Questions tagged [adjoint-operators]
For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).
1,070
questions
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Bound on spectral radii
Let $L$ be some linear and compact operator on some Hilbert space. And denote by $L^*$ the adjoint operator.
I need to bound for some $c>0$:
$$
\left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1
$$
...
- 35
0
votes
1
answer
60
views
Why a matrix has to be invertible for $\operatorname{adj} A^T=(\operatorname{adj}A)^T$ to be true?
I read a theorem
If $A$ is an invertible square matrix, then $\operatorname{adj} A^T= (\operatorname{adj} A)^T$.
But after attempting to prove it myself and also reading the proof I am unable to ...
2
votes
1
answer
44
views
Prove the following: If $T \in \mathcal{L}(V)$, and $v$ is an eigenvector of $T$, then $\overline{v}$ is an eigenvector of $T^*$.
I feel like this should be obvious, but I'm kind of stuck on how to prove it.
I know that $v \in \text{null}(T - \lambda I)$, and that $\overline{\lambda}$ is an eigenvalue of $T^*$, but I don't ...
- 57
2
votes
1
answer
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views
Show that for any $S$, if $ST=TS$, then $ST^*=T^*S$.
Let $V$ be a complex inner product space with $\dim(V)<\infty$. Let $T$ be a normal operator ($TT^*=T^*T$). Show that $V$ is the direct sum of $\text{null}(T)$ and $\text{range}(T)$. Show that for ...
- 934
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2
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84
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Show that the set is the convex hull of the spectrum of $T$.
Let $V$ be a complex inner product space with $\dim V<\infty$. Let $T$ be a normal operator $(TT^*=T^*T)$. Show that the set of $\{\langle Tv, v\rangle: v\in V, \|v\|=1\}$ is the convex hull of the ...
- 934
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1
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28
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Multiplication operator is self-adjoint on $L^2(\mathbb{R})$
I'm trying to do the following exercise:
Let $M: \mathcal{D}(M) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ be the multiplication operator $M f=x f$ with
$$
\mathcal{D}(M)= \{f \in L^2(\...
- 143
1
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2
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51
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Question about projections acting on dual space
Let $X$ be a complex Banach space, and let $P$ be a bounded linear operator acting on the dual $X^{*}$ such that that $P^2=P$.
I research for a bounded linear operator $Q$ acting on $X$ such that its ...
- 51
3
votes
0
answers
64
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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^{...
- 1,992
1
vote
2
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31
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Proving Equality Regarding the Adjoint of Bounded Linear Operator
I am proving the Proposition 2.13 from Elementary Functional Analysis by MacCluer, mainly on c) and d)
We're given that For any $A,B \in \mathscr{B}(\mathscr{H})$, we have
\begin{align*}
(\alpha A)^* &...
- 189
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votes
1
answer
28
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The proof of $||T|| = \sup\{|(T(f),f)| \;|\; ||f|| = 1 \}$ when $T = T^*$
I'm studying a property of symmetric linear operator in Hilbert space in Stein's Real analysis chapter 4.
When $T = T^*$, then $\Vert T\Vert = \sup\{|(T(f),f)| \;|\; \Vert f\Vert = 1 \}$
The following ...
- 133
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votes
1
answer
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Show that $S(\vec x) = \sum_{i=1}^n x_i\vec u_i$ is the adjoint of $[\vec x]_{\beta}$
Let $V$ be an inner product space over $\mathbb C$ and let $B = \{\vec u_1, \dots, \vec u_n\}$ be an orthonormal basis for $V$. Let $T \in \mathcal L(V, \mathbb F^n)$ and $S \in \mathcal L (\mathbb F^...
- 175
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votes
1
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$A^*+A\geq0$ if and only if $A\geq0$
Let H be a Hilbert space and $A:H\to H$ be a linear operator.
Definition (Positive) We say that A is positive (denoted by $A\geq0$) if $A=A^*$, i.e., $A$ is self-adjoint,
and $$\langle Ax,x\rangle\...
- 498
1
vote
0
answers
21
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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 ...
- 103
1
vote
1
answer
39
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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 ...
- 407
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votes
1
answer
32
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, ...
- 2,579
0
votes
1
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38
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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
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0
answers
31
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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}...
- 1
0
votes
3
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86
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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 ...
- 1,772
9
votes
1
answer
217
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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 ...
- 272
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0
answers
21
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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 ...
- 35
4
votes
1
answer
59
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: ...
- 41
2
votes
0
answers
57
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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(...
- 21
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vote
2
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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 ...
- 1,772
2
votes
0
answers
50
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 ...
- 3,439
0
votes
1
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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) ...
- 1
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1
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39
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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 ...
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1
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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 ...
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3
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1
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37
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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}(\...
- 453
1
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1
answer
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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, ...
- 125
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votes
2
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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 ...
- 1,145
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0
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25
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 ...
- 535
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0
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110
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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-...
- 21
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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 ...
- 1,281
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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&...
- 1,585
1
vote
1
answer
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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 ...
- 35
0
votes
1
answer
45
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 ...
- 525
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vote
1
answer
96
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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
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vote
1
answer
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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, \...
- 3,892
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votes
0
answers
91
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 ...
- 594
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votes
1
answer
92
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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 ...
- 594
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vote
2
answers
108
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 ...
- 4,401
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 ...
- 69
1
vote
1
answer
39
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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).
$$
I have showed that $A$ is bounded and symmetric, I have to prove only ...
- 19
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votes
1
answer
37
views
What does "normalised" mean in the following context?
The following is a question from a course on quantum mechanics:
Prove that $\hat P_i = \lvert i \rangle \langle i \rvert$ is a Projection operator as long as $\lvert i \rangle$ is normalized. ...
- 59
0
votes
1
answer
65
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Difference in Tikhonov regularization for linear and non-linear case?
the Tikhonov regularization for a linear operator $T: X \rightarrow Y, x \mapsto y$ means minimizing the least square problem
$$\begin{align*}
\lVert Tx - y \rVert^2_Y + \alpha \lVert x\rVert_X \...
- 151
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\...
- 11
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votes
1
answer
34
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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 $A$, $A^*$ ...
- 430
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votes
0
answers
35
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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 $L^p([0,1]) \subset L^r([...
- 175
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vote
2
answers
40
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Finding the adjoint of operator on P_1
The question asks to find the adjoint of the operator T on P$_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 ...
- 25
2
votes
2
answers
79
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Show that $T^*T$ is positive semi definite
I am struggeling with this one:
Let $X = \mathbb{R}^n$, $Y = \mathbb{R}^m$. We equipe $X$ with the scalare product $\langle x_1,x_2 \rangle_X = x_2^T \cdot M_x \cdot x_1 $, where $M_x\in \mathbb{R}^{...
- 151