Questions tagged [adjoint-operators]
For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).
1,073
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Showing that if $A$ is closed, then $A^\ast A$ is self-adjoint
Let $A$ be a closed linear operator on a Hilbert space $H$. Then I want to show that $B = A^\ast A$ is self-adjoint.
Now, $B$ is positive, i.e. $\langle f, B f \rangle \geq 0 \forall f \in D(B)$.
...
- 1,721
3
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3
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712
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Adjoint operator of $L^\infty$
Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
- 3,549
3
votes
1
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How to find adjoint operator?
Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by
$Tx:=\sum\...
- 903
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2
answers
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Clarifying the definition of essential self-adjointness
If a Hilbert space operator $T$ has a unique self-adjoint extension, must the extension be the closure of $T$?
- 7,193
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1
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800
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Arzela-Ascoli and adjoint of compact operator compact
I have seen in this thread a nice answer where it is shown that Thread
that the adjoint operator of a compact operator is compact by using the Arzela Ascoli theorem. Unfortunately, there is one thing ...
user66906
3
votes
1
answer
1k
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Eigenvectors of Adjoint Operators
Let $T \in \mathcal{L}(V)$ where $V \subseteq \mathbb{C}^n$. Furthermore, let $v, w \in V$ such that $v$ is an eigenvector of $T$ corresponding to eigenvalue $\lambda$ (i.e. $Tv = \lambda v$). It ...
3
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1
answer
40
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Adjoint of a linear mapping mapping into a product of Banach Spaces
I am self studying adjoint operators on Banach spaces. The adjoint of a linear mapping $L:X \rightarrow Y$, where $X,Y$ are Banach spaces, is a unique mapping
\begin{equation}
L^*:Y^{*} \rightarrow X^...
- 163
3
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1
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327
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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 ...
- 687
3
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3
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709
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Showing derivative operator is self-adjoint
Consider the Hilbert space $L^2(\mathbb{R})$, and unbounded operator $Au:=iu’$ with domain
$$D(A)= \{u \in L^2(\mathbb{R}) | u \text{ is absolutely continuous and } u’ \in L^2(\mathbb{R})\} $$
I’m ...
3
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1
answer
97
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What's the intuition behind $T^*T$?
Let $T : V \to W$ be a linear transformation across inner product spaces over some field $\mathbb{F}$ and $T^* : W \to V$ be its adjoint. When considering $T$ composed with $T^*$, we have some unique ...
- 1,857
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2
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254
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Exercise of compact self-adjoint operator
'Let $H$ be a Hilbert space. Find all compact self-adjoint operators $T:H \rightarrow H$ such that $T^{k}=0$ with $k>0, k \in N$.'
$ \ $
I have this idea. Consider $\lambda_n$ eigenvalue of T and ...
- 33
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1
answer
141
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OC of Adjoint Operator’s Image is subset of Kernel
Let $T \in B(H,K)$ when $H,K$ are Hilbert spaces and $T^{\star}$ is adjoint of $T$
Show that
$(ImT^{\star})^{\perp} \subseteq KerT$
( $ImT$ means Image of $T$ and $KerT$ means kernel of $T$)
My ...
- 1,227
3
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Calculating $T^*T$ when $T$ have direct sum codomain
Let \begin{equation*} \bigoplus_{ \ell_2} K_n := \{ (x_1,x_2,\cdots) \in \bigoplus_{n=1}^\infty : x_n \in K_n, \sum_{n=1}^{\infty} || x_n ||^2 <\infty \} \end{equation*}
where $K_n$ are Hilbert ...
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Is the operator $T((x_1,x_2,...,x_n))=(x_1,\frac{x_2}2,\frac{x_3}3,...)$ on $\ell^2$ self-adjoint and unitary
Consider the Hilbert space
$$\ell^2=\{(x_1,x_2,...,x_n),x_i\in\mathbb C\text{ for all }i\text{ and }\sum_{i=1}^\infty |x_i|^2<\infty\}$$ with the inner product
$$\langle(x_1,x_2,\dots,x_n)(y_1,y_2,...
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1
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Definition of adjoint operator
I understand that given a linear operator $A$, its adjoint is define to be another linear operator $A^\ast$ such that $(Au,v)=(u,A^\ast v)$ $\forall\, u, v$ in the vector space. I'm wondering whether ...
- 31
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Spectrum $\sigma(T)$ of $T:l^1 \to l^1$ given by $T((a_j))=\left( \sum_{j=2}^{\infty} a_j \right) e_1 + \sum_{j=2}^{\infty} a_{j-1} e_j$
I'm considering the bounded linear operator $T$ on $l^1$ (the space of all absolutely convergent complex sequences) given by (with $e_k=(\delta_{kj})_{j=1,2,...}$)
$$T((a_j))=\left( \sum_{j=2}^{\...
- 83
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1
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Confirm my understanding of adjoints
adjoints seem REALLY important and useful so I don't want to move onto the next topic without really understanding them; I have too many a times moved on and been lost because I don't have the ...
- 2,388
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0
answers
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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
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Is $L^*L$, where $L^*$ denotes the formal adjoint, positive semi-definite?
Let $E$, $F$ be vector bundles with metric over a smooth (not necessarily compact) manifold $X$.
Let $L:C^\infty(E) \rightarrow C^\infty(F)$ be a differential operator.
Let $L^*:C^\infty(F) \...
- 586
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If the boundary is adjoint to the differential, what is the "coboundary" adjoint to the codifferential in the continuum?
For a smooth manifold, Stoke's theorem says that the differential/exterior derivative $\mathrm{d}$ is adjoint to the boundary operator $\partial$, i.e.
$$\int_{\partial U} \omega = \int_{U} \mathrm{d}\...
- 1,141
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93
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Diffusion-Reaction problem $u_t = \int_{-\infty}^{\infty} K(x-y) u(y) dy . u + u^3$.
I have bee stuck in this problem since more than a week. During my study I kinda understand how to find the adjoint operator for the linearization. But I have no Idea how to find the linearization to ...
- 1,312
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1
answer
364
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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 ...
- 690
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0
answers
93
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adjoint matrix of an operator
considering a $2\times 2$ matrix $\bf S$,
\begin{equation}
{\bf S} = \begin{bmatrix} \frac{\partial}{\partial{t}} & \kappa\nabla .\\ 1/\rho \nabla . & \frac{\partial}{\partial{t}}
\...
3
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0
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101
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Range of $𝐴^*𝐴$ and range of $𝐴^∗$ have the same closure
I have the following lemma and proof in my lecture notes ($\mathcal{R},\mathcal{N}$ denote range, kernel, respectively, and $\mathcal{X}$ and $\mathcal{Y}$ are Hilbert spaces).
I was hoping there was ...
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If $T^m(\alpha)=0,$ with $\alpha \in V$ and $m \in \mathbb{N}$, then $T(\alpha)=0$ [duplicate]
I am not so sure how to prove this exercise, please help me:
Let $V$ be a finite dimensional vector space with inner product $\langle,\rangle$. Let $T$ be a linear normal operator on $V$, then:
If $...
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0
answers
276
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What is a duality argument?
I believe this is probably a question that can have a wide variety of answers, but i believe i'm still interested. The thing is, i have been in a couple pde talks, and i saw that in both of these ...
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0
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How to calculate a multiplication operator representation?
Let $H =\ \mathcal{l}^{2}(\mathbb{Z},\mathbb{C})$ and $A = R + L$,
where $L$ is the left-shift operator (and $R$ is the right-shift
$(Ra)_{n}=a_{n-1}$). Set
$$U : \mathcal{l}^{2}(\mathbb{Z},\...
- 31
3
votes
0
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Prove that for the defined $\langle .,. \rangle$ there exist $0 < a \le b$ such that $a\|x\| \le \|x\|_\ast ≤ b\|x\|$ for all $x \in H$.
Let $H$ be a Hilbert space over $\mathbb{R}$ with an inner product $(·, ·)$ and the norm $\|x\| = \sqrt
{(x, x)}$. Let $A$ be a bounded strictly positive definite linear operator on $H$ with $A^\ast = ...
- 1,341
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Null space equals annihilator of range of adjoint in general spaces
(It turns out that the claim is incorrect as pointed out in the comment. Will get back soon)
Let $A, B$ be normed linear spaces and let $S$ be a bounded linear map from $A$ to $B$. Define the map $S^*...
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0
answers
70
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Show that T is a surjective linear application
Let $X,Y$ be Banach spaces and $T:X\to Y$ a bounded linear application.
Let $T^*:Y^*\to X^*$ be its adjoint (i.e. $T^*g(x)=g(Tx)$, $\forall g\in Y^*, x\in X$ ).
If $||T^*g||\geq K||g||$ for some ...
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Showing that a map is open by usying lower boundedness of adjoint
Let consider the following exercise:
Let $V$ and $W$ be Banach spaces and $T\colon V \longrightarrow W$ be a closed linear operator. Then, the following assumptions are equivalent:
(i) R$(T)=W$;
(ii) ...
- 489
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2
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If $T$ is a densely-defined injective operator between Hilbert spaces with dense range, then $T^\ast$ is injective as well
Let $H_i$ be a $\mathbb C$-Hilbert space and $T$ be a densely-defined linear operator from $H_1$ to $H_2$.
How can we show that if $T$ is injective and $\operatorname{im}T$ is dense, then $T^\ast$ ...
- 13.4k
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Adjoint vs Self-adjoint operators represented by matrices
I want to see the difference between just adjoint and self-adjoint (hermitian) operator represented by matrices.
If I have a matrix $$A=
\begin{pmatrix}
1 & i \\
i & 1 \\
\...
- 1,453
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0
answers
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Check if a Differential operator is self-adjoint
Let $\Omega \in \mathbb{R}^n$ be a bounded domain with boundary of class $C^2$. Define
\begin{cases}
D(A) = H^2 \cap H_0^1(\Omega;\mathbb{C})\\
Au(x)=\Delta u(x)-V(x)u(x) \hspace{3mm} x \in \Omega \...
- 1,812
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Show that $T$ has an adjoint, and describe $T^*$ explicitly.
Let $V$ be an inner product space and $ \beta, \gamma$ fixed vectors in $V$. Show that
$T \alpha = (\alpha\mid\beta) \gamma$ defines a linear operator on $V$. Show that $T$ has an adjoint, and
...
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adjoint operators in the vector space of real polynomials
This problem is about the space $V$ of real polynomials in the variables $x$ and $y$. If $f$ is
a polynomial, $d_f$ will denote the operator $f(d/dx,d/dy)$ , and $d_f(g)$ will denote the result of
...
- 497
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votes
1
answer
380
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Positive and negative parts of an unbounded self-adjoint operator
This question is about the proof of lemma 14 in 1, which deals with the decomposition of an unbounded self-adjoint operator $A$ in positive and negative parts: $A = A^+ - A^-$. I have trouble ...
3
votes
0
answers
143
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Show that the integral-operator compact?
We have the operator $T$ on $L_2[0,1]$ which is defined as $Tf = y$ where $y$ is the solution to the ODE $y^{\prime \prime} + y^\prime = f$ with boundary conditions $y(0)=0, y(1) = 1$
Show that $T$ ...
- 171
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257
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Explicit form of generators of a Lie algebra in the adjoint representation
My question can be summarized as:
Generators of a Lie group in the adjoint representation can be written as,
$$
(T^a_\text{Ad})_{bc} = \text{i}f^{abc}, \tag{1}\label{adj}
$$
where $f^{abc}$ ...
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votes
1
answer
150
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How is the "adjoint representation" related to generic group representations?
I'm studying representation theory in order to have a basis to study quantum field theory.
I think the text (my professor's) i'm studying on is pretty confusing.
I don't really get the difference ...
- 131
3
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0
answers
276
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Derivative of Adjugate Matrix, $\frac{d^i}{dx^i}\operatorname {Adj}(x \mathbf I - \mathbf A)$ [duplicate]
Let $\mathbf A$ denote an $n \times n$ matrix with $r=\operatorname{rank}\mathbf A$ and define
$$ \mathbf B_i(x) \equiv \frac{d^i}{dx^i}\operatorname {Adj}(x \mathbf I - \mathbf A). $$
Conjecture:
...
- 1,885
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votes
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answer
230
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Operator $f \mapsto u(f)$ solution of non-homogeneous Laplace equation is compact and self-adjoint
Let $u : L^2_0(D) \to L^2_0(D): = \lbrace f \in L^2 : \int_D f = 0 \rbrace $ be the linear operator which associates $f$ to $u(f)$ the solution of
$$
\begin{cases}
\Delta u = f & \text{in } D \\
\...
- 31
3
votes
0
answers
27
views
Finding two adjoints, and showing boundedness of operators
Let $H = l_2$ and consider the following operators: $T,S:H \to H$ $Tx = (0,x_1,x_2,\ldots)$ and $Sx = (x_2,x_3,x_4,\ldots)$ Show they are bounded, and find the adjoint of both:
For $T$, I have $\|Tx\|...
- 31
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0
answers
332
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adjoint method for computing derivatives
I am curious if anyone has heard of this problem before:
Suppose that $u(x,p)$ is a function of $x$ and $p$. These arguments need not be scalars.
Let $u(x,p)$ satisfy some differential equation, say:...
- 3,883
3
votes
1
answer
558
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The adjoint of a linear isometry on normed space is surjective
I come across with the following question:
Let $X$ and $Y$ be normed spaces over the same field and $T:X \rightarrow Y$ be a linear isometry, with $\|Tx\|=\|x\| \, \forall x \in X$, then the adjoint ...
- 369
3
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1
answer
388
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Properties of adjoint matrix in a finite dimensional inner product space
let $V$ be a finite dimensional inner product space. Let $T$ be a linear operator on $V$.
Prove that there exists an invertible linear operator $U$ such that
$U^{-1}TT^*U = T^*T
$ where $T^*$ is ...
- 1,079
3
votes
0
answers
88
views
Do I have the correct mental map for adjoint operators for inner product spaces?
Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = <...
- 10.9k
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1k
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Linear map is diagonalizable iff its adjoint is diagonalizable
Problem
Let $V$ be a finite inner product space and let $T:V \to V$ be a linear transformation. Prove that $T$ is diagonalizable if and only if the adjoint transformation $T^{*}$ is diagonalizable.
...
- 2,551
3
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1
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107
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How to prove this is a self-adjoint operator?
I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$
I want to ...
- 5,173
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1
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$T \in B(X,Y)$ is an isometry if and only if $T^*$ is an isometry
I would like to prove that $T \in \mathscr{B}(X,Y)$ is an isometry of $X$ onto $Y$ if and only if $T^*$ is an isometry of $Y^*$ onto $X^*$. I am not really sure what to do. I started the argument as ...
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