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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Question regarding conjugate operators and the harmonic operator.

Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$ I'...
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Spectrum analysis

I have a problem: Given the Hamiltonian $ H_n= -\frac{d|}{d^2}+V_n $ where $ V_n= a n e^{-n|x|}; a\in\Bbb R, n=1,2,....$ 1)Domain in which the operators are self adjoint; 2)study and list the ...
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Orthogonal projection + self-adjoint exercise (Axler Chapter 7 Exercise 4)

I think I am misunderstanding something about this exercise (and about adjoints in general). Exercise: Suppose $P \in L(V)$ is such that $P^2 = P$. Prove that if $P$ is an orthogonal projection then $...
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Formal adjoint of the exterior derivative on a manifold with boundary

Let $(M,g)$ a compact Riemmanian manifold with boundary. I want to define the formal adjoint of the exterior derivative $\mathrm d\colon C^\infty(M)\to \Omega^1(M)$. If $\partial M=\emptyset$ and $\xi$...
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Is the operator $(-\Delta)^{-1} : C^\beta(\overline{\Omega}) \to C^\beta(\overline{\Omega})$ compact and self-adjoint?

Well, from variational methods theory we know that the operator $(-\Delta)^{-1}:L^2(\Omega) \to L^2(\Omega)$ is compact and self-adjoint. Since $C^\beta(\overline{\Omega})$ is a subset of L^2(\Omega), ...
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show that operator $A$ that $(A x)(y)=(A y)(x)$ is continuous [duplicate]

Let $X$ be a Banach space and let $A: X \rightarrow X^{*}$ be a linear operator satisfying $$(A x)(y)=(A y)(x)$$ Show that $A$ is a continuous operator, i.e. $A \in L\left(X, X^{*}\right)$. I know ...
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meaning of "adjoint" in ordinary differential equations

I am reading the book "Ordinary Differential Equations" By Birkhoff and Rota (1989). I am a little confused about the meaning or definition of "adjoint operator" and "adjoint ...
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Proof that T*T is non-negative on linear transformations

I've been doing a few problems and ran upon one I've tried to solve with no luck: Let $T: V \to W$ be a linear transformation and $T^*$ be its adjoint, prove that $T^*T$ is non-negative. Most of the ...
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Theorem 1.1 of The mathematics of computerized tomography

In the book "The mathematics of computerized tomography" by Natterer comes the theorem: Theorem 1.1 $f=A^+g$ is the unique solution of $A^*Af=A^*g$ in $range(A^*)$. where $A:H\rightarrow K$ ...
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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}\...
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How to Find the Adjoint of a Differential Equation?

I am a bit confused about the entire topic of the adjoint and am not totally sure how to apply the definition of it to a given differential equation in order to obtain the adjoint equation. I have ...
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On proving Ker $(T^*T)$=Ker $((T^*T)^\frac{1}{2})$

Can we prove Ker $(T^*T)$=Ker $((T^*T)^\frac{1}{2}))$ without using functional calculus? One inclusion Ker $((T^*T)^\frac{1}{2}))\subset$ Ker $(T^*T)$ is clear. For the other inclusion I want to prove ...
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Prove that T is self-adjoint

Define the linear operator $T:\ell^2 \to\ell^2 $ by $$T(x_1,x_2,x_3,...)= \left(x_1,\frac{1}{2}x_2,\frac{1}{3}x_3,...\right)$$ I need to show that T is self-adjoint. I know that T is self-adjoint iff $...
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Differential of a function in the inner product does not have an adjoint

Given two elements $f,g$ from the vector space $\mathbb{R}[x]$, we define the inner product to be $$\langle f,g \rangle = \int ^1 _0 fg \,\,dx.$$ If $Df$ is the derivative of $f$, prove that $D$ doesn'...
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Computing adjoint of linear operator

I am preparing for my exam in functional analysis and came across the following question: Let $n\in \mathbb{N}$. Find the adjoint to A: $\ell^1\to\ell^1 $ (where $\ell^1=\{x=(x_1,x_2,x_3,...),\,\sum^{...
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Is my proof that $\text{rank}(A + B) = \text{rank } A + \text{rank }B$ correct?

Problem Statement. Let $A, B \in \mathcal{M}^{m \times n}(\mathbb{C})$ such that $\text{range }A^* \cap \text{range }B^* = \{0\}$ and $B^*A = 0$. Prove that $\text{rank}(A + B) = \text{rank } A + \...
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Proving Theorem 10.3 on Steven Roman's Advanced Linear Algebra

I want to prove the (3) and (4) of theorem which says $\textrm{im}(\tau \tau ^{*})=\textrm{im}(\tau )$ and $(\rho _{\textrm{S,T}})^{*}=\rho _{\textrm{T}^{\perp },\textrm{S}^{\perp }}$, here $\tau ^{*}$...
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How to prove the adjoint of this operator is the $-T-I$

Suppose $\mathbb R$ is a finite-dimensional real inner product space, and $T$ is an operator on it with the following properties $1. T^2=T^*$ $2. T$ is invertible $3. T-I$ is invertible Prove that $T^*...
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Finding adjoint and norm of projection

We consider the Hilbert space with standard inner product, $0<\phi<\frac{1}{2} \pi$. We consider the projection P: $ P \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} x_1-x_2cot(\phi)\\...
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How to find adjoint?

where From this question I concluded that L^T = L*. But when I find the transpose and then adjoint of the matrix I get as the adjoint L* of L. But it is wrong. What am I doing wrong?
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The linear operator $L:V\to V$ is defined by $L(p(x)) = -6p'(x) - 6p(x)$. How can I find the adjoint $L^{*}$ of $L$?

If I'm given an inner produce: $$\langle p(x),q(x)\rangle = \int_{0}^{1}p(x)q(x)\mathrm{d}x$$ on the vector space $V$ of real polynomials with degree less than $2$. The linear operator $L:V\to V$ is ...
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For $ [T]_{\mathcal{B}}=\left(\begin{array}{lll} a & & \\ & b & \\ & & c \end{array}\right) $, Prove that $T$ is self-adjoint if and only if $a=b$.

Problem: Let $T$ denote a linear map defined on $V=\mathbf{R}^{3}$, and let $(\cdot, \cdot)$ denote an inner product defined over $V$. Let $\mathcal{B}=\left\{v_{1}, v_{2}, v_{3}\right\}$ denote a ...
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Solving Simple PDE by Green's Function, Very Confused By Some Mistake

Suppose I want to solve $u_{xy} = xy$ via Green's Function. This will correspond to the associated PDE $G_{xy} = \delta(x - x_G,\ y - y_G)$, and I want my boundary conditions for this Green problem ...
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if range $A^* \cap$ range $B^* = 0$ and $B^*A = 0$ then $\text{rank }(A + B) = \text{rank }A + \text{rank }B$

Problem. Let $A, B \in \mathcal{M}_{m,n}(\mathbb{C})$ such that $\text{range }(A^*) \cap \text{range }B^* = \{0\}$ and $B^*A = 0$. Prove that $\text{rank }(A + B) = \text{rank }A + \text{rank }B$. My ...
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Derivation of adjoint poisson equation with convective boundary condition

I am trying to derive the adjoint Poisson equation for the following problem to find the sensitivity of an objective function with respect to a decision variable, but I get stuck in the middle of the ...
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Finding Boundary Conditions That Make This Green's Function Nice

This question used to be included here as a second point, but I figured the two were better asked as two separate questions. When people use the phrase "the ____ operator is self-adjoint" (i....
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Computing the Adjoint of an Operator

Can a computer take in a linear operator and set of boundary conditions, and compute the adjoint operator/conditions? I can find calculators/Mathematica code/etc. to compute the adjoint of a matrix, ...
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$TT^* = T^*T$ iff $T = A + iB$ with $A, B$ self-adjoint

I'm working through the ideas in this document from Berkley. They claim (I'm paraphrasing), "$TT^* = T^*T$ if and only if $T$ is of the form $T = R + iM$ where $R$ and $M$ are self-adjoint." ...
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Fourier transform of a spectrum with operators using Dirac delta functions

I have the operator $h(t)$ and its Fourier transform $h(\omega)$ defined such that $$ h[\omega]=\mathcal{F}\{h(t)\}=\int h(t)e^{i\omega t}dt\,\,\,\,\,\,\,\,\,\,(1) \\ h(t) = \mathcal{F}^{-1}\{h[\omega]...
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example in spectral theory

Consider $X=l_{2}.$ Let $T : l_{2}\longrightarrow l_{2}$ be defined by : $T(x_{1},x_{2},....)= (x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},...).$ And $S=I$ , the identity operator. Here $N(T)=N(S)=\lbrace{...
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Column space of $A^*$ is row space of $A$?

I have seen that range $A^*$ = $(\text{null }A)^\perp$, for example here. And I have seen that $(\text{null A})^\perp$ = row $A$, for example here. Does it therefore follow that range $A^*$ = row $A$? ...
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Is it possible to obtain an expression like $\langle T^*Tx,x\rangle_H\ge c\|x\|_H^2$ for $c>0$?

Let $H$ be a Hilbert space and let $S\in B(H)$ be a selfadjoint linear operator. We say that $S$ is positive if for all $x\in H$, $$\langle Sx,x\rangle_H\ge0.$$ If $V\in B(H)$ a not necessarily ...
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Is the positive operator $TT^*-T^*T$ on a finite complex inner product space is a zero operator?

Suppose the vector space is a complex finite-dimensional inner product space $V$, and the operator $TT^*-T^*T$ is a positive operator on it. Can this information be sufficient to conclude that $TT^*-T^...
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Fourier Transform definitions for operator

I am following the convention of FT for operators, normalized to a length scale $L$, whereby $$ a_{k}=\frac{1}{\sqrt{L}}\int dx\,e^{-ikx}a(x)\,\,\,\,\,\,\,\,\,\,\,(1) \\ a^{\dagger}_{k}=\frac{1}{\sqrt{...
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Adjoint of unbounded integral operator

Consider a Borel-measurable function $a:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}$, and let $T_a$ be the linear operator on $L^2(\mathbb{R})$ with domain \begin{equation} \mathcal{D}(T_a)=\left\{...
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About the proof of invertible quadratic expressions in Linear Algebra Done Right

The theorem is that if $ T\in L(V)$ (and $V$ is a real finite-dimensional vector space) is self-adjoint and b,c $\in \mathbb R$ such that $b^2<4c$. Then the operator $$ T^2 + bT +cI$$ is invertible....
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Geometric interpretation of adjoint

Given the inner product $\langle x, y \rangle = y^HMx, \, x,y\in\mathbb{C}^n$ where $y^H = \bar{y}^T$ is the conjugate transpose and $M^H=M$ positive definite, I find the adjoint $A^*$ of $A\in\mathbb{...
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Construct the Green's function for the following boundary value problem and use it to find the solution $y''-y'=t^2$

Construct the Green's function for the following boundary value problem and use it to find the solution $$y''-y'=t^2,\quad y(0)=0,\: y(1)=0$$ I know that, $$ \begin{aligned} &\langle L u, G\...
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Prove that in a complex Hilbert space $\|Tx\|\le\|T\|^\frac{1}{2}\langle Tx,x\rangle^\frac{1}{2}$

Let $H$ be complex Hilbert space. Prove that $\|Tx\|\le\|T\|^\frac{1}{2}\langle Tx,x\rangle^\frac{1}{2}$. I am trying to prove this result by using the following result. Let $T:H\to H$ be a bounded ...
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Proving 2 properties of the adjoint map $T^*$ on a Hilbert space

Let $(\mathcal{H}, \langle \cdot, \cdot \rangle)$ denote a seperable Hilbert space with orthonormal basis $(e_n)_{n \in \mathbb{N}}$ with induced norm $\Vert \cdot\Vert$. Let $T:\mathcal{H} \...
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Self-adjoint operators conditions

I know that a system with $\mathfrak{L}\equiv\mathfrak{L}$* and the boundary conditions also hold then the problem is known as fully self-adjoint, and if only the first statement holds it is formally ...
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2 answers
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Define a linear map $T:V \to V$ by $T(v)=\left(v,u_{1}\right)u_{2}$. Here $v,u_{1},u_{2}\in V$. Write down a formula for the adjoint map $T^{*}$.

Problem: Let $V$ denote an inner product vector space. Define a linear map $T:V \rightarrow V$ by $T(v)=\left(v,u_{1}\right)u_{2}$. Here $v,u_{1},u_{2}\in V$. Write down a formula for the adjoint map $...
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If there exists $v\in L$ s.t. $ \|\phi(v) \| < \| \phi^*(v) \| $, then there exists $u\in L$ s.t. $\| \phi(u)\|>\|\phi^*(u) \|$

Problem: Given the operator $ \phi : L \to L $ in an inner-product space $L$ over $\mathbb{C} $. Prove: If there exists $ v \in L $ s.t. $ \left\lVert \phi(v) \right\rVert < \left\lVert \phi^*(v) \...
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How to prove T is a closable operator

Let $T:H\to H$ a densely defined operator, with $H$ a Hilbert space such that: $$Re(x,Tx)\geq 0, \forall x\in Dom(T) $$ I want to prove that $T$ is a closable operator, that means... that there exists ...
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If $A^\star$ is surjective, then there is $c>0$ such that $|u| \leq c |A u|$ for all $u \in D(A)$

I'm trying to prove Theorem 2.21 in Brezis' book of Functional Analysis. The author leaves the proof as an exercise. Could you have a check on my attempt? Let $E, F$ be Banach spaces. Let $A: D(A) \...
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Let $A^\star:\operatorname{dom}A^\star\subseteq F^\star\to E^\star$ be the adjoint of $A$. Then $\operatorname{ker}A=(\operatorname{im}A^\star)^\perp$

I'm trying to do exercise 2.18 in Brezis' book of Functional Analysis. Could you have a check on my attempt? Let $(E, |\cdot|_E), (F, |\cdot|_F)$ be Banach spaces and $A: \operatorname{dom} A \...
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Why do we restrict unbounded linear operator to Banach spaces?

Let $(E, | \cdot|_E)$ and $(F, | \cdot|_F)$ be Banach spaces. An unbounded linear operator $A$ from $E$ to $F$ is a linear map of the form $A: D(A) \to F$ where $D(A)$ is a subspace of $E$. Let $A$ be ...
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Proving orthonormal basis for linear operators

given \begin{equation*} E_{ij} \mathbf{e}_k = \langle \mathbf{e}_j,\mathbf{e}_k \rangle \mathbf{e}_i, \quad 1 \leq k \leq n. \end{equation*} prove that for each $1 \leq i,j \leq n$ let $E_{ij} \...
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The meaning of a symbol in a paper

I have some trouble figuring out the meaning of this symbol $\upharpoonright$ in this paper. I have never seen this. The paper is "Inverse Scattering on the Line". Sorry for the author, my ...
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Spectral representation intuitive explanation

I am following a course in functional analysis and in the lectures we encountered the following theorem: Theorem: Let H be a Hilbert space and $T:H\to H$ a self-adjoint and compact operator. Then: $T ...
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