have $H\psi = E_n\psi$ with $E=(n+\frac{1}{2})\hbar\omega$ and $H = \frac{P^2}{2m} + \frac{1}{2}m\omega^2 X^2$ where $X$ is position operator and $P$ is ...
1
vote
1answer
53 views
Is it true that $\text {ran} (T^*)$ is closed whenever $\text {ran} (T)$ is closed?
Let $\mathcal H$ be a Hilbert space and $T \in \mathcal L (\mathcal H).$ Suppose that $\text {ran} (T)$ is closed. Does it always mean that $\text {ran} (T^*)$ is also closed?
I am trying to prove ...
0
votes
0answers
25 views
equivalent characterization of anti-symmetric operator in complex Hilbert space
Assume that $\mathcal{H}$ is a complex Hilbert space and $B$ is a bounded linear operator such that $B^* = -B$. In some notes I found the claim that $$B^* = -B \iff \forall h \in \mathcal{H},~ \langle ...
1
vote
2answers
56 views
Is the image of closed unit ball under the linear operator closed?
I have the linear operator $A:l_1 \rightarrow L_3(0,+\infty)$ which is defined with the formula $(Ax)(t)=\sum\limits_{k=1}^{+\infty} \frac{x(k)}{\sqrt{k}+\sqrt{t}} $ for all $t\in (0,+\infty)$ and for ...
0
votes
0answers
27 views
Adjoint of a matrix operator
In my master course of operator theory of infinite dimensional linear systems, I am studying the equation $\mathcal{M}x=0$, where
$$
\mathcal{M}:=\left(
\begin{array}{rcccc}
\frac{d}{dt}& 0 ...
