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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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1answer
20 views

$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

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 ...
1
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1answer
28 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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0answers
23 views

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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0answers
25 views

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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0answers
29 views

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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0answers
20 views

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 ...
4
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2answers
130 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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2answers
48 views

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 ...
2
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2answers
51 views

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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0answers
30 views

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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0answers
18 views

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
24 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
44 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 ...
0
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1answer
37 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
32 views

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 $Tf=f'$, ...
2
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1answer
35 views

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^...
0
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1answer
29 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
27 views

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
33 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 ...
1
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1answer
31 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
34 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
13 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<...
3
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0answers
32 views

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) :...
0
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1answer
33 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
21 views

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
30 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 ...
0
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1answer
58 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 ...
1
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1answer
39 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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2answers
48 views

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 ...
0
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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. ...
1
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1answer
39 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 ...
4
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4answers
46 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
26 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
27 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
35 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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0answers
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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 ...
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1answer
52 views

Proving adjoint operator identity $(I-\lambda T)^*=I^*- \bar{\lambda}T^* $

Let $T\in \cal{B}$$(H,H)$, where $H$ is a Hilbert space and $\lambda \in \mathbb{C}$. I need to show that $(I-\lambda T)^*=I^*- \bar{\lambda}T^* $ . My most likely wrong attempt: $\langle y,(I-\...
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1answer
26 views

Finding the polar decomposition of a $2\times2$ matrix.

Specifically, the question is as follows: Let $B$ be the standard basis of $\mathbb{R}^2$. Let $T\in\mathcal{L}(\mathbb{R}^2)$ be such that $$\mathcal{M}(T,B)=\begin{bmatrix}2&3\\0&2\end{...
1
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1answer
31 views

Eigenbases in the infinite dimensional case

Let $H$ be a Hilbert space and $T \in \mathcal{L}(H)$ be self-adjoint. I know that this is a basic question, but I do not understand the infinite dimensional case of linear algebra very well. My ...
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2answers
32 views

Normal operator in Hilbert Complex space share an eigenvalue.

Everyone, I get stuck in an exercise of Functional Analysis. Let $T \in B(H)$ (H a complex Hilbert space) and $T^*$ adjoint of $T$. Supose $T$ is a normal operator. 1) Prove that $Ker(T)= Ker(T^*) = ...
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1answer
49 views

Relation between range of adjoint of T and kernal of T.

I was given two problems in homework Let $V$ be a finite dimensional inner product space and $T$ be a linear operator on it then, Prove the following (1) $ \operatorname{range}$$ ({T^{\dagger}})^{...
0
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1answer
110 views

Orthogonal complement of a vector space

I am working on a problem where I have to find the orthogonal complement of a specific vector space. Let $V = \{f: \mathbb{R} \rightarrow \mathbb{C}$, such that f is continuous and periodic with ...
0
votes
1answer
42 views

Proof regarding self-adjoint linear operators.

Let $H$ be a Hilbert space, and let $T : H \rightarrow H$ be a bounded self-adjoint linear operator, with $T \neq 0.$ I need to show that $T^{2^k} \neq 0$ $\forall k \in \mathbb{N}$. Here's what I'...
0
votes
1answer
25 views

Linear Map Adjoint or Inverse?

In my lecture notes, we have a linear map $\mathcal{A}(X)=X_{11}$ that maps $X\in S^2$ to its first element. It then claims that $\mathcal{A}^*(X_{11})= \begin{bmatrix} X_{11}&0\\ 0&0 \end{...
2
votes
0answers
35 views

Find $T^*(x,y)$ with the given inner product

Let $T:\mathbb{C}^2 \to \mathbb{C}^2$ be a inner product space, $T(x,y)=(2ix+y, -x-iy)$ and the inner product $<(x_1, x_2), (y_1, y_2)>=4x_1\overline{y_1}+9x_2\overline{y_2}$. Find $T^*(x,y)$ ...
3
votes
2answers
60 views

$Av=\lambda v \Rightarrow A^*v=\bar \lambda v$ (general case)

Suppose $V$ is a finite-dimensional complex inner product space and $v_1,v_2,...,v_3$ is an orthonormal basis of $V$. Define $A:V \to V$ by $Av_i=\lambda_i v_i$ for some $\lambda_i \in \mathbb{C}.$ ...
0
votes
1answer
60 views

If U and T are normal operator which commute with each other then U+T is normal

we had question in our end-semester exam. State true or false with explanation. If U and T are normal operator which commute with each other then U+T is normal. On inner product space. It is not ...
3
votes
2answers
62 views

Is this differential operator Hermitian?

The operator is $$\hat{A} = -i \left(x \frac{d}{dx} + \frac{1}{2} \right).$$ Is it true that $$\langle \hat{A} \psi_1(x)|\psi_2(x)\rangle = \langle \psi_1(x)|\hat{A}\psi_2(x)\rangle\ ?$$ Here, $\...
1
vote
0answers
30 views

$<L(v), v> = 0 $ for all $v$ if and only if $L + L* = 0$.

How do I prove that <$L(v), v$>$ = 0$ for all $v$ if and only if $L + L$*$ = 0$. Here, $L$ * is the adjoint. I did something like <$L(v), v$> = <$v, L*(v)$> then added the two to get the $L +...