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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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 $\langle Tf,g\rangle = \langle f, T^{\ast}g\rangle$ \...
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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^{\...
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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 ...
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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 ...
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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 ...
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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^{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $V^*$
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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 ...
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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 ...
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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 $\int_{-\infty}^\infty (\hat{A}\psi)^*(\hat{A}\psi)dx$...
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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 ...
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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?
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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 ...
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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
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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}{\...
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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 ...
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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: $\...
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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 $...
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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^...
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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)...
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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(\...
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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^{...
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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 & ...
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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 $$...
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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 ...
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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....
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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 ...
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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 ...
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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 $\mathcal{A}_1^{adj}:S^n \to S^n$?
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...

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