Dear friends, I would like to find the spectrum of the linear operator ##A\in\mathscr{L}(\ell_2,\ell_2)##, which I have verified to be compact and eigenvalueless, defined by(adsbygoogle = window.adsbygoogle || []).push({});

##A(x_1,x_2,x_3,...,x_n,...)=(0,x_1,\frac{1}{2}x_2,...,\frac{1}{n-1}x_{n-1},...)##but my book does not give examples of how to do so. Could anybody help me in finding its (continuous) spectrum, i.e. the set ##\sigma(A)=\{\lambda\in\mathbb{C}\quad|\quad\nexists B\in\mathscr{L}(\ell_2,\ell_2):B=(A-\lambda I)^{-1}\}##?

##\infty## thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Spectrum of ##(x_1,x_2,x_3, )\mapsto(0,x_1,2^{-1}x_2, )##

**Physics Forums | Science Articles, Homework Help, Discussion**