All Questions
Tagged with operator-theory adjoint-operators
321
questions
2
votes
1
answer
40
views
Can an unbounded operator and its adjoint both have full domains?
Let $H$ be a complex Hilbert space. This post shows that there exist unbounded (which I will use to mean “not bounded”) operators on $H$ whose domain is all of $H$, i.e., $\mathcal D(T) = H$ (although ...
- 6,672
1
vote
0
answers
63
views
Computing the adjoint of $-\Delta$
In B. Helffer's Spectral Theory and its Application, Remark 2.7 p. 16 the author is considering the two following operators
$T_0=-\Delta $ with $D(T_0)=C^{\infty}_{c}(\mathbb{R}^N)$
$T_1=-\Delta $ ...
- 545
0
votes
0
answers
37
views
Reconstruction of an operator given the eigenfunctions and eigenvalues
I am interested in operator theory, in particular if I know the sequence of eigenvalues $\{\lambda_n\}_{n=1}^\infty \subset \mathbb{R}$ and eigenfunctions $f_n \subset X$ of a self-adjoint ...
0
votes
1
answer
35
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Untiunitary operator on a Hilbert space
A bijective linear (antilinear) operator $A$ on a Hilbert space $\mathcal{H}$ is called unitary (untiunitrary) if $\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle$ (resp. $\langle A\psi |A\...
- 763
3
votes
3
answers
112
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If $\langle Lx,y\rangle = \langle x,Ry \rangle$ then $L$ is bounded
Suppose $L,R$ are not necessarily bounded operators on a hilbert space $H$. Show that, if $L,R$ satisfy $$
\langle Lx,y \rangle = \langle x,Ry\rangle
$$ for all $x,y \in H$, then $L$ is bounded.
I ...
- 914
0
votes
0
answers
37
views
Adjoint of an operator on scaled Euclidean spaces
For $N\in \mathbb N$, equip $\mathbb C^N$ with the inner product $\langle\mathbf x,\mathbf y\rangle_N := N^{-1}\sum_i \overline x_i y_i$. Let $A$ be an $N\times M$ complex matrix. As an linear ...
- 1
0
votes
1
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71
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Why does this prove that the span of the eigenvectors is dense in $\text{Im } T$?
Let $H$ be a Hilbert space with the inner product $(\cdot,\cdot)_H$ and let $T:H\to H$ be a bounded compact and self-adjoint operator on $H$. In this case there exists an $H$-orthonormalsystem $(\...
1
vote
0
answers
78
views
If $A\in B(H)$ commutes with all self-adjoint operators, then $A=\lambda I$ for some $\lambda\in\mathbb{R}$
Let $H$ be a complex Hilbert space and $A\in B(H)$. Could anyone explain why the following assertion is true: if $A$ commutes with all self-adjoint operators, then $A=\lambda I$ for some $\lambda\in\...
- 573
2
votes
1
answer
106
views
Determining whether an operator is trace-class
Let $H$ be a separable Hilbert space, and let $A$ and $B$ be trace-class operators on $H$ such that $A^{-1/2}B$ is a Hilbert-Schmidt operator. Then is it possible to know whether the operator $B'A^{-1}...
- 513
1
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0
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79
views
Confusion about adjoint of $\nabla$
Let $\Omega$ be an open set in $\mathbb{R}^n$.
Consider $\nabla: H^1(\Omega) \to L^2(\Omega)^n$. It is a bounded linear operator.
Consider its Hilbert adjoint $\nabla^*: L^2(\Omega)^n \to H^1(\Omega)$...
- 57
2
votes
1
answer
75
views
Normal operators, verification
I have been reading some article and I came across this equality
$$\big\| |T|+|T^*|\big\|^2 = \big\|(\,|T|+|T^{*}|\,)^2\big\|\,,$$
where $|T|=(T^{*}T)^{\frac12}$. The only way this can hold is if the ...
- 537
1
vote
0
answers
141
views
Why do we need the operator to be densely defined for defining adjoint?
Suppose $T$ is an operator with domain and range in the Hilbert space $\mathcal{H}$. The usual way of defining the adjoint $T^*$ of $T$ uses density of $dom(T)$. But cannot we use this same definition ...
- 2,870
1
vote
1
answer
55
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Finding the conjugate of an operator between the Banach spaces $\ell_{p}$
I am working with conjugate operators acting between Banach spaces. I am doing the following exercise.
Let $(\beta_{n})_{n \in \mathbb{N}}$ be a bounded sequence of complex numbers. Define the ...
- 106
0
votes
1
answer
67
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Properties of Cesàro Operator in $L^2$
Put $L^2=L^2(0,\infty)$ relative to Lebesgue measure, and the Cèsaro operator $C$
is defined as follows:
$$(Cf)(s)=\frac{1}{s}\int_0^s f(t)dt$$
we can find its adjoint operator:
\begin{align*}
\langle ...
- 27
2
votes
1
answer
167
views
Domain of the adjoint operator of a bounded operator (on a Hilbert space). (Experimental physicist)
Let $A$ be a bounded operator on a Hilbert space $H$ and $D_A$ its domain. We can define the following functional on $D_A$: $$f_\eta(\xi)=(\eta,A\xi).$$
Then we have that: $$||f_\eta(\xi)||=||(\eta,A\...
- 71