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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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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29 views

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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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\...
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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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48 views

Inclusion operator on half-integer weight modular forms and its adjoint

We have an inclusion $\iota: S_{k+1/2}(8N) \hookrightarrow S_{k+1/2}(16N)$, whose adjoint with respect to the Petersson scalar product is apparently given by $$\iota^{*} = Tr: \begin{pmatrix} 1 & ...
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46 views

Is this operator surjective?

Let $H$ Hilbert space, $A \colon D(A) \subset H \to H$ be a linear, closed, densely defined operator. In [Fabbri, Giorgio, Fausto Gozzi, and Andrzej Swiech. "Stochastic optimal control in ...
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33 views

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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123 views

Fixed point of an adjoint operator

Let $H$ Hilbert space and $T: H → H$ linear operator continuous. We define $T^*: H → H$ such as $\langle T^*x\,,y\rangle = \langle x\,,Ty\rangle $ We suppose $||T|| \lt 1 $, I want to proof that $T$ ...
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Operators on tensor product of Hilbert spaces

I was reading about teleportation which had the following Bell counterpart in $N$ dimensions $|{\phi}>=\sum_{i=0}^{N-1}|i>|i>$. The next line was $$(U\otimes I)|\phi>=\sum_{i,j=0}^{N-1}|j&...
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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 ...
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Eigenvalue of self-adjoint

I am solving an exercise: Let $T: V \rightarrow W$ be a linear transformation. $V$ and $W$ are finite-dimensional inner product spaces. Prove $T^*T$ and $TT^*$ are semidefinite. This is a solution ...
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249 views

Geometric intuition for adjoint

Let $V$ be a finite-dimensional inner product space, and let $T$ be a linear operator on $V$. Then $T^*$ ($T$ adjoint) is defined as the unique function such that $\langle T(x), y \rangle = \langle x, ...
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Proving operator is self-adjoint w.r.t. given inner product

Let $s$ be a nonnegative half-integer and $\mathscr P_s$ be the space of complex polynomials $p(z)$ of degree at most $2s$ in the formal variable $z \in \Bbb C$, equipped with the sesquilinear product ...
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When does a graded map of finite dimensional Hilbert space have a graded adjoint?

Suppose I have two finite dimensional graded inner product spaces $$V = \oplus_\alpha V_\alpha \hspace{3em} W = \oplus_\beta W_\beta$$ and a map $f : V \to W$. I can represent $f$ as a direct sum of ...
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If $T$ is a normal operator on vector space $V$, then is $T^k$ also normal?

By definition, it seems to be true that $$ T^*T=TT^*\implies (T^k)^*T^k=(T^*)^kT^k=T^k(T^*)^k $$ since $(TS)^*=S^*T^*$. I wonder if the above proof (hence the proposition) is true?
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Problem on adjoints of Linear Operators.

Could you help me check my solution to the question- Suppose $T\in L(V)$ and $U$ is a subspace of $V$, show that $U$ is invariant under $T$ if $U^\perp$ is invariant under $T^*$. Solution: Let $U$ ...
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57 views

$(Af)(t)=\int_{0}^{1}\min\{s,t\}f(s)ds$ is compact in $L_2[0,1]$

Show that the operator $(Af)(t)=\int_{0}^{1}\min\{s,t\}f(s)ds$ is compact in $L_2[0,1]$. (not $L_2[0,1]^2$). Our definition of compact operator is: operator $K$ is compact if for bounded sequence $(...
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I have a hard time interpreting the adjoint of operators defined over Banach spaces

Let $T$ be an operator defined by $$T:X \to Y$$ where X and Y are Banach spaces. The adjoint of $T$, $T^*$ is defined by $${T^*}:{Y^*} \to {X^*}$$ where ${X^*}$, ${Y^*}$ are the dual spaces. In other ...
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Trying to understand an example about the domains of unbounded operators

This is from the handbook of spectral theory... Take $H = L^2(\mathbb{R})$. Consider $Dom(T) = H^1(\mathbb{R})$ and $T = i \partial_x $. What is $(Dom(T^*), T^*)$? And if we choose $Dom(T) = C_0^{\...
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Why $||(T-\lambda I)^*v||=||(T^*-\bar{\lambda}I)v||$ where $T$ is normal operator

I was reading Sheldon Axler. In the proof of Suppose $T\in\mathcal{L}(V)$ is normal and $v\in V$ is an eigenvector of $T$ with eigenvalue $\lambda$. Then $v$ is also and eigenvector of $T^*$ with ...
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Is this a correct method for determining the domain of the adjoint?

Someone told me about this condition ("definition of the domain of the adjoint") for an operator $T$ acting on $C^\infty(L^2(\mathbb{R}))$ functions: $$f\in D(T^*) \iff (\exists M_f>0 \;s....
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115 views

Eigenvalues of a linear operator from $\mathbb R\to\mathbb C$

My problem is as follows (full problem): Let $V$ be the vector space of continuous functions $f:\mathbb R\to\mathbb C$ with period $1$, i.e. $$f(t)=f(t+1),\ \forall t\in\mathbb R$$ Let $V$ have the ...
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Adjoint operator for a system of nonlinear PDEs

I have this system of nonlinear reaction diffusion equations $\begin{cases} & u_t = u_{xx} + \partial_u W(u,v),\\ & v_t = d v_{xx} - \partial_v W(u,v). \end{cases}$ I found the linearization ...
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Is the map sends $T$ to $T^*$ adjoint of $T$ surjective?

Let $B(X)$ denotes the set of all bounded linear operators from $X$ to $X$, where $X$ is a Banach space. Same is defined for the set $B(X^*)$, where $X^*$ denotes the set of all bounded linear ...
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Number of linearly independent solutions of adjoint equation

Let us consider the variational equation $$\dot{u}=A(t)u,$$ with $u\in\mathbb{R}^n$. Then, the adjoint variation equation is $$ \dot{v}=-A^T(t)v, $$ with $v\in\mathbb{R}^n$. Now, if $X(t)$ is the ...
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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 ...
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31 views

Closure of a set is compact in the bidual

Let $F$ be a Banach space, $J:F\to F^{**}$ the canonical embedding. Let $B\subset F$ be such that $\overline{J(B)}$ is compact. Can I conclude that $\overline{B}$ is compact? I need this to show that $...
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Understanding solutions given by transposition

I'm studying solution given by transposition and i would like to clear somethings. Assume we have the following, let $T < \infty$ and $Q=\Omega \times] 0, T[$, let $\varphi$ given in a Hilbert ...
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34 views

Weak solution on Banach spaces.

Let $X$ and $Y$ Banach spaces. Let $A:D(A)\subset X\to Y$ with $D(A)$ dense in $X$ and $y\in Y$. Let $x\in X$ a weak solution of $Ax=y$, i.e. $(x,A^*y')=(y,y')$ forall $y'\in Y^*$ where $A^*:D(A^*)\...
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34 views

Inverse of adjoint action?

Given the adjoint action $\text{ad}_AX=AX-XA$, is there an inverse adjoint action $\text{ad}^{-1}_A$ such that $$\text{ad}^{-1}_A(\text{ad}_AX)=X?$$
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Showing the adjoint of $A = A^{-3}$ iff $A^* = A$ and $A^2 = I$

I'm trying to show that $A^* = A^{-3}$ iff $A^* = A$ and $A^2 = I$ given $A$ is invertible and $A\in M_n(\mathbb{C})$ but I'm having trouble with manipulating $A$ and just don't know what's legal or ...
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Adjoint of reduce sum operation

Consider the linear map $f:\mathbb{R}^{n\times m}\to \mathbb{R}^{n\times 1}$ defined as follows $$ f(X) = X 1_{m\times 1} $$ This is essentially a reduce operation that collapses the rows of $X$ into ...
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Find Adjoint of linear map

Given two fixed matrices $A\in\mathbb{R}^{n\times m}$ and $B\in\mathbb{R}^{n\times p}$ I have a linear map $F:\mathbb{R}^{pm\times 1}\to \mathbb{R}^{n\times 1}$ that can be described as follows: $$ F(...
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Adjoint of Hadamard Product

0I have a linear map $f:\mathbb{R}^{n\times m}\to \mathbb{R}^{n\times m}$ defined as $$ f(X) = A \odot X \qquad \qquad \text{ for some } A\in\mathbb{R}^{n\times m} $$ where $\odot$ is the Hadamard ...
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The resolvent of self-adjoint operators and Herglotz function

I am reading about the spectral theorem, and I came across the following claim: If $A$ is a self-adjoint operator and $R_{A}$ is the resolvent then $\left<\psi|R_{A}(.)\psi \right >$: $\mathbb{C}...
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Why does the adjoint exist for densely defined operators?

It is my understanding that when an operator is densely defined, then the adjoint operator exists. But, if $X,\, Y$ Banach spaces and $A:D(A)\subset X\to Y$ linear operator. Why is it necessary for $...
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If $T$ is an invertible linear operator, then so is $T^*$

Suppose that $V$ is an arbitrary vector space and that $T \in L(V)$ is invertible and admits an adjoint. Is it true that, in this case, $T^*$ is invertible? It is well-known that this is true in ...
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What is the definition of formal adjoint?

What is the definition of formal adjoint? In An introduction to the pseudo differential operators by Wong. Let $\sigma\in S^m$ and $T_{\sigma}:\mathcal{S}\subset L^p\to L^p$ the pseudo differential ...
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Operator norm of the matrix of eigenvectors of a non-normal matrix

Let $A$ be a non-normal, square, diagonalizable matrix. In view of the spectral theorem, since $A$ is non-normal but diagonalizable, we can write $$ A = R \Lambda R^{-1} $$ where $R$ is not unitary. ...
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Example for trivial intersection of domains

For a bounded operator $A$ in a Hilbert space $\mathcal H$ the real part of $A$ is defined by $\operatorname{Re}(A) = \frac 12(A+A^*)$. However, if $A$ is unbounded, this operator is defined on the ...
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What is the motivation and intuition behind the Adjoint and Dual Operator?

Those concepts were introduced, i.e. Adjoint Operator It's an operator $T^{*} \: : \: H_2 \rightarrow H_1$ such that $\forall _{x\in H_1,\:y\in H_2} \: \langle Tx, y \rangle_2 = \langle x, T^{*} y \...
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Why is the Compactness of an Operator so important? What is the use of compact operators in Mathematics?

Compact Operators have been the major topic in our Operator Theory course for the past few weeks. All the theorems which tell us whether a operator is compact or not are clear to me, but I still don't ...
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properties of adjoint operators and anti-hermitian operators

If $A$ is an operator on $H$. Show that : (a) $A$ is anti-Hermitian if and only if $iA$ is self-adjoint. (b) $A-A^{*}$ is anti-Hermitian. I'm search what means a anti-Hermitian operator and I found ...
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Hermitian vector spaces, dual and adjoint maps

I am having a slight problem regarding the specific relationship between adjoint and dual maps in the context of complex vector spaces endowed with Hermitian metrics. I know that a Hermitian metric ($\...
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Computing the adjoint of an operator between $W^{1,p}$ and $L^p$

I was reading some lectures notes and at some point we needed to look at the operator $L_S:= \partial_s+J_0\partial_t+S: W_0^{1,p}(\mathbb{R}\times ]0,1[,\mathbb{R}^{2n})\rightarrow L^p(\mathbb{R}\...
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How to prove that $(\tau^v)^* = \tau^{-v}$ for distributions?

I remember reading once that the translation operator $\tau^v f := f(\cdot + v)$ satisfies $(\tau^v)^* = \tau^{-v}$ in the sense of distributions. I believe $*$ operator refers to the adjoint, and in ...
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Finding the adjoint of $L$ where $Lf = \int_0^T \exp(A(T-t))\pmb b f(t)\;dt$

$L:L_2[0,T] \to \mathbb R^n$ $$ Lf = \int_0^T \exp(A(T-t))\pmb b f(t)\;dt $$ I'm supposed to show that $L^*=\pmb b^T \exp(A^T(T-t))$. The inner product is over $\mathbb R^n$ since $Lf \in \mathbb R^n$....
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Question 2.22 from Brezis' book of Functional Analysis

Question: $2.22$ The purpose of this exercise is to construct an unbounded operator $A: D(A) \subset$ $E \rightarrow E$ that is densely defined, closed, and such that $\overline{D\left(A^*\right)} \...
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Adjoint Operators Between Posets are Unique

Exercise Let $(X, \leq)$ and $(Y, \leq)$ be partially ordered sets and let $f:X \to Y$. Prove that the right adjoint of $f$ (if it exists) is unique. Proof. Suppose $f$ has a right adjoint $g: Y \to ...

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