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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Example 4.5-3 in Kryeszig's Functional Analysis: What is the relation between the matrix of a linear operator and that of its adjoint?
Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \longrightarrow Y$ be a linear operator. (Then by Theorem 2.7-8 in Kreyszig $T$ is bounded ...
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Selfadjoint Endomorphism
Question:
Let $p >1$ be an integer, let $G = \mathbb{Z}/(p)$, and let $V = \mathbb{C}^G$, which is an inner
product space over $\mathbb{C}$ with inner product defined by $\langle f, g\rangle =\...
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Find the inverse of the $n\times n$ matrix whose entries are given by $a_{ij} = \max (i,j)$
The actual question on the past papers is "Let $n\ge 1$ be an integer and consider the $n\times n$ matrix $A$ whose entries are given by $a_{ij} = \max(i,j)$ for all $1\le i,j\le n$. Show that $A$ is ...
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eigenfunctions of the adjoint of an operator
If the eigenfunctions of a linear operator are known, is there a way to calculate the eigenfunctions of the corresponding adjoint operator based on the known eigenfunctions? In other words, what's the ...
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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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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 $D$ be the differentiation operator on $V$. Find $D^*$.
Let $V$ be the vector space of the polynomials over $R$ of degree less than or
equal to $3$ with the inner product space $(f|g)=\int_{0} ^{1}f(t)g(t) dt$, and let $D$ be the differentiation operator ...
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Find the operator norm: $T\in B(L^{2}(\mathbb{R},e^{-x^{2}}dx)) $, $Tf(x) = f(x+1)$. $||T|| = $?
Let $L_{e}(\mathbb{R})$ denote the Hilbert space with inner product $\langle f, g \rangle = \int_{-\infty}^{\infty}f(x)\overline{g(x)}e^{-x^2}dx$.
Let $T: L_{e}(\mathbb{R}) \to L_{e}(\mathbb{R})$ be ...
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Given $T:E^*\to E^*,$ does there exist $S:E\to E$ such that $\langle Te^*,e \rangle = \langle e^*,Se\rangle?$
Let $E$ be a Banach space and $T:E\to E$ be a bounded linear operator.
Denote $E^*$ a dual space of $E,$ that is, the space of all bounded linear functional $e^*:E\to\mathbb{R}.$
It is well known ...
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Prove $||T^n||=||T||^n$ if $T\in B(H)$ is self-adjoint
Let $H$ be a complex Hilbert space and let $T\in B(H)$ be self-adjoint. I have already proven that $||T^{2^k}||=||T||^{2^k}$ for $k=0,1,2,\dots$.
Now I have to prove the more general statement that ...
user428890
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What is the intuition behind the kernel of $A$ being orthogonal to the range of $A^*$?
We know that, if $A$ is a linear bounded operator, then $\operatorname{Ker}(A) \perp \operatorname{Range}(A^*)$, where $A^*$ is the adjoint of $A$.
I have no troubles understanding the proof of this (...
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self-adjoint operator, such that $T^2=T^3,$ is orthogonal projection.
Let $T$ be a self-adjoint operator on a Hilbert space $H$, such that $T^2=T^3.$
Prove that $T$ is an orthogonal projection. Is that true if we had $T^4=T^3$?
Thank you for the help.
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Integration by parts to find the adjoint operator
On the interval $(0,1)$ consider the differential operator $Lu=u''''+u'$ with boundary conditions
$u(0)+u'(1)=u(1)+u'(0)=0$
$2u(0)+u''(1)=2u(1)+u''(0)=0$
$(1)$
I want to find the adjoint ...
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Is it a unitary, self adjoint and normal operator?
Let $A\colon H\to H$ be a bounded linear operator on a complex Hilbert space such that $\|Ax\|=\|A^*x\|\forall x$, given that there is a nonzero $x$ for which $A^*x=(2+3i)x$. Then I need to know ...
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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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Counter example for infinite dimensional vector space
Consider a linear operator O acting on a Hilbert space H. If the dimension of H is finite, I have shown that: dim(ker O) = dim(ker O†). Where O† is the Hermitian conjugate of O.
Does this hold when ...
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Definition of Adjoint Operator for Quantum Mechanics
While learning about adjoint operators for quantum mechanics, I encountered two definitions.
The first definition is given by Shankar in The Principle of Quantum Mechanics:
Given a ket
$$ A\lvert ...
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Self-adjoint operator with increasing sequence of eigenvalues
The spectral theorem says that if $A$ is a self-adjoint operator on a Hilbert space $H$ with compact inverse, then the eigenvectors of $A$ form a complete orthonormal basis of $H$. Furthermore each ...
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Proof of " The set of all self-adjoint operators are closed ".
The set of all self-adjoint operators are closed in Hilbert space.
Let $T \in B(H)$ where $H$ is the Hilbert space and $B(H)$ denotes the set of linear operators from $H \mapsto H$.
We are working ...
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Operator $T^*$ surjective iff $T$ topologically injective
Let $T : E \to F$ operator between Banach spaces, and $T^* : F^* \to E^*$ - adjoint operator. I want to proof next proposition: $T$ is topologically injective (or equivalently: injective with closed ...
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1-Lipschitzian Linear Operators on Hilbert Spaces and Fixed Points
$\textbf{Question}$
Let $(\mathcal{H}, \langle \cdot \, | \, \cdot \rangle)$ be a real Hilbert space with induced norm $\|\cdot\| = \sqrt{\langle \cdot \, | \, \cdot \rangle}$ and let
$$\mathscr{B}(\...
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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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Unitary operators, inner products and unit operators with Dirac notation
From my notes, I have that for a unitary matrix:
$$\underline{\underline{U}}^\dagger=\underline{\underline{U}}^{-1},\qquad \underline{\underline{U}}^\dagger=\big(\underline{\underline{U}}^{T}\big)^*$$
...
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Understanding the defintion of dual operators
I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators.
I'm having some difficulties understanding the following definition -
Why $A^*$ is $Y^*\rightarrow ...
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norm of positive semi-definite complex matrix
Suppose $0\neq X_n\in \mathbb{M}_{k(n)}(\mathbb{C})$,if $\lim_{n \to \infty}tr(X_n^*X_n)=0$,can we conclude that $\lim_{n \to \infty}\|X_n^*X_n\|=0$,where $tr$ is the standard trace on complex matrix,$...
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Is the kernel of the adjoint operator equal to the kernel of the operator ($\ker (A)=\ker (A')$)?
I am in a middle of a proof where I asked myself about the following:
Is the kernel of the adjoint operator equal to the kernel of the operator ($\ker (A)=\ker (A')$)?
Theorem:Let $X,Y$ be Banach ...
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Reference request for the following theorem in Von Neumann algebras.
I know the following result is true for a von Neumann algebra $A$:
Theorem: Let $A$ be a von Neumann algebra. Then every self-adjoint element $h\in A$, i.e $h=h^*$, is the limit of a sequence of ...
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Is $\text{ker} (\delta_{\nabla^E}d_{\nabla^E})$ always non-zero?
Let $E$ be a vector bundle over a smooth manifold $M$, equipped with a metric $\eta$ and a metric-compatible connection $\nabla$.
Denote by $\delta_{\nabla^E}:\Omega^1(M,E) \to \Omega^{0}(M,E)=\...
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Proving that an operator is self-adjoint.
Let $(\lambda_{n})_{n \in \mathbb{N^{*}}}$ be a sequence of real numbers converging to $0$ and let $(u_{n})_{n \in \mathbb{N^{*}}}$ be an orthonormal family in a Hilbert space $H$. Define $T:H \...
user426277
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Show that the powers of a nonzero Hermitian matrix is always nonzero
Let $0 \neq A \in M_{3×3}(\mathbb{C})$ be a Hermitian matrix. Show that $A^k \neq 0$ for all positive integers $k$.
I'm stuck on this problem. I am not sure why the question is only about $ M_{3×3}(\...
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Finding the adjoint of $\alpha: p(X)\to(X+1)p(X)$
Let $V = \mathbb{C}[X]$ and define an inner product on $V$ by setting
$$
\left\langle \sum_{i=0}^{\infty}a_iX^i,\sum_{i=0}^{\infty}b_iX^i\right\rangle=
\sum_{i=0}^{\infty}a_i\bar{b_i}.$$
Let $\alpha$ ...
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Self-adjoint bounded operators in a Hilbert space
Let $A$ be a bounded operator on complex Hilbert space $H$ that is not self-adjoint. Let $\epsilon>0$ and define $R_{\epsilon}=\{T\in B(H):||A-T||<\epsilon\}$.
How do I prove that there ...
user402675
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Adjoint matrix as pseudo-inverse
I'm very new to signal processing (seismic 1,2 and 3D-signal) and have read many papers recently. One thing I encounter quite often is the use of adjoint matrix.
If $d = Am$ where $d$ is the data, $...
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Adjoint Operator and Subspaces of Hilbert Spaces
Let $H_1$ , $H_2$ Hilbert Spaces with $T:H_1 \to H_2$ adjoint operator and $M_1 < H_1$ , $M_2 < H_2$ their subspaces. Show that:
$T(M_1) \subset M_2 $ if, and only if, $T^*({M_2}^{\perp}) \...
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What is the adjoint of an inverse matrix? [duplicate]
What is the adjoint of an inverse matrix? Is $(T^{-1})^{*} = (T^{*})^{-1}$?
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Eigenvalues of compact operators and his adjoint.
Let $T: H \to H$ be a compact operator with $H$ a Hilbert space. Let then $\lambda \neq 0$ be an eigenvalue of $T$ with eigenfunction $v$.
Is then $\lambda$ an eigenvalue for the adjoint $T^*$ ...
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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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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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Are the following two inner products on differential forms equal?
There are two inner product on differential forms:
$\langle \alpha,\beta\rangle$ induced from Riemannian metric $g$ by defining on 1-forms as dual of vector fields then extending to all differential ...
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Adjoint problem for convection-diffusion-reaction equation
I have given the convection-diffusion-reaction equation
$$-\Delta u+\beta\cdot\nabla u+\gamma u=f\ \text{in}\ \Omega\\
\quad u=0\quad\quad\quad\quad\quad\quad\quad\quad \text{on}\ \partial\Omega
$$
...
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Involution is not strongly continuous
Let $H$ be a Hilbert space and consider on $B(H)$ the strong topology, i.e. the topology induced by the seminorms
$$x \mapsto \Vert x \xi \Vert, \quad \xi \in H$$
I.e. a net $(x_\alpha)_\alpha$ ...
user745578
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On the existence of the adjoint of a densely defined unbounded operator
I am reading proofs on how the adjoint of an unbounded operator $T$ on a Hilbert space $\mathcal{H}$ is well defined provided its domains is a dense linear subspace. The main idea is that if two ...
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Relation between the transpose and adjoint of a linear transformation?
This question is not about matrices.
Kunze, Linear algebra (1971) defines transpose of a linear transformation and adjoint of a linear transformation.
Kunze states the relation between the adjoint ...
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$T=AU \iff T $ is a normal operator on Hilbert space
This is Exercise 16.(c) from Conway's Functional Analysis book.
Suppose $H$ is a Hilbert space and $T$ is a compact operator on $H$. Assuming the result that $\exists A$ positive operator and $U$ a ...
user422112
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The ranges of $T+iI$ and of $T-iI$.
In Rudin's book: Functional analysis. P386.exercises 9.
Let $H^2$ be the space of all holomorphic function $f(z)=\sum_{n=0}^{\infty} c_{n}z^n$ in the open unit disc that satisfy
$$\Vert f\Vert ^2=...
user469065
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Residual spectrum of the adjoint on Banach space
This is a homework question and I find myself ending up with the "opposite" of what I am supposed to prove. Below is the question and my attempt:
Suppose that $\lambda \in \mathbb{R}$. Show that if $\...
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Prove that a regular $\phi : E \to E$ can be uniquely decomposed as a composition of self-adjoint map and a rotation(Unitary trick of Weyl)
In the book(archive.org) of Linear Algebra by Greub, at page 226 it is asked that:
Note: $E$ is a $n$-dimensional real vector space.
Prove that a regular linear transformation $\phi$ of a ...
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Matrix of self-adjoint operator such that every element of the diagonal is $0$.
Let $V$ be a finite dimensional $\mathbb R$-vector space and let $T:V\rightarrow V$ be an self-adjoint operator such that $\text{trace}(T)=0$. Show that there exists an orthonormal basis $B$ such that ...
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Adjugate Matrix and Characteristic Polynomials in Wikipedia
The following is written in the Adjugate Matrix Wikipedia page:
If
$$ p(t)~{\stackrel {\text{def}}{=}}~\det(t\mathbf {I} -\mathbf {A} )=\sum _{i=0}^{n}p_{i}t^{i}\in R[t],$$
is the characteristic ...
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Let $T:\ell_1 \to c_0$ be linear operator defined as $x_n \to \sum_{k\geqslant n} x_k$. Then $T \in B(\ell_1, c_0)$.
I'm solving some exercises for my Functional Analysis Exam. Here is one on which I am stuck:
Let $T: l_1 \to c_0$ be linear operator defined as $x_n \to \sum_{k\geqslant n} x_k$. Show that $T \in B(...
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