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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45 views

Infinite-dimensional analogue of orthogonal matrices

In functional analysis one learns that self-adjoint operators are the infinite dimensional generalisation of symmetric matrices and the dual operator is the generalisation of the transposed matrix ...
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Question about Friedrichs Extension when reading Applied Functional Analysis

In the book I read recently(Eberhand Zeidler, Applied Functional Analysis: Applications to Mathematical Physics (Applied Mathematical Sciences 108)), when consider the extension of symmetric operator ...
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48 views

Relation between Ad and ad

Let $h(t)$ be a smooth curve in a matrix Lie group G, with $h(0)=e$. Let $v \in T_e G$. Then it holds that $$\left.\frac{d}{dt}\left(Ad_{h(t)}(v)\right)\right|_{t=0} = ad_{\left.\frac{d}{dt} h(t) \...
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Positivity conditions for $N^\dagger N$ where $N$ is a linear map

Let $N: \mathcal{A} \rightarrow \mathcal{B}$ be a positive linear map between two Hilbert spaces, $\mathcal{A}$ and $\mathcal{B}$. Define the adjoint map $N^\dagger$ as the linear map which satisfies ...
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20 views

Verifying the equivalence between the adjoint transformation and adjoint matrix

I'm trying to understand the relation between the adjoint as an operation $T^* $ and the adjoint as a matrix manipulation $T^\dagger$ (transposing and conjugating). So far, I understand that these ...
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T and T*T are simultaneously diagonalisable

I suspect the following might be true but I can't prove it. Suppose $T \in \text{End}(V)$ for some finite-dimensional complex inner product space $V$, such that $T^*T = TT^*$ (i.e. $T$ is normal). ...
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30 views

Show that the norm of this operator is equal to 1

Let $H$ be a Hilbert space and $P$ a projection $H \rightarrow H$ ( a bounded linear operator on $H$ such that $P^2=P$ and $P$ is not equal to $0$) I showed that $ ||P|| \ge 1$ and that $P$ is auto ...
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21 views

Why is a map defined as $A \rightarrow V^\dagger A V$ for all V completely positive?

In this paper by M. D. Choi, he claims, For each $n * m$ matrix $V$, it is evident that the map: $M_n \rightarrow M_m$ with $A \rightarrow V^\dagger A V$ is completely positive. $M_x$ denotes all ...
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39 views

Operator normality for different inner products

I have a wave equation in PDE form, defined as $$ \frac{\partial P}{\partial t} + div(u) = 0, \\ \frac{\partial u_i}{\partial t} + \nabla_i P - \mu \Delta u_i = 0. $$ Here $(P, u_i)$ are the ...
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Existence of a pre adjoint in $C_0(\mathbb{R}^d)$?

Suppose $A^* : D(A^*)\subset C_0(\mathbb{R}^d)\rightarrow C_0(\mathbb{R}^d)$ is the generator of a strongly continuos semigroup. Does there exists an operator $A:D(A)\subset X \rightarrow X$ for some ...
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Matrix of the operator $A(x,y)=(y-x+iy,x-2y-ix)$

I got an operator $A: \Bbb C^2 \to \Bbb C^2$ given as $A(x,y)=(y-x+iy,x-2y-ix)$ and I want to represent it as a matrix, so I could find then the orthonormal basis to which would have this operator a ...
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39 views

Eigenvalue expansion of Green's function of non self-adjoint operator

I have a general non-self adjoint operator $L$, and a set of equations with some inhomogeneous boundary conditions, $$ Lq= 0 \quad in \quad \Omega, $$ $$ q = q_0 \quad on \quad \partial\Omega. $$ I ...
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Self Adjoint and Skew Adjoint Linear Transformations

I'm studying adjoints, and I'm confused as to how I prove this. I have a definition of a self-adjoint $T$, such that $T^*=T$, where $T^*$ is the adjoint. I then have that the definition of a skew-...
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Self-adjoint operator is diagonalisable

I am revising adjoints for a linear algebra exam and am confused as to how to prove this. Suppose that $T: V \rightarrow V$ has the property that $T^*=aT$ for some complex a. How then do you prove ...
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1answer
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Example of an unbounded operator whose adjoint is not densely defined

In his book "Quantum Theory for Mathematicians", B. C. Hall mentions that there are some pathological examples of unbounded operators on separable Hilbert spaces whose adjoint is not densely defined (...
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68 views

Find the adjoint of the right shift operator in $\ell^1$.

Find the adjoint of the right shift operator $T$ in $\ell^1$. More specifically, $T: X \longrightarrow Y$ defined by $$Tx= T(x_1, x_2, \dots, x_n, \dots) = (0, x_1, x_2, \dots, x_n\dots) = y$$ ...
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1answer
24 views

Find adjoint operator defined on element

Let $E = L^2 (0, 1)$. Given $u ∈ E$, set $Tu(x)=\int_0^x u(t)dt$. Find $T^*$. Solution says only $(u, T^* v) = \int_t^1 v(x)dx$. I dont understand. adjoint operator is defined by $(Tu,v)=(u,T^*v)$. ...
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associativity property of convolution with non-constant function

Given two functions $f=f(x)$ and $g=g(x)$ and a constant $a$, we all know from the associativity property that $$a(f\ast g)(x)=((af) \ast g)(x)$$ Let's assume that $a=a(x)$, then I would like to ...
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properties about adjoint operators

Let $T:V \rightarrow V$ a linear map such that $Tv = <v,u>w$ then the adjoint linear map $T^*$ is $T^*v = <v,w>u. \forall u,v,w \in V $. My professor defined the linear map $T^*$ as ...
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About the adjoint of some differential operator on $L^2(0,1)$

If $T=-d^2/dx^2$ is defined on the domain $D(T)=\{f\in C^2[0,1]: f(1)=f'(0)=f'(1/2)=0\}\subset L^2(0,1)$. What's the Hilbert adjoint operator $T^*$? Many many thanks for your answers. Math.
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If $T$ is a topological isomorphism, then so is $T^*$

The question comes from the following link on page 25: https://www.ucl.ac.uk/~ucahad0/3103_handout_3.pdf They prove $(T^{-1})^*=(T^*)^{-1}$, but I don't see how it proves $T^*$ is a topological ...
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To show an operator is symmetric

Suppose I want to show that the two operators $\mathcal{L}$ and $\mathcal{A}$ where they are respectively: \begin{align} &\mathcal{L} = \begin{pmatrix} -J & 0 \\ 0 & 0 \end{pmatrix} \...
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Are Hermitian operators positive?

I can only find the proof for the reverse statement (i.e. here). However Nielsen Chuang Quantum Computation and Quantum Information p. 90 states the following: Suppose we define $$E_m = M^\...
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Does $\operatorname{Tr}(e^x(\lambda,\infty) x) =\operatorname{Tr}(px)$ imply that $e^x(\lambda,\infty) \le p \le e^x[\lambda,\infty)$?

Let $H$ be a Hilbert space and $\operatorname{Tr}$ be the standard trace on $B(H)$. Let $x$ be a self-adjoint operator in $B(H)$. Let $e = e^x(\lambda,\infty)$ be the spectral projection. Assume that $...
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48 views

how to find adjoint of this operator on the space of C[0,1]?

We are given $T$ is an operator on $C[0,1]$ as follows $T(g(x))=\sum\limits_{k=1}^{m}p_kg(f_k(x)), p_k\in [0,1], f_k\in C[0,1]$, could anyone tell me how to show adjoint of this operator is as ...
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1answer
18 views

domain of the adjoint of the momentum operator.

I was reading the book by Roman where he discusses the linear momentum operator on the half line: $pf= -if’$, this operator is densely defined and in the book he finds the adjoint by using the inner ...
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1answer
32 views

Is an operator between Hilbert spaces bounded iff the adjoint is bounded?

Let $X,Y$ be Hilbert spaces. Let $D(A)\subset X$ be a dense subspace and let $A:D(A)\to Y$ be a linear operator. Define the following (not necessarily dense) subspace of $Y$: $$D(A^*):=\{y\in Y|\...
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$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ are orthogonal projections, then show that : $$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$$ Seeking a formal ...
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1answer
33 views

$a=cc^*c$ for some $c$. $a \in A$ a $C^*$ algebra.

Let $A$ be a $C^*$ algebra. Let $a \in A$, then there exists $c \in A$ such that $a=cc^*c$. This fact is used from example (1) of Prop 4.25. How does one show this?
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Continuous adjoint of the one-dimensional Laplace equation

Say I have a problem given by the 1D Laplace equation, $$ R (T(\alpha), \alpha) = \frac{d^2 T(x)}{dx^2} - \alpha(x) T (x) = 0, $$ with $x \in [0,1]$, Dirichlet boundary conditions on $x=0$ and $x=1$, ...
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Show that graph of operator with adjoint operator is closed

Let $X,Y$ be inner-product spaces. Let $T\in L\left(X,Y\right)$ be a linear operator with adjoint operator $S\in L\left(Y,X\right)$ such that $$\langle Tx,y\rangle_Y=\langle x,Sy\rangle_X\quad\forall (...
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Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, ...
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Taking fourier transform of operators with different exponents

I have a coupled differential equation that I wish to take the Fourier transform (FT) of. However, they consists of different operators which also includes an exponent (more will be shown below). This ...
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156 views

$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 ...
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Uniqueness of the Adjoint operator

So I was just stuck in the middle of proving the uniqueness of the adjoint operator. Known theorem(I already know how to prove it): Assume V is a finite dimensional inner product space over a field ...
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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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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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Find $T^* $, where $T$ is the linear operator defined by $T \epsilon_1 = (1, - 2), \,\,T\epsilon_2 =(i, - 1)$.

Let $V$ be the space $\mathbb C^2$, with the standard inner product. Let $T$ be the linear operator defined by $T \epsilon_1 = (1, - 2), \,\,\ T\epsilon_2 = (i, - 1)$. If $ \alpha = (x_1, x_2)$, find ...
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1answer
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Prove that $\ker P(A^\ast) $ is an invariant subspace of $A$.

Let $A$ be a normal linear operator on a finite dimensional unitary space and $P(x)$ a polynomial. Prove that $\ker P(A^{*})$ is an invariant subspace of $A$ (where $A^{*}$ is its adjoint operator). I ...
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Is this Sturm-Liouville problem self-adjoint?

We are interested in determining whether the problem $\begin{cases}xu''-u'+u = \cos(x)\\u(0) = 0 \\ u(1) = u'(1)\end{cases}$ is self-adjoint. This is not a Sturm-Liouville problem, the corresponding ...
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1answer
176 views

Linear operator is compact if and only if its adjoint is compact

Let $H$ be a Hilbert space, and $A:H\rightarrow H$ a linear operator. Prove that $A$ is compact if and only if $A^*$ is compact. I saw the following proof in my book - What I don't understand ...
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computing the adjoint of operator $T$ on the space $P_2(\mathbb{R})$

Suppose that the inner product on $P_2(\mathbb{R})$ is defined by $$\langle f,g \rangle:= f(-1)g(-1)+f(0)g(0)+f(1)g(1).$$ Consider the operator $T \in B(P_2(\mathbb{R}))$ which is defined as $Tf=f'$, ...
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1answer
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Does the concept of “adjoint map” determine the metric up to scaling?

Let $V$ be a real finite-dimensional vector space, and $g$ and inner product on $V$. $g$ induces a concept of "adjoint map" , i.e. a linear map $\text{Hom}(V,V) \to \text{Hom}(V,V)$ given by $S \to S^...
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1answer
41 views

Inner product of a vector field and gradient - Adjoint of the gradient

On page 9 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.5952&rep=rep1&type=pdf it is being shown why the negative divergence is the adjoint of the gradient. $V: \mathbb R^n \...
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1answer
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How can the adjoint be defined if $f$ is not one to one

Let $f \in \mathcal{L}(V, W)$. Moreover let's suppose $(e_1, ..., e_n)$ is a basis of $V$ and $f(e_i) = v_i$ where the $v_i$ aren't distinct (so there is at least $i \ne j$ such that $v_i = v_j$) so ...
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1answer
38 views

How to use operators on tensor products

This problem is from physics, but I have trouble understanding the math. I have some problem understanding how to use tensors. Let's say in Quantum Optics if I have the state in mode $b$ (where I can ...
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1answer
39 views

Computing the Adjoint using the Definition $\langle Tv,w\rangle = \langle v, T^*w\rangle$

Let $T$ be the linear operator on $\mathbb{C}^2$ defined by $$T(a,b)=(2ia+3b,a-b).$$ I am trying to compute the adjoint. The answer is $$T^*(c,d)=(-2ic+d,3c-d)\tag{$1$},$$ which can be seen by (i) ...
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1answer
23 views

Defining Hermitian Adjoints Non-degenerate Hermitian Forms that are NOT positive definite.

I was looking around different textbooks and websites for the definition of a Hermitian adjoint. All the resources that I have checked including the one I am studying at the moment (Jeevanjee's Intro. ...
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1answer
36 views

$\underset{_{a\rightarrow 0}}{\lim }L_{a}^{\ast }\left( L_{a}L_{a}^{\ast }\right) ^{-1}L_{a}=?$

Let $\varepsilon >0$ and let $L$ be a bounded operator acting on a Hibert space such that $L_{\lambda }=L-\lambda I$ is surjective of every $\lambda $ such that $\varepsilon >\left\vert \lambda \...
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15 views

Existence of solution of some functional equation involving integral

I want to prove that there exists always solutions to this equation $$g(x) = \int\limits_a^b {f(s,x)ds} $$ where $g \in {L^2}(0,1)$ is given and $f \in {L^2}((a,b) \times (0,1))$ is the uknown, $0<...