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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).

2
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1answer
817 views

Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...
4
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2answers
164 views

System of equations wrt self-adjoint operators

$X = \left( \begin{matrix} 2&s\\ 8&2 \end{matrix} \right)$ and $Y = \left( \begin{matrix} 2&-1\\ 2&2 \end{matrix} \right)$ are two operators wrt the same orthonormal basis $B$ in a 2D ...
2
votes
1answer
103 views

Operator inequalities: $0 \leq A \leq B \Rightarrow Tr(A^p) \leq Tr(B^p)$?

It is trivial to show that $0 \leq A \leq B \Rightarrow Tr(A^2) \leq Tr(B^2)$, but does this generally hold for all $p >$ 2 as well?
1
vote
1answer
521 views

Prove that every self-adjoint unitary linear operator can be expressed in the form $U\alpha = \beta - \gamma$

This problem is from Kunze Hoffman book. I think I go in the right direction to solve this but I miss some point to finish it. Can anyone help me? Suppose $U$ is a self-adjoint unitary linear ...
0
votes
1answer
468 views

Adjoint operator

This is about, a question I answered. Now there is an additional question that I cannot answer and do not want to spend any more time on. I feel like the question will not get any attention, as I ...
2
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0answers
299 views

Find the adjoint operator

I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator $$ (Ax)(t)=x(at), x\in L^2(0,\infty), a>0. $$ My calculation is the following; I use the ...
1
vote
0answers
39 views

Multiplicationoperator self-adjoint on given domain

I have a question. Given the Hilbertspace $H=l^2(\Bbb{N})$ with multiplicationoperator $T_f$, with $f:\Bbb{N}\rightarrow\Bbb{C}$ and $T_f\psi=f\psi$. I want to prove the following statement: Suppose ...
2
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0answers
109 views

Verify a given SVD of an operator

Show that the Singular Value Decomposition of the operator $$ A\colon L^2([0,1])\to L^2([0,1]), x\mapsto\int\limits_0^t x(s)\, ds $$ is given by $$ \sigma_j=\frac{1}{(j-1/2)\pi},~~~~~v_j(x)=\sqrt{2}...
5
votes
2answers
527 views

Find adjoint operator of an operator T

I would like to find the adjoint operator of $$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$ Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$. I tried to find ...
5
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1answer
3k views

Why is every selfadjoint operator closed?

I've read this theorem multiple times, but never seen a proof: Every selfadjoint operator is closed. But it's always been stated without a proof. Is it somehow obvious? I can't see it immediately ...
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2answers
53 views

Using “adjunction” to refer to the act of taking adjoints of operators

I have an especially flabby terminology question. How acceptable is it, in your opinion, to use the word "adjunction" to refer to the process of taking adjoints of operators on a Hilbert space? ...
2
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2answers
1k views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
1
vote
1answer
456 views

The convergence of the adjoint operator

If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
0
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1answer
207 views

Verifying the Transpose of a Linear Operator

I have a linear operator $\Phi$ in $\mathbb{R}^n$ and have created another operator that I believe to be its adjoint (transpose) $\Phi^T$. What is the most direct way to verify that my $\Phi^T$ is ...
4
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2answers
1k views

Hilbert Adjoint Operator from Riesz Representation Theorem - $T^{*}y=\frac{\left\langle y,Tx\right\rangle }{\left\langle z_{0},z_{0}\right\rangle}z_0$

Kreyszig's Functional analysis seems to introduce the hilbert adjoint operator by means of an explicit representation. I haven't seen this anywhere else and I would like to confirm this explicit ...
1
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1answer
209 views

Invertibility of matrix with each element equal to cofactor

I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
0
votes
1answer
544 views

self-adjoint operator and unitary orthogonal matrix

Please offer a solution to the following problem. It was offered in class by my professor as an additional exercise to try on one's own. Let $V$ be the inner product space, and assume that $\alpha \...
1
vote
1answer
562 views

Find an adjoint; inner product

Let V be the space of real polynomials of degree at most 1 with inner product defined by $\langle p,q \rangle = \frac12\int_{-1}^1p(t)q(t)dt.$ Define $\alpha\in End(V)$ by $$\alpha(p)=p(0)+p(1)t.$$ ...
6
votes
2answers
958 views

inner product and adjoint operator

This is a problem I found in Schaum's Outlines: Linear Algebra, and I was wondering if someone knew how to solve it. I began using integration by parts, but that approach did not lead to any ...
1
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1answer
6k views

Cofactor matrix 4x4, evaluated by hand

I am currently studying linear algebra. So far I have studied method for aquiring the inverse of a matrix A. Now I would like to evaluate the inverse of a $4\times 4$ matrix using the following ...
2
votes
1answer
119 views

Is $\text{rk}L=\text{rk}L^*L $ true for finite rank operators?

Let $L$ be a compact linear operator in an infinitedimensional space that has finite rank. Do the equations $$\text{rk}L=\text{rk}L^*L\ \text{and} \ \text{rk}L^*L=\text{rk}R,$$ where $R$ is the (...
8
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0answers
3k views

Gradient operator the adjoint of (minus) divergence operator?

Recently I found this statement -- the gradient operator is the adjoint of the minus divergence operator -- in one of my lecture notes. Knowing only a little about functional analysis, I'm looking for ...
12
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2answers
2k views

Motivation for adjoint operators in finite dimensional inner-product-spaces

Given a finite dimensional inner-product-space $(V,\langle\;,\rangle)$ and an endomorphism $A\in\mathrm{End}(V)$ we can define its adjoint $A^*$ as the only endomorphism such that $\langle Ax, y\...
5
votes
0answers
865 views

Sum of operator and adjoint is self-adjoint

In abstract Hodge theory there is the following lemma: Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
1
vote
1answer
264 views

Norm of conjugate Hardy operator

For the classical Hardy operator $T\colon \ell^p\to \ell^p \quad (Tx)_n=\frac{1}{n}\sum_{k=1}^n x_k$ or the integral type $S\colon L^p\rightarrow L^p \quad (Sf)(x)=\frac{1}{x}\int_0^x f(t) dt \ \ $ ...
0
votes
1answer
102 views

The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).

Let $A\in B(\mathbb{C}^n) \cong \mathbb{M}_n(\mathbb{C} )$ Prove: The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).
3
votes
2answers
779 views

eigenfunctions of the adjoint of an operator

If the eigenfunctions of a linear operator are known, is there a way to calculate the eigenfunctions of the corresponding adjoint operator based on the known eigenfunctions? In other words, what's the ...
3
votes
1answer
589 views

Eigenvalues of compact operators and his adjoint.

Let $T: H \to H$ be a compact operator with $H$ a Hilbert space. Let then $\lambda \neq 0$ be an eigenvalue of $T$ with eigenfunction $v$. Is then $\lambda$ an eigenvalue for the adjoint $T^*$ ...
3
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2answers
490 views

Exponential Law for based spaces

I realize most people work in "convenient categories" where this is not an issue. In most topology books there is a proof of the fact that there is a natural homeomorphism of function spaces (with ...