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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 $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$.

Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$. (a) Show that from $A^4 = 0$ it follows $A = 0$. (b) Find the eigenvalues of the operator $A$. (c) ...
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Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such a normal operator that $A^2 = -A$. [duplicate]

Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such a normal operator that $A^2 = -A$. Show: (a) $\langle A^* A x, A^* y \rangle = - \langle A^* x, A^* y \rangle$, $x, y \in H$, (b) $A = ...
hd1's user avatar
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$E$ is finite-dimensional Euclidean space over $\mathbb{R}, x \in E, x \neq 0$. Then $\{Ax: A = A^* \succeq 0, \|A\| \leq 1\}$ is a closed ball

Let $E$ be a finite-dimensional Euclidean space over $\mathbb{R}$ and let $x \neq 0$ be a vector in $E$. Show that the set $K=\{Ax: A = A^* \succeq 0, \|A\| \leq 1\}$ is the closed ball of radius $\...
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Let $H$ be a Hilbert space, $A, B \in \mathcal{B}(H)$, and let $C = A^*A + B^*B$.

Let $H$ be a Hilbert space, $A, B \in \mathcal{B}(H)$, and let $C = A^*A + B^*B$. Show: (a) $\ker C = \ker A \cap \ker B$. (b) The eigenvalues of the operator $C$ are non-negative real numbers. ...
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Let $H$ be a Hilbert space and let $P \in \mathcal{B}(H)$ be a projector.

Let $H$ be a Hilbert space and let $P \in \mathcal{B}(H)$ be a projector. Show that $\text{Im} P$ is a closed subspace. Also, show the implications (a) $\implies$ (b) and (b) $\implies$ (c): (a) $P = ...
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Let $ H $ be a Hilbert space and let $ A \in \mathcal{B}(H) $. Show that $ H = \ker A \oplus \overline{\operatorname{Im} A^*}. $

Let $ H $ be a Hilbert space and let $ A \in \mathcal{B}(H) $. Show that $ H = \ker A \oplus \overline{\operatorname{Im} A^*}. $ Attempt: Since $ \operatorname{Im} A^* \subseteq \overline{\...
good12's user avatar
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derivatives of Frobenius norm with respect to complex matrix

I encountered a math problem when reading some optimization papers. In one of the paper, it minimize a function of Frobenius norm of complex matrices: $\min_{U} \frac{\beta}{2}\mathrm{\left\| H-UV_{}^{...
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Question about uniqueness of adjoint operator

Let $S, S'\in L(V)$ be such that $$\langle Tx,y\rangle=\langle x, Sy\rangle=\langle x,S'y\rangle$$ for all $x,y\in V$. Defining $C=S-S'\in L(V)$, we have $$\langle x, Cy\rangle=\langle x, S-S'y\rangle ...
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Is this differential operator self-adjoint?

If $V$ is the space of functions in $C^{\infty}(\mathbb{R})$ such that $f(x)=f(x+2)$ for all $x\in \mathbb{R}.$ Equipping $V$ with the inner product of the space $C[-1,1]$, is the operator $D:f\mapsto ...
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How to represent a point's velocity seen from a moving coordinate system?

let us consider a three-dimensional point $x$ that is fixed in the world coordinate system, ie. its velocity is zero, $\dot{x}=0$. Consider another coordinate system that is defined by a rotation ...
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Computing the adjoint of $-\Delta$

In B. Helffer's Spectral Theory and its Application, Remark 2.7 p. 16 the author is considering the two following operators $T_0=-\Delta $ with $D(T_0)=C^{\infty}_{c}(\mathbb{R}^N)$ $T_1=-\Delta $ ...
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Reconstruction of an operator given the eigenfunctions and eigenvalues

I am interested in operator theory, in particular if I know the sequence of eigenvalues $\{\lambda_n\}_{n=1}^\infty \subset \mathbb{R}$ and eigenfunctions $f_n \subset X$ of a self-adjoint ...
mathematurgist's user avatar
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How to Verify the Conjugate transpose

The operator $A$ is defined by the matrix $$A_B:=\begin{pmatrix} 1 & 1 & 3 \\ 0 & 5 & -1 \\ 2 & 7 & 3 \end{pmatrix} $$ in the basis $b_1=(1,2,1), b_2=(1,1,2), b_3=(1,1,0)$ of ...
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$(T^{*})^{+} = (T^{+})^{*}$

I'm trying to prove why $(T^{*})^{+} = (T^{+})^{*}$ only using properties of matrix operations (I'm considering $()^{*}$ and $()^{+}$ operations as well, just to be clear). However, I assume I cannot ...
pseudobulbose's user avatar
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How should we characterize the relationship between two matrix representations of a linear operator with respect to two different orthonormal bases?

Nielsen / Chuang remark on page 71 of "Quantum Computation and Quantum Information" that, if $| v_i \rangle$ and $|w_i \rangle$ are orthonormal bases, then the operator $U$ defined by $\sum_{...
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If $U $ is a unitary linear operator, how can I show that any matrix representation of $U$ must be a unitary matrix?

Nielsen / Chuang "Quantum Computation and Quantum Information" states on p. 70: "A matrix $ U$ is said to be unitary if $U^\dagger U = I$. Similarly, an operator $U$ is unitary if $U^\...
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Can be use $u$-substitution for calculating the adjoint of an operator in Schwartz space?

I only have seen that for calculating the adjoint of an operator in $\mathcal S$, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for ...
Daniel Muñoz's user avatar
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$S \in \mathcal{L}(L^1(\Omega))$, find $T^* \in \mathcal{L}(L^\infty(\Omega))$ with $T^*g = Sg \forall g \in L^1(\Omega) \cap L^\infty(\Omega)$

Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this ...
Meta-chan's user avatar
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What guarantees that the adjoint of a suitable integral operator, e.g. a Hilbert-Schmidt operator, is again an integral operator with a kernel?

This is likely a silly question, but I was wondering if $T$ is some nice integral transform, e.g. a Hilbert-Schmidt integral operator, with an, say, $L^2(\mathbb{R}^n)$ kernel, what then guarantees ...
Cartesian Bear's user avatar
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$T^{**} = T$ if the space is reflexive

Assuming $X$ is reflexive, that is the linear map $J_X:X \mapsto X^{**}$, $J_X(x)(f) = f(x)$ for all $f \in X^*$ is bijective. Is it true that $J_X^{-1}T^{**}J_X = T$? Here's my attempt: Let $x \in X$,...
hteica's user avatar
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How does the boundary term appear when taking the transposed form of the inner product with a linear operator

I'm trying to figure out how the second term appears here: $$\int_\Omega \mathcal{L}(u) w \;d\Omega = \int_\Omega u \mathcal{L}^*(w) \; d\Omega + \int_\Gamma \left[S^*(w) G(u) - G^*(w)S(u)\right] d\...
Cedric Martens's user avatar
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Untiunitary operator on a Hilbert space

A bijective linear (antilinear) operator $A$ on a Hilbert space $\mathcal{H}$ is called unitary (untiunitrary) if $\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle$ (resp. $\langle A\psi |A\...
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Ask a question on adjoint operator?

Let $H_1,H_2$ be Hilbert spaces with inner products $\langle \cdot, \cdot \rangle_1$ and $\langle \cdot, \cdot \rangle_2$. Corresponding to every $T \in \mathcal B(H_1,H_2)$, there is a unique element ...
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Adjoint operator and random variable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$. I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ ...
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Is there a name for this bilinear form which compares a linear operator with its adjoint?

Let $X$ be a reflexive Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $A:X\to X$. Denote the adjoint of $A$ by $A^\ast$. Define the bilinear form $B:X\times X\to\mathbb R$ by $$ ...
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If $\langle Lx,y\rangle = \langle x,Ry \rangle$ then $L$ is bounded

Suppose $L,R$ are not necessarily bounded operators on a hilbert space $H$. Show that, if $L,R$ satisfy $$ \langle Lx,y \rangle = \langle x,Ry\rangle $$ for all $x,y \in H$, then $L$ is bounded. I ...
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Adjoint of an operator on scaled Euclidean spaces

For $N\in \mathbb N$, equip $\mathbb C^N$ with the inner product $\langle\mathbf x,\mathbf y\rangle_N := N^{-1}\sum_i \overline x_i y_i$. Let $A$ be an $N\times M$ complex matrix. As an linear ...
AdamNie's user avatar
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Why does this prove that the span of the eigenvectors is dense in $\text{Im } T$?

Let $H$ be a Hilbert space with the inner product $(\cdot,\cdot)_H$ and let $T:H\to H$ be a bounded compact and self-adjoint operator on $H$. In this case there exists an $H$-orthonormalsystem $(\...
Max Stuthmann's user avatar
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How to calculate the adjoint operator of the projected intensity differential (based on the Fresnel propagator)?

Let $G(\phi,\epsilon) = 2Real \left ( (e^{-B+i\phi} * P_z \right ) \cdot ([-i\epsilon \cdot e^{-B-i\phi}]*\overline{P_z})$ with $\phi, \epsilon, B \in L^2(\Omega,\mathbb{R})$ and $P_z \in L^2(\Omega,\...
Loric's user avatar
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Adjoint, self-adjoint, Which is the proper noun?

I'm currently learning linear algebra, and I was confused by these two terminologies. It seems that adjoint is the tranpose of cofactor matrix, and a self-adjoint operator has a matrix representation ...
AL-CEL's user avatar
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Domain of sum of momentum operators

Given the tensor product of Hilbert spaces $\otimes_{i \in \mathcal{Z}} (\mathcal{H}_i, \psi_i)$ (here $\mathcal{Z}$ is the set of integer numbers, $\mathcal{H}_i = \mathcal{L}^2(\mathcal{R}, dx)$ and ...
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If $A\in B(H)$ commutes with all self-adjoint operators, then $A=\lambda I$ for some $\lambda\in\mathbb{R}$

Let $H$ be a complex Hilbert space and $A\in B(H)$. Could anyone explain why the following assertion is true: if $A$ commutes with all self-adjoint operators, then $A=\lambda I$ for some $\lambda\in\...
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Brezis' exercise 8.17: the kernel of $A^*$ where $A u=u''-xu'$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
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Brezis' exercise 8.17: the domain of $A^*$ where $A u=u''-xu'$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
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Brezis' exercise 8.17: the domain of $A^*$ where $A u=u''$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
Akira's user avatar
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Determining whether an operator is trace-class

Let $H$ be a separable Hilbert space, and let $A$ and $B$ be trace-class operators on $H$ such that $A^{-1/2}B$ is a Hilbert-Schmidt operator. Then is it possible to know whether the operator $B'A^{-1}...
metric's user avatar
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Confusion about adjoint of $\nabla$

Let $\Omega$ be an open set in $\mathbb{R}^n$. Consider $\nabla: H^1(\Omega) \to L^2(\Omega)^n$. It is a bounded linear operator. Consider its Hilbert adjoint $\nabla^*: L^2(\Omega)^n \to H^1(\Omega)$...
vampip's user avatar
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Surjectivity and injectivity of Hilbert space operators

Let $V, W$ be two Hilbert spaces, and $T : V \rightarrow W$ be a surjective bounded linear operator. It is easy to check that $T^{\ast} : W \rightarrow V$ is an injective bounded linear operator. Then ...
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Normal operators, verification

I have been reading some article and I came across this equality $$\big\| |T|+|T^*|\big\|^2 = \big\|(\,|T|+|T^{*}|\,)^2\big\|\,,$$ where $|T|=(T^{*}T)^{\frac12}$. The only way this can hold is if the ...
Vuk Stojiljkovic's user avatar
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Let $V$ be finite dimensional complex vector space, $d:V\to V$ be linear map such that $d^2=0$ and $\Delta=dd^{\star}+d^{\star}d$. Prove the following

Let $V$ be finite dimensional complex vector space, $d:V\to V$ be linear map such that $d^2=0$ and $\Delta=dd^{\star}+d^{\star}d$, where $d^{\star}$ is the adjoint of $d$. Prove that (a) $dd^{\star}x=...
OneLamp's user avatar
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Why do we need the operator to be densely defined for defining adjoint?

Suppose $T$ is an operator with domain and range in the Hilbert space $\mathcal{H}$. The usual way of defining the adjoint $T^*$ of $T$ uses density of $dom(T)$. But cannot we use this same definition ...
sigma's user avatar
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Finding the conjugate of an operator between the Banach spaces $\ell_{p}$

I am working with conjugate operators acting between Banach spaces. I am doing the following exercise. Let $(\beta_{n})_{n \in \mathbb{N}}$ be a bounded sequence of complex numbers. Define the ...
liamsi Meean's user avatar
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Properties of Cesàro Operator in $L^2$

Put $L^2=L^2(0,\infty)$ relative to Lebesgue measure, and the Cèsaro operator $C$ is defined as follows: $$(Cf)(s)=\frac{1}{s}\int_0^s f(t)dt$$ we can find its adjoint operator: \begin{align*} \langle ...
Luis De Oro's user avatar
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1 answer
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What is the adjoint of the curl operator?

Let $\Omega \subseteq \mathbb{R}^3$ be a bounded, connected domain, and set $$\mathcal{V} = \{ \vec\phi = (\phi_1, \phi_2, \phi_3) \in C_c^\infty(\Omega): \nabla\cdot \vec\phi=0\}.$$ Denote $V$ to be ...
Matt E.'s user avatar
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Action of the adjoint operator in tangent space

I'm reading a paper on the computation of covariant Lyapunov vectors (https://arxiv.org/pdf/1212.3961.pdf) and, as I have a Machine Learning background, I have some gaps concerning dynamical systems. ...
Pepper08's user avatar
2 votes
1 answer
156 views

Domain of the adjoint operator of a bounded operator (on a Hilbert space). (Experimental physicist)

Let $A$ be a bounded operator on a Hilbert space $H$ and $D_A$ its domain. We can define the following functional on $D_A$: $$f_\eta(\xi)=(\eta,A\xi).$$ Then we have that: $$||f_\eta(\xi)||=||(\eta,A\...
davise's user avatar
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Bounded orbit in Adjoint operator $T^*$ implies bounded orbit in $T$?

Let $\mathcal{T}=\{T(t)\}_{t\geq0}$ be a $C_0$-semigroup of linear operators of Banach space $X$.The Adjoint of $T(t)$ is denoted by $T^*(t)$. It is known that $T^*(t)$ is bounded linear operator and $...
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3 votes
1 answer
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Eigenvectors of any arbitrary matrix is same as its adjoint.

Recently I came across Normal matrices and their properties, one of which states that their eigenvectors are the same as their adjoint and are orthogonal. I've gone across some proofs and I understand ...
Apoorv Mishra's user avatar
1 vote
1 answer
110 views

Convolution and difference convolution adjoint

I'm reading this paper (https://arxiv.org/pdf/2302.07211.pdf) and the authors mention a simple identity that looks like it should be trivial to prove but I'm having trouble with a sign that I can't ...
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If $T$ is a symmetric densely defined operator its adjoint $T^*$ is a closed extension of $T$

In this question Why is a densely defined symmetric operator $T$ extended by its adjoint $T^*$? the accepted answer proves that for a densely defined symmetric operator $T$ on a Hilbert space $H$, its ...
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