# The spectra of weighted shifts

Since weighted shifts are like the model-operators in operator theory and people have been studying them for so long, I think there should be quite a large literature on the spectra of such operators. However, after some search I hardly found anyone which gives a complete picture of what the spectra of a weighted shift.

Most of the papers I found deal with some specific properties of their spectra, but the question I have in mind is what the spectra are, exactly.

I wonder whether there is some good reference on this.

Thanks!

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One nice fact about weighted shifts is that given weights, $a_n,b_n$, if $|a_n|=|b_n|$ for all $n$ (either in $\mathbb{N}$ or $\mathbb{Z}$ depending on whether it is the unilateral or bilateral shift), then the weighted shifts determined by $a_n$ and $b_n$ are unitarily equivalent (This is exercise 89). A direct corollary to this is that the spectrum of a weighted shift is radially symmetric about zero.