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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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24 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) :...
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1answer
22 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
17 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 = \...
2
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1answer
23 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
53 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
34 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
43 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 ...
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1answer
33 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
37 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
42 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
21 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
17 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 $x\in H$, $y\in K$, $z\in L$...
1
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1answer
25 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 ...
2
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0answers
24 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 ...
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1answer
44 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-\...
1
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1answer
22 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
22 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
31 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
27 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
102 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
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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
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1answer
18 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
34 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
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2answers
57 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
49 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
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2answers
59 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, $\...
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0answers
25 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 +...
1
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1answer
21 views

Trivial Kernel and density of range

On the wiki page for unbounded operators, it states that since: $$ \text{ker}~T = (\text{range} T^*)^{\perp} $$ Then we may conclude that if $T^*$ has trivial kernel then $T$ has dense range. Does ...
0
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1answer
19 views

Specific non self-adjoint operator for $\mathcal{P}^2$

Given the inner product space $\mathcal{P}^2$ with inner product defined as $(<p,q>) =\int_0^1 p(x)q(x) dx$ and an operator $$T(a_0 + a_1x + a_2x^2) = a_1x$$I need to show that the operator is ...
0
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2answers
20 views

Self-adjoint operator or not?

let be the operator on $L^2(\mathbb{R})$ define by $$ T[f](x)=f(-x) $$ and I tried to determinate the adjoint with a change of variables: $$ (T^*g,f)=(g,Tf)=\int_{\mathbb{R}}g(x)f(-x)\,dx=[y=-x; dx=-...
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1answer
22 views

Kernel of an operator

can you help me to solve this exercice? Show that the kernel of the operator $M:L^2((-1,1))\to L^2((-1,1))$ has infinite dimension. The operator is: $$ M[f](x):= \int_{(-1,1)} \sin(xy)f(y)\,\,\,dy. $$ ...
1
vote
1answer
48 views

Given $T$ an idempotent operator, why $R(T)=R(T^*)\implies T=T^*$?

The context of the problem I have is that $T$ is supposed to be an idempotent/projection operator. (Actually I'm trying to re-understand an answer[at 1. fifth line "If..."] I got about a month before) ...
0
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0answers
20 views

Self-Adjoint Integral Operator

I'm dealing with a new problem of functional Analysis: Let $f(t) = (1-\frac{e^{it}}{2})^{-1} \in L^2(-\pi,\pi)$ and let $T$ be the integral operator with integral kernel $K(x,y) = f(x-y)$. a)Show ...
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1answer
60 views

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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0answers
28 views

Find $ T^*\left( \begin{smallmatrix} a&b\\ c&d\end{smallmatrix} \right)$

Let $T:\mathbb{C^3} \to \mathcal{M}_{2x2}$ and $T(x,y,z)=\begin{pmatrix} 3x+iy & y-iz\\ iy+z & 2x \end{pmatrix}$ with the inner product $<A, B>=tr(\bar{B}^t, A)$. Find $T^*\begin{...
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1answer
34 views

Determining Adjoint Operator

I was dealing with this exercise of my functional analysis course : Let $\mathcal{H} = l^2(\mathbb{Z})$, $\hspace{2mm}$ $U:\mathcal{H} \rightarrow \mathcal{H}$ such that $(Ux)_k = x_{k+1}$ a)Find $U^...
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0answers
13 views

Finding the Weyl sequence for two operators.

Consider the operator $T=i(x\partial_y -y\partial_x)$ with $i^2=-1$. It is known that the spectrum of $T$ is essential and $\sigma(T)=\sigma(T)_{p}=\sigma(T)_{essential}=\Bbb{Z}$[ note that in ...
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2answers
52 views

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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2answers
38 views

Find adjoint of a linear transformation

Let $T:\mathbb{R^3} \to \mathbb{R^3}$ be a linear transformation, and $T(1,1)=(2,4)$ $T(1,-1)=(0,-2)$ Find $T^*(x,y)$. I "found" (I mean, I think it's wrong...) the general form of the linear ...
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1answer
81 views

$T^{\ast}T$ unitary then $T$ isometry

An operator $T$ is an isometry if $||Tf||=||f||$ for all $f$ in a Hilbert space $H$. (a) Show that if $T$ is an isometry, then $<Tf,Tg>=<f,g>$ for every $f,g \in H$. Prove as result that $...
2
votes
1answer
53 views

Operator bounded then adjoint is bounded

Why if an operator $A$ bounded then $A^{\ast}$ is bounded? I have already proven that $\|A\|=\|A^{\ast}\|$ but if $A$ is bounded then exists $M$ such that $\|A(f)\|\leq M||f||$ for all $f$ in $H$ ...
0
votes
1answer
50 views

Property of a partial isometry on a Hilbert space

Let $\,V\,$ be a partial isometry on a Hilbert space $\,\mathsf H$. This means by definition that $\,V^*V\,$ is idempotent which is equivalent to $\,VV^*$ being idempotent. Thus both $\,V^*V\,$ and $\...
2
votes
1answer
61 views

$U$ invertible and $U^{\ast}U=1$ then $UU^{\ast}=1$ (Barry Simon)

(a) Let $U$ map $H\to K$ for two Hilbert spaces. Prove that $|U\varphi|=|\varphi|$ for all $\varphi$ if and only if $U^{\ast}U=1$ (b) If $U$ is invertible and $U^{\ast}U=1$, prove that $UU^{\ast}=1$ ...
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0answers
41 views

Adjoint of operators with dense range

Let $X$ and $Y$ be Hilbert spaces and $A:X\to Y$ be a linear bounded operator. Assume further that range of $A$ is dense in $Y$. Is the operator $A^*$ bounded?
3
votes
1answer
42 views

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,$...
0
votes
0answers
18 views

Adjoint operator for $C: \mathring{W}_2^1(\Omega) \to L_2(\Omega)$

I am a bit confused about the writing the explicit form of adjoint operator from one Hilbert space to another. Let we have a domain $\Omega$ and a subdomain $\Omega_0 \subset \Omega$ (everything is ...
0
votes
1answer
46 views

Are these two definitions of adjoint identical to each other?

I read a lecture note in where the definition of adjoint is Let $X$ and $Y$ be normed linear spaces and $T \in \mathcal{B}(X,Y)$. The Banach space adjoint (or simply adjoint) of $T$, denoted by ...
2
votes
2answers
47 views

Adjoint of T $\in B(\ell^{2})$

I was wondering if you guys can help me with this one! Let $T \in B(\ell^{2})$ a continuous operator. Also, \begin{align*} (Ts)(n)=\frac{1}{n}\sum^{t}_{m=1} s(m),\hspace{1cm}n\in \mathbb{N}, s\in \...
0
votes
0answers
13 views

Example: $F(t)$ is strongly continuous on $X$ but $F^*(t)$ is not

Let $X$ be a reflexive Banach space. I am looking for examples of the strongly continuous operators $F(t):t\mapsto \mathcal{L}(X)$ such their adjoint on $X$ for every $t$ is not a strongly continuous ...
0
votes
0answers
13 views

Find Adjoint of Product & Sum of Differential Operator

I am asked to find the adjoint of the following differential operators: \begin{equation} L_{1} = a(x)\frac{\partial}{\partial x}b(x)\frac{\partial}{\partial x}, \end{equation} and \begin{equation} ...