Questions tagged [adjoint-operators]

For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).

Filter by
Sorted by
Tagged with
1 vote
0 answers
8 views

Eigenvalues of an operator and its adjoint

I would like to verify the proof given in an answer #2 form here. The claim to prove was that $\lambda$ is an eigenvalue of $T$ if and only if $\overline{\lambda}$ is eigenvalue of the adjoint ...
2 votes
1 answer
81 views

Change of basis and diagonalization

For my notation here I'm going to use $'$ to indicate a new basis. Let $U$ be a unitary operator that takes a basis vector $|i\rangle$ and transforms it to a new basis vector $|i'\rangle$. So let's ...
1 vote
1 answer
35 views

Find self-adjoint of $P=|a \rangle \langle b |$

Let $P$ be an operator s.a. $P=|a \rangle \langle b |$ and $P|f \rangle = \langle b | f \rangle | a \rangle $. Find the self-adjoint and the $P^2$ operator. My attempt: We know to find the self ...
0 votes
1 answer
30 views

Question about operators from $\mathbb{R}^\mathbb{N}$ to/from a real separable Hilbert space.

Let $X \equiv \mathbb{R}^\mathbb{N}$ denote the space of real sequences with the product topology. Suppose that $T \colon \mathcal{H} \to X$ is a linear operator, where $\mathcal{H}$ is a real, ...
0 votes
1 answer
32 views

Show that if $u\circ u^{\star} = u^{\star}\circ u$ then $\ker(u^{\star})=\ker(u)$

Let $E$ be a Euclidean Space and $u\in\mathcal L(E)$. We let $u^{\star}\in\mathcal L(E)$ such that $\forall x,y\in E, \langle u(x)|y\rangle = \langle x|u^{\star}(y)\rangle$. We want to show that if $u\...
0 votes
0 answers
28 views

Derivative self-adjoint operator

1)I have the symbol $\circ$ ? In $-(\mathcal{L} \psi)\circ \psi^{-1}$ How to read it in math in this case ? 2)The derivative of $q(y)=-(\mathcal{L} \psi(x)) $ is $q^\prime(y)= \frac{-{[\mathcal{L}...
9 votes
1 answer
196 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 ...
1 vote
1 answer
1k views

$T^{\ast}T$ unitary then $T$ isometry

An operator $T$ is an isometry if $||Tf||=||f||$ for all $f$ in a Hilbert space $H$. (a) Show that if $T$ is an isometry, then $<Tf,Tg>=<f,g>$ for every $f,g \in H$. Prove as result that $...
0 votes
3 answers
85 views

Prove that $\{T_v: \|v\|\leq 1\}$ is a family of pointwise uniformly bounded functionals

Let $V,W$ be Hilbert spaces and $A:V\rightarrow W$, $B:W^*\rightarrow V^*$ linear operators such that $l(Av)=(Bl)(v)$ for all $v\in V$ and for all $l\in W^*$. For $v\in V$ define $T_v(l):=l(A(v))$ for ...
4 votes
1 answer
2k 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\...
0 votes
0 answers
20 views

Showing that $\Vert T^*T \Vert = \Vert T \Vert^2$ for $T$ a linear and bounded operator from a Hilbert space to itself [duplicate]

I am reading a book in which it is mentioned that for a bounded and linear operator $T: H \to H$, with $H$ a Hilbert space we have $\Vert T^{*}T\Vert = \Vert T \Vert^2$. The proof is left as an ...
4 votes
1 answer
54 views

Find adjoint to integral operator from $H^1$ to $L_2$

Let $k(x, y): \mathbb{R}^2 \to \mathbb{R}$ be a kernel and $T: H^1(a, b) \to L_2(c, d)$ $$ T u(x) = \int\limits_{a}^{b} k(x, s) u(s) ds$$ Find the adjoint operator $T^*$. It is easy to see the if $T: ...
2 votes
0 answers
40 views

Differential Operators on Bessel Functions

I am currently reading through Advanced Real Analysis by Anthony Knapp, and I've been stuck on a bit of a computation that the author seems to glance over. The larger problem is proving $\int_0^1J_0(...
1 vote
2 answers
96 views

How can I finish my proof that $||A^*A||=||A||^2$?

Let $A:X\rightarrow X$ be a bounded linear operator on a Hilbert space $X$. I want to show that $||A^*A||=||A||^2$. My idea was the following: Let me consider $x\in X$ such that $||x||\leq 1$, then ...
-1 votes
2 answers
246 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 ...
2 votes
0 answers
46 views

Spectral Representation of $T$

the spectral family of the operator $T:\ell^2\rightarrow\ell^2$ defined by $$T(\xi_1,\xi_2,\xi_3,....)=(\xi_1/1,\xi_2/2,\xi_3/3,....)$$ I am trying to get the application of Spectral theorem of ...
4 votes
2 answers
1k views

How to show that $\|T\|^2=\|T^*T\|$ for a bounded linear operator $T$?

I need to show that for a bounded linear operator, $T$, on a Hilbert space: \begin{align*} \|T\|^2=\|T^*T\| \end{align*} All I have so far: \begin{align*} \|T^*T\|&=\sup\{|\langle T^*Tf,g \rangle ...
0 votes
1 answer
50 views

Bessel sequence ,frame sequence in Hilbert Space

This question was asked in my assignment of Functional Analysis and I am not able to make progress in few parts. Question: Let H be an Hilbert space and let $(f_i)_{i\in I}$ be a (finite or infinite) ...
2 votes
2 answers
200 views

Why are the eigenfunctions of my Hermitian operator not orthogonal?

I am finding that the eigenfunctions of my Hermitian differential operator are not orthogonal and I do not know why. Consider the differential operator $$ \mathcal{L} = x^2 \frac{d^2}{dx^2} + 2x \frac{...
3 votes
3 answers
1k views

What is the intuition behind the kernel of $A$ being orthogonal to the range of $A^*$?

We know that, if $A$ is a linear bounded operator, then $\operatorname{Ker}(A) \perp \operatorname{Range}(A^*)$, where $A^*$ is the adjoint of $A$. I have no troubles understanding the proof of this (...
0 votes
1 answer
37 views

Different answers for adjoint of $T : \ell^2 \to \ell^2$ such that $T(x_1, x_2, \dots) = (c_1 x_1, c_2 x_2, \dots),$ where's the mistake?

$c = \{c_n\} \in \ell^{\infty}$ and $T : \ell^2 \to \ell^2$ is defined as $T(\{x_n\}) = \{c_n x_n\},$ all spaces are over $\mathbb{C}.$ It's easy to prove that $||T|| = ||c||,$ but I'm getting 2 ...
4 votes
1 answer
103 views

Can someone explain why we define adjoint?

Recently I've been reviewing linear algebra. The definition of adjoint of a linear map on an inner product space seems not really natrual. Looks like people use this to define normal and self-adjoint ...
3 votes
1 answer
36 views

1-Lipschitzian Linear Operators on Hilbert Spaces and Fixed Points

$\textbf{Question}$ Let $(\mathcal{H}, \langle \cdot \, | \, \cdot \rangle)$ be a real Hilbert space with induced norm $\|\cdot\| = \sqrt{\langle \cdot \, | \, \cdot \rangle}$ and let $$\mathscr{B}(\...
1 vote
1 answer
42 views

Proof that linear combination of self adjoint maps is also self adjoint.

I want to show that if $V$ is an inner product space and $S,T\in \mathcal{L}(V)$ are self-adjoint linear maps, then $aS+bT$ is a self-adjoint linear map for all $a,b\in \mathbb{R}$. From what I tried, ...
2 votes
0 answers
70 views

Let $T$ be a linear operator in a vector space $V$ such as $T$ admits an ajoint. Prove that if $T^*T=0$ then $T=0$.

Let $T$ be a linear operator in a vector space $V$ such as $T$ admits an ajoint. Prove that if $T^*T=0$ then $T=0$. I tried to do the following: If $v\in V$ then $0=\left< v,T^*T(v)\right>=\left&...
3 votes
2 answers
109 views

Definition of the formal $L^2$-adjoint $T^*$ of a linear operator $T:C^\infty(T^*M\odot T^*M)\to C^\infty(M)$

Let $(M,g)$ be a Riemannian manifold, $C^\infty(T^*M\odot T^*M)$ the space of all smooth symmetric $2$-tensor fields on $M$, and $C^\infty(M)$ the space of all smooth functions on $M$. I'd like to ...
0 votes
0 answers
23 views

A question about weak solution of the adjoint operator in Evans' PDE

In Partial Differential Equations (Evans, 2nd edition) $\S$6.2.3, the author discusses about the Fredholm alternative w.r.t the second order elliptic PDE. My question may require that you are familiar ...
3 votes
1 answer
291 views

Matrix of self-adjoint operator such that every element of the diagonal is $0$.

Let $V$ be a finite dimensional $\mathbb R$-vector space and let $T:V\rightarrow V$ be an self-adjoint operator such that $\text{trace}(T)=0$. Show that there exists an orthonormal basis $B$ such that ...
0 votes
1 answer
254 views

Proving $\dim\ker(I-T)=\dim\ker(I-T^*)<\infty$ for finite-rank operator on a Hilbert space

In my functional analysis class, I have encountered the following problem Let $H$ be a Hilbert space and $T$ a finite-rank (its range is finite-dimensional) and bounded linear operator on $H$. We are ...
1 vote
0 answers
16 views

Compute the adjoint of the derivative operator with different domains

Let $T:=\frac{d}{d x}$ acting on $L^2[0,1]$. Compute $\operatorname{Dom}(T^*)$ for the following choices of domain: $\operatorname{Dom}(T):=C_0^{\infty}(0,1)$. $\operatorname{Dom}(T):=\{f \in C^{\...
2 votes
0 answers
96 views

Prove that the domain of unbounded self-adjoint operator is strictly larger than that of its square

This is the problem from the textbook Unbounded Self-adjoint Operators on Hilbert Space by Konrad Schmudgen Let $A$ be a self-...
0 votes
1 answer
65 views

Show that the adjoint of two operators is the sum of the adjoints

Problem Show that for any two operators $\hat{A}$ and $\hat{B}$, the adjoint $(\hat{A} + \hat{B})^\dagger = \hat{A}^\dagger + \hat{B}^\dagger$. Do so using the integral form of the definition of ...
1 vote
1 answer
64 views

Mappings of Adjoint Linear Operators

Let $$ \begin{align*} \mathsf{A}:~ \mathbb{R}^n & \rightarrow \mathbb{R}^m \\ \boldsymbol{x} & \mapsto \boldsymbol{y} = \boldsymbol{A} \boldsymbol{x}. \end{align*} $$ The ...
0 votes
1 answer
42 views

uniqueness of adjoint operator

I'm trying to prove the uniqueness of the adjoint operator between two finite-dimensional Hilbert spaces $E$ and $F$. My idea is to simply use the isomorphism between finite-dimensional Hilbert spaces ...
1 vote
1 answer
87 views

A question to show that the integral operator is a bounded operator

This question was asked in my assignment of Functional analysis and I was not able to solve this particular problem. Question:(a) Show that the formula $Af(x) = \frac{1} {x} \int_{0}^x f(y) dy$ ...
1 vote
1 answer
73 views

Disintegration theorem: how to obtain $\|\pi_\sharp f\|_{L^\infty(Y)}\leq \|f\|_{C(X)}$ for all $f\in C(X)$?

Theorem $4$ of this blog entry of Terrence Tao states the following: Let $X$ be a compact metric space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$. $(Y, \...
0 votes
1 answer
87 views

Bounded below adjoint operator on the dual of an ordered Banach space

Suppose $X$ is a real Banach space with a $\textit{generating}$ closed cone $X_+$. That is $X=X_+ - X_+$. Let $B\subset X$ be the $\textit{open}$ unit ball, and denote $B_+:=B\cap X_+$. Show that the ...
0 votes
0 answers
59 views

Closed range of adjoint operator: Closed range theorem and a theorem in Conway's Functional Analysis book

Recently I came across the "Closed range theorem, which applies to densely defined closed operators. However, I'm also aware of a similar theorem (VI.1.10) in Conway's A Course in Functional ...
1 vote
2 answers
78 views

Proof that the orthonormal projection $P$ onto $W$ in a Hilbert space satisfies $P^2=P$ and $P*=P$.

Let $H$ be a Hilbert space. Given a closed subspace $W\subseteq H$, the orthogonal projection onto $W$ is the unique bounded linear operator $P$ such that $\text{Im}(P)=W$ and $\ker(P)=W^{\perp}$. The ...
0 votes
0 answers
16 views

If $A \in \Bbb C^{n\times n}$, Show that $L(A, b) \neq \emptyset$ iff $b \in L(A^H, 0)^\perp$

If I put the problem into other words, then $Ax = b$ has a solution iff b is in the orthogonal complement of the kernel of $A^*$ i.e. the complex transpose of A. I know how to do these problems when ...
3 votes
2 answers
2k views

Definition of Adjoint Operator for Quantum Mechanics

While learning about adjoint operators for quantum mechanics, I encountered two definitions. The first definition is given by Shankar in The Principle of Quantum Mechanics: Given a ket $$ A\lvert ...
1 vote
1 answer
36 views

Compact operator in $L^2(0,2\pi)$

I have to prove that an operator is compact. The operator is $A:L^2[0,2\pi]\to L^2[0,2\pi]$ given by $$ Au(x)=\cos x\,u(x). $$ I have showed that $A$ is bounded and symmetric, I have to prove only ...
2 votes
1 answer
36 views

What does "normalised" mean in the following context?

The following is a question from a course on quantum mechanics: Prove that $\hat P_i = \lvert i \rangle \langle i \rvert$ is a Projection operator as long as $\lvert i \rangle$ is normalized. ...
0 votes
1 answer
58 views

Difference in Tikhonov regularization for linear and non-linear case?

the Tikhonov regularization for a linear operator $T: X \rightarrow Y, x \mapsto y$ means minimizing the least square problem $$\begin{align*} \lVert Tx - y \rVert^2_Y + \alpha \lVert x\rVert_X \...
1 vote
0 answers
21 views

Finding adjoint of a Matrix equation

For $\alpha_i \in \mathbb{C}$, write down the adjoint of $$\alpha_1\left|V_1\right\rangle=\alpha_2 \Theta \Pi\left|V_2\right\rangle\left\langle V_3 \mid V_4\right\rangle+\alpha_3^*\left|V_5\right\...
0 votes
1 answer
26 views

Adjoint of a Densely Defined Unbounded Operator is Unique

Let $\mathcal H$ be a Hilbert space and $A: \mathcal D_A \to \mathcal H$ be an unbounded linear operator. Suppose also that $\mathcal D_A$ is dense in $\mathcal H$. We define the adjoint of $A$, $A^*$ ...
9 votes
1 answer
1k views

Intuition of Adjoint Operator

A linear operator $T$ on an inner product space $V$ is said to have an adjoint operator $T^*$ on $V$ if $⟨T(u),v⟩=⟨u,T^*(v)⟩$ for every $u,v\in V$ I know how to proof "why this operator exist$(\text{...
2 votes
0 answers
27 views

Bounded surjective linear map from $L^p$ to $L^r$ with $r < p$

I am thinking about the following problem. Let $1 \leq r < p \leq \infty$, and consider the spaces $L^r([0,1])$ and $L^p([0,1])$. A routine application of Holder shows that $L^p([0,1]) \subset L^r([...
3 votes
1 answer
334 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 ...
1 vote
2 answers
38 views

Finding the adjoint of operator on P_1

The question asks to find the adjoint of the operator T on P$_1$([0,1]), which is the space of polynomials with degree no greater than 1 over the field [0,1], which contains all numbers between 0 and ...

1
2 3 4 5
22