All Questions
Tagged with operator-theory self-adjoint-operators
162
questions
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answers
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Commutators of unbounded operators on Hilbert spaces
Commutation seems to be a tricky business when it comes to unbounded operators, because of the domain questions. I have some trouble understanding the usual material about commutators of unbounded ...
- 57
1
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1
answer
62
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Basis of eigenfunction of nonself-adjoint operator
If $H$ is a separable Hilbert space, $A$ is a bounded nonself-adjoint operator, $\{\lambda_n\}_{n\in\mathbb{Z}}$ are the eigenvalues of $A$, and the corresponding eigenfunctions are $\{\psi_n\}_{n\in\...
- 169
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28
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On self-adjoint extensions and multiplicity of eigenvalues
I hope you can help me with the following question.
Let $B$ be a densely defined closed symmetric operator on a infinite-dimensional separable complex Hilbert space $\mathcal{H}$ with deficiency ...
- 43
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46
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Asking for reference about self adjointness
Is there someone that knows and can share any reference about the compactness and the self-adjointness of the operator
\begin{equation}
\pmb{\sigma}\cdot L
\end{equation}
on $H^1(\mathbb{R}^{3},\...
- 545
2
votes
1
answer
42
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On functions of a self-adjoint operator of the form $U^{-1} A U$
I found in Reed Simon that, since $\mathcal{F}$ (Fourier Transform) is unitary in $L^2(\mathbb{R}^n)$ and the self-adjoint operator (in a suitable domain) $-\Delta = H_{0}$ can be expressed as $H_{0} =...
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24
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Inequality for inverse of an unbounded self-adjoint operator
Given an unbounded self-adjoint operator $A$ on some Hilbert space $\mathcal{H}$, and $\mu$ a non zero real number: I want to show that
\begin{equation}
\lVert (\mathbf{A} + i\mu \mathbf{I})^{-1} \...
3
votes
1
answer
84
views
exponential of operators - Trotter product formula
Let $H$ be a self-adjoint operator, bounded from below, with spectrum consisting of isolated eigenvalues $E_n (E_0 < E_1 < ...)$ with finite multiplicities $d(n)$, and $Tr[exp(- \beta H)] < \...
- 181
2
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1
answer
111
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Weinstein's bound on eigenvalues of self-adjoint operators
Consider a complex, separable Hilbert space $H$ and a (densely defined) self-adjoint operator $A: \mathcal D(A)\to H$. Assume that $A$ admits an orthonormal basis of eigenvectors $(\varphi_n)_{n\in \...
- 428
2
votes
2
answers
87
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Operator norm of powers of bounded normal operators and self adjoint operators on Hilbert space
I saw this problem in Sheldon Axler's Measure,Integration and Real Analysis Ex. $10B$, problem $17$ and $18$.
Let's just restrict to the self adjoint case. What I am trying to show is that $||T^{n}||=|...
- 1,285
3
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2
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64
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If $P$ is a selfadjoint operator on a Hilbert space satisfying $P^4 = P$, is $P$ an orthogonal projection?
I’ve been trying to solve this problem to no avail. I have proved already that the spectrum is a subset of $\{0,1\}$ just like for orthogonal projections. I’ve also proved $\lVert P \rVert = 1$. ...
- 1,445
1
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1
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94
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Is the composition of positive, self-adjoint operators on a Hilbert space positive?
Let $A,B$ be self-adjoint, bounded, linear operators on a Hilbert space $H$ over $\mathbb{R}$ such that $\langle Ax,x\rangle\geq 0$ and $\langle Bx,x\rangle\geq 0$ for all $x\in H$. Does it also hold ...
- 11
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91
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Decomposition of normal operators into selfadjoint ones: $T=A+iB$
Assume we are given a densely defined and closed operator $T$ on a Hilbert space such that $D(T)\cap D(T^*)$ is dense. I am looking for sufficient conditions for $T+T^*$ to be selfadjoint as well as ...
- 789
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78
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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
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27
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Property of being Fredholm after applying the Helmholtz operator
Consider a bounded self-adjoint operator $L$ on $L^2(\mathbb{R})$. I want to study its essential spectrum, thus I need to determine when is $L-\lambda I$ Fredholm. Am I right with the statement
$$
L-\...
- 1,139
3
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182
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Reed and Simon, Fourier Analysis and Self-Adjointness, second corollary to Theorem X.$25$: how to show that $D(A^2)$ is dense in $D(A)$ for its norm?
This question arose while trying to figure out the proof of the second corollary to Theorem X.$25$ in Reed and Simon's Fourier Analysis, Self-Adjointness, stated as follows:
Theorem X.$25$: Let $A : ...
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