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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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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
23 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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1answer
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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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How do you call operators $T$ that satisfy $\langle Tx, y \rangle = - \langle x, T y \rangle$

First, some definitions I know an linear bounded operator $T: \mathscr{H}_1 \to \mathscr{H}_2$ between to Hilbert spaces always has an so called adjoint operator $T^*: \mathscr{H}_2 \to \mathscr{H}_1$ ...
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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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1answer
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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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1answer
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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
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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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1answer
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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
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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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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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$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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2answers
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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
28 views

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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1answer
66 views

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
120 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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2answers
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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
35 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
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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
32 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
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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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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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0answers
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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<...
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Adjoint of a polynomial in a closed linear operator.

Let $ H $ be a Hilbert space and let $ T $ be a closed densely defined linear operator in $ H $ with domain $ D(T) $ and with nonempty resolvent set. We define the following polynomial in T: $ P(T) :...
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1answer
59 views

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^*\...
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0answers
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Adjoint operator on $\mathbb{L} ^2 _\mathbb{R}$ of $\frac{d}{dx}$

If A is a linear operator defined on $\mathbb{L} ^2 _\mathbb{R}$ of $\frac{d}{dx}$ then its adjoint must satisfy the property: $$(A^*f,g) = (f,Ag) \\ f,g \in \mathbb{L} ^2 _\mathbb{R}$$$ Now if $A = \...
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1answer
34 views

Prove that there is no self-adjoint extension using deficiency indices

Consider an operator $P =-i\frac{d}{dx} : dom(P) \to L^2(\mathbb{R}^+)$ where $$ dom(P) = \{ f \in \mathcal{D}(\mathbb{R}^+) : f(0)=0\}$$ where $\mathcal{D}(\mathbb{R}^+)$ - smooth compactly ...
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1answer
61 views

if $AA^*=BB^*$ what are the relations between A and B [closed]

I'm wondering if we have two linear operators $A, B \in \ell(V)$. and we know that $AA^*=BB^*$. then what informations can this give to us about relationships between $A$ and $B$? I think they have ...
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1answer
40 views

Prove that $(Au)(t)=\frac{d^{2}u(t)}{dt^{2}}$ is self-adjoint

Let $D(A)=\{ u \in L_2(0,T)| u, \frac{du}{dt}$ are absolutly continuous with $\frac{du}{dt} \in L_2(0,T)$, $u(0)=u(T)=0\}$ and, $(Au)(t)=\frac{d^{2}u}{dt^{2}}$ prove that $A$ is self-adjoint. Trial ...
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prove $\dim(\operatorname{range}(T)) = \dim(\operatorname{range}(\sqrt{T^*T}))$

I'm a student and I'm studying linear algebra. in Polar Decomposition we have: for a linear operator $T$, there exist a linear isometry $S$ that: $$ T =S\sqrt{T^*T}$$ so if $S$ is a linear ...
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1answer
36 views

Prove $Tx=x$, for $x\in H$, if and only if $(Tx,x)=\|x\|^2$ and $\ker(I-T)=\ker(I-T^*)$

Let $H$ be a complex Hilbert space and $T:H\rightarrow H$ an operation such that $\|T\|\leq 1$. Show that $Tx=x$ if and only if $(Tx,x)=\|x\|^2$ $\ker(I-T)=\ker(I-T^*)$. My attempt 1. ...
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1answer
40 views

Existence and uniqueness of adjoints with respect to pairings

Let $V,W,L$ be $R$-modules over a commutative ring $R$. A pairing is an $R$-linear map $V\otimes W\to L$. An adjoint of an endomorphism $f:V\to V$ w.r.t a pairing $V\otimes W\overset{g}{\to}L$ is an ...
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4answers
47 views

Showing that is a normal operator

Let $H$ is a Hilbert space $I$ is unit operator, $T \in B(H)$ and $\lambda \in \mathbb C$ $T$ is normal operator $\Rightarrow$ $T-\lambda I$ is a normal operator too. I could only write : I must ...
3
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1answer
30 views

OC of Adjoint Operator’s Image is subset of Kernel

Let $T \in B(H,K)$ when $H,K$ are Hilbert spaces and $T^{\star}$ is adjoint of $T$ Show that $(ImT^{\star})^{\perp} \subseteq KerT$ ( $ImT$ means Image of $T$ and $KerT$ means kernel of $T$) My ...
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1answer
34 views

Adjoint of operator composition in Hilbert spaces

Let $H,K,L$ are Hilbert spaces, $R \in B(H,K)$ and $T \in B(K,L)$. Show that $(TR)^\star = R^\star T^\star$ (where $\star$ denotes the adjoint operator). My attempt: Let $x\in H$, $y\in K$, $z\in L$...
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1answer
55 views

Existence of adjoint in normed space

In Luenbeger's book Optimization by Vector Space Methods, chapter 6, the adjoint of a linear operator is defined in the following way: Let $X$ and $Y$ be normed spaces and let $A: X \mapsto Y$ be a ...
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39 views

adjoint operators in the vector space of real polynomials

This problem is about the space $V$ of real polynomials in the variables $x$ and $y$. If $f$ is a polynomial, $d_f$ will denote the operator $f(d/dx,d/dy)$ , and $d_f(g)$ will denote the result of ...