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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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 $c>0$: $$ \left\|(L^*L+c I)^{-\frac{1}{2}}L^*\right\|\leq1 $$ ...
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 ...
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2 votes
1 answer
153 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 ...
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0 votes
2 answers
84 views

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
0 votes
1 answer
28 views

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(\...
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1 vote
2 answers
51 views

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 ...
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3 votes
0 answers
64 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^{...
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1 vote
2 answers
31 views

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)^* &...
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0 votes
1 answer
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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 ...
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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 $S \in \mathcal L (\mathbb F^...
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0 votes
1 answer
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$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\...
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1 vote
0 answers
21 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 ...
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1 vote
1 answer
39 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 ...
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0 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, ...
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0 votes
1 answer
38 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
31 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}...
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3 answers
86 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 ...
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9 votes
1 answer
217 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 ...
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0 votes
0 answers
21 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
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: ...
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2 votes
0 answers
57 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 ...
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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 ...
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0 votes
1 answer
55 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) ...
0 votes
1 answer
39 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 ...
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4 votes
1 answer
106 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
37 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
59 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, ...
3 votes
2 answers
115 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 ...
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0 votes
0 answers
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 ...
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2 votes
0 answers
110 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-...
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0 votes
1 answer
95 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 ...
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2 votes
0 answers
73 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&...
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1 vote
1 answer
66 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
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 ...
1 vote
1 answer
96 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
78 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, \...
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0 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 ...
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0 votes
1 answer
92 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 ...
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1 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 ...
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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 ...
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1 vote
1 answer
39 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
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. ...
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0 votes
1 answer
65 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 \...
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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\...
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0 votes
1 answer
34 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 $A$, $A^*$ ...
2 votes
0 answers
35 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([...
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1 vote
2 answers
40 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 ...
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2 votes
2 answers
79 views

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}^{...
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