# Questions tagged [self-adjoint-operators]

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### Proof Explanation: Proving that $f$ has at least one real eigenvalue

I'm having some trouble understanding one part of the following proof. We want to prove the following statement: Let $V$ be an inner-product, finite-dimensional real vector space. If $f:V \to V$ is a ...
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### How to prove eigenvalues are real when given self-adjoint and positive operators?

Let's say I have some T, self-adjoint operator, as well as S, which is a positive operator on a complex finite-dimensional inner product space. How could I go about proving that all eigenvalues of ST ...
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### Spectral decomposition of a self-adjoint operator. Working only with a part of it.

A symmetric matrix can be decomposed as $\boldsymbol{\Sigma}=\boldsymbol{U} \boldsymbol{\Lambda} \boldsymbol{U}^{T}$ where $\boldsymbol{U}$ contain the eigenvectors and $\boldsymbol{\Lambda}$ the ...
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### True or false Self adjoint operator and trace

I have a problem trying to prove the next statement or giving a counterexample If $T:\mathbb{R}^3 \to \mathbb{R}^3$ is self-adjoint such that trace($T^2$)=$0$, then $T=0$. I tried with the canonical ...
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### How to find the self-adjoint extension of an unbounded symmetric operator?

I have an unbounded symmetric operator, and I would like to find its self-adjoint extension if possible. First off, what properties does such an operator need in order to have a self-adjoint extension,...
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### Are the following two inner products on differential forms equal?

There are two inner product on differential forms: $\langle \alpha,\beta\rangle$ induced from Riemannian metric $g$ by defining on 1-forms as dual of vector fields then extending to all differential ...
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### Comparing Hamiltonians - Quantum harmonic oscillator

For standard 1D quantum harmonic oscillator we have $H\psi = E_n\psi$ with $E=(n+\frac{1}{2})\hbar\omega$ and $H = \frac{P^2}{2m} + \frac{1}{2}m\omega^2 X^2$ where $X$ is position operator and $P$ is ...
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### Is the log of a diagonalizable operator also diagonalizable

Let $\mathscr{H}$ be a separable complex Hilbert space and $H$ be an (unbounded) self-adjoint operator on $\mathscr{H}$ bounded from below, e.g., $H\ge 0$. Suppose that $e^{-H}$ is of trace-class so ...
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### Spectrum and self-adjointness

For bounded operator on a Hilbert space, if its spectrum is a subset of $\mathbb{R}$, then is this operator self adjoint? If not, what is a counterexample?
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### Essentially self adjoint operators: a verification procedure

$\langle \ .,. \rangle : \scr H \! \times \! \scr H \rightarrow \mathbb C$ is the inner product. Let $T : {\cal D}(T) \rightarrow \scr H$ be a linear symmetric operator, so: ${\cal D} (T)$ is dense ...
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### Spectrum of a self-adjoint integral operator

Trying to figure out the ingredients for the spectral representation of the following self-adjoint integral operator: $(Tf)(s)=\int_0^1 (s-t)^2 f(t) dt$ A hint in the exercise was to consider $im T$ ...