Questions tagged [adjoint-operators]
For questions about adjoint operators in inner product spaces. For adjoint functors from category theory, use the tag (adjoint-functors).
783
questions
0
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0answers
15 views
Range of restriction of a Hilbert adjoint operator
Let $V, \hat V$ be Banach spaces with $H, \hat H$ Hilbert spaces such that $V \subset H$ and $\hat V \subset \hat H$, both continuous, compact and dense injections.
They have different norms.
Suppose ...
5
votes
2answers
90 views
a linear map without an adjoint
Define $W = \{(a_1, a_2,\cdots) : a_i \in \mathbb{F}, \exists N\in\mathbb{N}, \forall n \geq N, a_n = 0\},$ where $\mathbb{F} = \mathbb{R} $ or $\mathbb{C}$ and $W$ has the standard inner product, ...
0
votes
0answers
19 views
Positivity of an eigenvalue of a composition of a differential operator and its formal adjoint
For $q:\mathbb{R}\rightarrow\mathbb{R}$ let $\delta$ be a differential operator of the form
$$
\delta=\frac{d}{dx}+q(x),
$$
defined on some dense subspace of $L^2((0,1),d\mu)$, where $\mu$ is a finite ...
1
vote
2answers
70 views
Infinite-dimensional inner product space: if $A \geq 0$ and if $\langle Ax, x\rangle = 0$ for some $x$, then $Ax = 0$.
Exercise 8, Section 82 from PR Halmos's Finite-Dimensional Vector Spaces, 2nd Edition
If $A$ is a positive semidefinite operator, and if $\langle Ax, x\rangle = 0$ for some vector $x$, show that $Ax = ...
2
votes
2answers
46 views
Adjoint operator for $l^2(\mathbb{N})$ [closed]
I am working with a problem from my exercise sheet (not mandatory). I hope that someone could held me to get started with the problem. I do not know how to get started.
Problem: Consider the Hilbert ...
0
votes
0answers
11 views
Closed, densely defined operator and its spectrum.
When I was reading this book: Positive Operator Semigroups
From Finite to Infinite Dimensions by András Bátkai. In the appendix A.8 there is a proposition A.32 said that:
For a closed and densely ...
1
vote
2answers
59 views
prove that an operator is unitary
I have $V$ an inner product space above $\mathbb{C}$ and a linear operator $T$ such as $T=T^{*}$ on $V$ (surjective).
I need to prove that $U=\left(I-iT\right)\left(I+iT\right)^{-1}$ satisfies the ...
1
vote
1answer
58 views
Show the operator $I-iT$ is invertible and that $(I-iT)(I+iT)^{-1}$ is unitary operator
I have $V$ an inner product space above $\mathbb{C}$ and a linear operator $T$ such as $T=T^{*}$ on $V$ (surjective).
I need to prove that the operators $I+iT$, $I-iT$ are invertible and that $U=\left(...
0
votes
1answer
22 views
show that Norm of v+iT(v) equals to the Norm of v-iT(v)
So I have $V$ an inner product space above $C$ and a linear operator $T$ such as $T=T^*$ on $V$
I need to prove that:
$$||v+iT(v)||=||v−iT(v)||$$
I tried to write it by definition, but I didn't get ...
0
votes
1answer
23 views
find the hermitian operator of an operator which attaches a matrix and its iverted
$V=M_n\left(\mathbb{C}\right)$ with the standard inner product $<A,B>=tr\left(B^{*}A\right)$ and let $P\in V$ be an invertible matrix. We define an operator $T_p:V->V$ such that $$T_p\left(A\...
1
vote
1answer
23 views
Bounded linear operator of $\ell^{2}(\mathbb{N})$ which is normal but not self-adjoint
My question is : Does there exist a bounded linear operator $T:\ell^{2}(\mathbb{N})\rightarrow\ell^{2}(\mathbb{N})$ which is normal but not self-adjoint?
Just to be clear, if $H$ is an Hilbert space, ...
2
votes
1answer
14 views
Prove that $v = 0 \iff v + iT(v) = 0$ for a self adjoint operator
Let $T:V\rightarrow V$ be a self adjoint operator over $C$.
Prove that $v = 0 \iff v + iT(v) = 0$.
I've tried to use a norm and equate it to $0$, and also to use $\langle v + iT(v),u\rangle = 0$ ...
2
votes
3answers
45 views
Infinite-dimensional inner product spaces: if $A^k = I$ for self-adjoint $A$ and for integer $k > 0$, then $A^2 = I$
Exercise 5(d), Sec 80, Pg 162, PR Halmos's Finite-Dimensional Vector Spaces:
If $A^k = I$ where $A$ is a self-adjoint operator and $k > 0$ is some positive integer, show that $A^2 = I$. The ...
1
vote
1answer
20 views
Prove that T-cI is normal
Can anyone help me proving this:
Let $(V,<,>)$ be a finite dimensional vector space , Let $T$ be a normal linear operator on $V$. Then $T-cI$ is normal for every $c \in \mathbb{F}$, $c \neq 0$
...
0
votes
1answer
19 views
Finite-dimensional unitary spaces: are $A^*A$ and $AA^*$ are always unitarily equivalent?
A variant of Ex 2(b), Sec 79, Pg 158 from PR Halmos's Finite-Dimensional Vector Spaces:
If $A$ is an arbitrary linear transformation on an $n$-dimensional complex inner product space $V$, does it ...
1
vote
1answer
32 views
Complex inner product spaces: are $A^*A$ and $AA^*$ always unitarily equivalent?
Problem 2(b) from Sec 79, pg 158 of PR Halmos's Finite-Dimensional Vector Spaces:
If $A$ is an arbitrary linear transformation on a complex inner product space $V$ (not given to be finite-dimensional)...
4
votes
1answer
62 views
If $T^m(\alpha)=0,$ with $\alpha \in V$ and $m \in \mathbb{N}$, then $T(\alpha)=0$ [duplicate]
I am not so sure how to prove this exercise, please help me:
Let $V$ be a finite dimensional vector space with inner product $\langle,\rangle$. Let $T$ be a linear normal operator on $V$, then:
If $...
0
votes
1answer
38 views
compute the adjoint operator
Suppose there is a bounded linear operator $T:L^2[-1,1]\to L^2[-1,1]$ given by $Tf(t)=\int_{-1}^0f(s)ds+(\int_{-1}^1f(s)ds)t^2$. We need to compute the adjoint of the operator $T$.
When I calculate ...
1
vote
1answer
46 views
Compute the adjoint of an operator in $L^2[-1,1]$
Consider a bounded linear operator $A:L^2[-1,1]\to L^2[-1,1]$ given by $Af(t)=\int_{-1}^0f(s)ds+(\int_{-1}^1f(s)ds)t^2$.
Question: Compute the adjoint of the operator $A$.
I know we need to ...
2
votes
1answer
45 views
Proving the map $f-i\cdot \text{id}_v$ is invertible with $f:V\to V$ a self-adjoint linear map
I am struggling with the following question.
Let $V$ be a finite dimensional complex inner product space and let
$f:V\to V$ be a self-adjoint linear map. Show that the map $f-i\cdot
\text{id}_v$ ...
1
vote
0answers
21 views
Adjunction pair on exterior algebra
Let $(V,h)$ be real Euklidean vector space and $(V \otimes_{\mathbb{R}}\mathbb{C}, \overline{h})$ complex extension by complex linearity. We consider the exterior algebra space $E= \wedge^{\bullet} V= ...
0
votes
1answer
30 views
unitary operator explanation [closed]
the unitary operator definition is $ ⟨Tv,Tw⟩ = ⟨v, w⟩$
for every $v, w$ in $V.$
can you please explain the intuition and what the formal definition actually means?
why unitary operator preserves the ...
1
vote
1answer
36 views
self-adjoint operator intuition
can someone please explain self-adjoint operator intuition to me.
And why when $T^* = T^{-1}$, $T$ preserves the inner product and therefore preserves the the orthonormal basis and the length and ...
3
votes
0answers
40 views
What is a duality argument?
I believe this is probably a question that can have a wide variety of answers, but i believe i'm still interested. The thing is, i have been in a couple pde talks, and i saw that in both of these ...
0
votes
1answer
45 views
adjoint transformation intuition
I can't find the connection between the Riesz Representation Theorem and inner product spaces and the adjoint transformation.
what I understood that dual spaces enables us to have an transpose ...
1
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0answers
31 views
An operator of Banach spaces is an isomorphism if its adjoint is [duplicate]
For an operator $T:E\rightarrow F$ of Banach spaces $E, F$ I have to show that $T$ is an isomorphism iff its adjoint $T'$ is an isomorphism.
I have already shown the "$\Rightarrow$" direction and now ...
1
vote
1answer
33 views
Compute the adjoint of an operator in Hilbert space
I have the following question. In the Hilbert space $l^2$, consider the operator $Tx=(\frac{x_n+x_{n+1}}{2})_n$ and $x=(x_n)_n$. Compute the adjoint of operator $T$.
I tried to find $T^*$ such that $(...
0
votes
0answers
19 views
Show that if $A$ is a symmetric operator then its domain is a subset of domain of its adjoint operator
Let $A \in L(D,H)$ and $A$ is symetric operator show that $D \subset D(A^*)$.
My attempt:
Domain of $A^*$ is defined as follows
Let $v\in H$, consider a linear form $\phi_v \in L(D, \mathbb{C}), \ ...
1
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0answers
33 views
Eigenvalues and eigenvectors of $a^*a$ and $aa^*$
I have an operator $a$ in Euclidean space. How can i show that $a^{*}a$ and $a a^{*}$ have the same characteristic polynomials.
I've understood that these operators are self-adjoint and that their ...
1
vote
2answers
27 views
Orthogonality relation of eigenvectors for a self-adjoint operator
So everyone knows eigenvectors corresponding to different eigenvalues are orthogonal to each other, given that the operator is self-adjoint.
If we have a self-adjoint operator, say $L$, is it ...
0
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0answers
33 views
Complexified of the adjoint is the adjoint of the complexified
Given a real Hilbert space H, I can construct the complexified space $H_{\mathbb C}$, which is now a complex Hilbert space. Similarly, given a (possibly unbounded) densely defined operator $A:D(A)\...
3
votes
2answers
60 views
Exercise of compact self-adjoint operator
'Let $H$ be a Hilbert space. Find all compact self-adjoint operators $T:H \rightarrow H$ such that $T^{k}=0$ with $k>0, k \in N$.'
$ \ $
I have this idea. Consider $\lambda_n$ eigenvalue of T and ...
1
vote
1answer
36 views
A problem about operators on an Euclidean space
Let $A, B$ be self-adjoint positive operators, $C, D$ be orthogonal
operators on an Euclidean space, and $AC = DB$.
Prove that $C=D$.
I know many properties of self-adjoint and orthogonal ...
1
vote
1answer
42 views
Operators on an Euclidean space
Let $A, B$ be self-adjoint positive operators, $C, D$ be orthogonal
operators on an Euclidean space, and $AC = DB$.
Prove that $A=B \Leftrightarrow AC$ is normal.
I know many properties of ...
3
votes
0answers
31 views
How to calculate a multiplication operator representation?
Let $H =\ \mathcal{l}^{2}(\mathbb{Z},\mathbb{C})$ and $A = R + L$,
where $L$ is the left-shift operator (and $R$ is the right-shift
$(Ra)_{n}=a_{n-1}$). Set
$$U : \mathcal{l}^{2}(\mathbb{Z},\...
2
votes
3answers
83 views
Does $P \circ P =P$ and $\langle Px, y \rangle = \langle x, Py \rangle$ imply $P$ is linear?
Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space and $P:H \to H$. Suppose that
$P \circ P =P$
$\langle Px, y \rangle = \langle x, Py \rangle$ for all $(x,y) \in H^2.$
I ...
0
votes
1answer
11 views
Adjoint of right shift operator on orthonormal basis $(e_{n})_{n\in\mathbb{N}}$ of $\ell^{2}(\mathbb{N})$
I'm sorry if this question is a duplicate.
Suppose $(e_{n})_{n\in\mathbb{N}}$ is the usual orthonormal basis of $\ell^{2}(\mathbb{N})$. We can define an operator $v\colon H\to H$ by $ve_{n}:=e_{n+1}$....
1
vote
2answers
29 views
Question on adjoint operator
Theorem Let $V$ and $W$ be finite dimensional inner product spaces over $F$ and let $T\in L(V,W)$. Then there is a unique function $T^*:W\to V$, defined by the condition $<T(v),w>=<v,T^*(w)&...
2
votes
1answer
22 views
Involution is not strongly continuous
Let $H$ be a Hilbert space and consider on $B(H)$ the strong topology, i.e. the topology induced by the seminorms
$$x \mapsto \Vert x \xi \Vert, \quad \xi \in H$$
I.e. a net $(x_\alpha)_\alpha$ ...
4
votes
1answer
37 views
Linearity of an adjoint map
In my class I was asked to solve the following task:
Suppose the map $\varphi : L \rightarrow M$ where L and M are linear spaces. Is the adjoint map $\varphi^* : M^* \rightarrow L^* $ linear?
This ...
1
vote
1answer
23 views
Exponential series of bounded Hilbert space operator is convergent
Let $A \in B(H)$ be a bounded Hilbert space operator. For $z \in \mathbb{C}$ exponential is defined as follows:
$$e^{zA} = \sum_{k=0}^{+\infty}\frac{z^kA^k}{k!}$$
Show that series defined above is ...
0
votes
0answers
56 views
Any more explanations for proof $(Lu,v)_{L^{2}}=(U,L^*v)_{L^{2}}-(A_du|_{x=0},v|_{x=0})_{L^{2}} $?
I come accross the below identity which is montioned in the proof of Lemma2.2. 1 page 35 of this paper entitled Stability of Small Viscosity
Noncharacteristic Boundary Layers by Guy Me ́tivier here, ...
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0answers
22 views
Hilbert space problem of orthogonal space
could anyone help me with this problem. ¡Thankyou very much!
Let H be a Hilbert space.
(a) Let M be a closed subspace of H and let P be a orthogonal projection on M. Prove that P² = P and P is an ...
1
vote
1answer
35 views
Limit of $H^1(\mathbb{R})$ functions
Let $H^1(\mathbb{R})=\{f\in L^2(\mathbb{R}): f\in AC([a,b])\,\, \text{for all}\,\, [a,b]\subseteq\mathbb{R}\,\,\text{and}\,\, f^{\prime}\in L^2(\mathbb{R}) \}$, where $AC([a,b])$ denotes the set of ...
0
votes
1answer
23 views
If $I$ is identity operator on Hilbert space $H$ then show that $I^* =I$
If $I$ is identity operator on Hilbert space $H$ then show that $I^* =I$
My attempt:
For any two bounded linear operators $S$ & $T$ on $H$ we have, $(ST)^∗=T^∗S^∗$ taking $T=I$ & $ S=T^∗$ we ...
3
votes
2answers
35 views
Infinite-dimensional inner product spaces: if $A$ is a skew operator, does it follow that $A-I$ is invertible?
I am trying to find an answer to this question: if $A$ is a skew-Hermitian operator (i.e., $A^* = -A$) on an infinite-dimensional inner product space, does it follow that $A-I$ is invertible? The ...
1
vote
2answers
56 views
If two matrices have the same characteristic polynomial, they need not be unitarily equivalent. Why?
Fact: Two matrices are unitarily equivalent, then they have the same characteristic polynomials.
I find both of the matrices $\begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & ...
1
vote
2answers
28 views
Question concerning proof of unique existence of adjoint operator
Here, a proof is given on p. 50. However, one crucial moment is when they define
$$T^{\star}k := y.$$ Okay, the next step reads:
$$\langle h, T^{\star}\left( \lambda_1k_1 + \lambda_2k_2 \right)\...
2
votes
2answers
71 views
Adjoint of integral operator
On the space of $L^2([0,1])$ consider the following operator; $\Lambda u(x)= \int_0^xu(s)ds$ , I want to find the adjoint of this operator.
\begin{align}
(\Lambda u , g)
& = \int_0^1\Big(\int_0^...
0
votes
0answers
11 views
Computing the adjoint of a Lie derivative w.r.t. the Petersson inner product
Consider $X\in\mathfrak{sl}_2(\mathbb{R})$, and an irreducible smooth representation $(\pi,V)$ of $SL_2(\mathbb{R})$. Assume $u,v\in V$, and let $<\cdot,\cdot>_P$ be the Petersson inner product, ...
