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3 exercise(s) shown, 0 hidden
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Find an explicit orthonormal basis for the Hilbert space
[[math]]
H=L^2[0,1]
[[/math]]
by starting with the algebraic basic [math]f_n=x^n[/math] with [math]n\in\mathbb N[/math], and applying Gram-Schmidt.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that for the usual matrices [math]A,B\in M_N(\mathbb C)[/math] we have
[[math]]
\sigma^+(AB)=\sigma^+(BA)
[[/math]]
where [math]\sigma^+[/math] denotes the set of eigenvalues, taken with multiplicities.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Clarify, with examples and counterexamples, the relation between the eigenvalues of an operator [math]T\in B(H)[/math], and its spectrum [math]\sigma(T)\subset\mathbb C[/math].