629 questions
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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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### What is the Adjoint of inverse of a matrix?

I recently read that when we take the Adjoint of the inverse of a matrix we get the matrix itself i.e. $\operatorname{adj}(A^{-1})=A$ I am unable to prove the result. Help!
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### 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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### 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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### 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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### How do I prove that the angular momentum is a Hermitian operator?

Confirm that the operator $$\hat I_z= \left(\frac hi\right)\frac{d}{dφ},$$ where $\varphi$ is an angle, is Hermitian.
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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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### 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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### Find a general 2x2 matrix where A = adj(A)

I know how to find the adjoint of $2\times 2$ matrix but I'm at a loss for finding a general $2\times 2$ matrix where $A = adj(A)$. Thanks for your help!
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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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### 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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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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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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### 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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### 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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### 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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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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### $T=AU \iff T$ 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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### $PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

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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### $a=cc^*c$ 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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### 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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### 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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### 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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### Express in terms of $E$ a self-adjoint operator $T$ such that $T^2 = I+E$

I was trying the following problem: Let $V$ be a finite dimensional inner product space. Let $E: V \to V$ be an orthogonal projection onto some subspace of $V$. Express in terms of $E$ a self-adjoint ...
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### 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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### Prove that $T^*$ is injective iff $ImT$ Is dense

Let X,Y be two normed spaces, and $T:X\rightarrow Y$ a bounded linear operator. prove that the adjoint operator $T^*$ ($T^*f(x)=f(Tx)$ is injective iff $ImT$ is dense any help would be great guys. I ...
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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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### 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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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 $S \to S^... 1answer 38 views ### Inner product of a vector field and gradient - Adjoint of the gradient On page 9 in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.639.5952&rep=rep1&type=pdf it is being shown why the negative divergence is the adjoint of the gradient.$V: \mathbb R^n \...
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 ...
### 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 ...