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fa.useful evaluation – Spectrum decomposition of the scaling operator on weighted areas

Consider the bounded linear operator $M_a$ outlined by $M_au(x)=frac{1}{sqrt{a}}uleft(frac{x}{a}privilege)$, for $a>1$. On $L^2(mathbb{R})$, it’s simple to behold that this can be a unitary operator and that (both straight or by an utility of Stone’s theorem to the continual one-parameter group) it has spectrum the gross unit coterie:$$sigma(M_a)=sigma(M_a^*)==1.$$ Furthermore, this spectrum is all steady spectrum, as it isn’t tough to display that there are not any eigenvalues.

My query considerations what occurs once you deem this operator on a weighted house, equivalent to $L^2(mathbb{R},e^{-x^2}dx)$ for instance. Here the operator $M_a$ is quiet bounded however is not regular. We might compute the spectrum by computing the norm $$|M_a|=sqrt=sqrt{sup_{xinmathbb{R}}e^{-(a^2-1)x^2}}=1$$ and remarking that $z$ are eigenvalues comparable to eigenvectors of the figure $x^s$.

That leaves the border $=1$. Is this steady or residual spectrum? And what are the $sigma_r$ and $sigma_c$ of $M_a^*$?

(Apologies if that is patent, I’m fresh to spectral concept.)

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