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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.


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I've found that some good material on this subject can be found in Paul Halmos' A Hilbert Space Problem Book. Chapter 11, entitled Spectral Radius, contains some of the basic results in operator theory along with some specific problems about weighted shifts. In particular, the exercises work through the norm, spectral radius, point spectrum and the approximate point spectrum of a weighted shift. Some other exercises in this section utilize weighted shifts to construct examples and counterexamples to various questions presented.

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.

The questions contained here don't answer in its entirety the question: "what is the spectrum of a weighted shift." However, they do contain some particularly illuminating facts and examples. A Hilbert Space Problem Book contains exercises, hints, and solutions. I highly recommend it for a better understanding of many aspects of operator theory.

Edit: After doing a little light reading, I came across an article that gives a much more complete picture of the the spectra of weighted shifts. The paper is entitled Approximate Point Spectrum of a Weighted Shift by William C. Ridge. It gives a very explicit breakdown of the parts of the spectra of weighted shifts based on different conditions on their weights. But, for the big picture, the spectrum of a unilateral weighted shift is always a closed disk centered at the origin; the spectrum of a bilateral weighted shift is always connected.

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After reading halmos' book, I can understand that the spectrum of a weighted is centered at the origin and has rotational symmetry around the origin. But why is it always a disk? Could you please explain a little bit? – Hui Yu Jun 7 '12 at 0:22
I'm sorry, I was inherently thinking of unilateral shifts when I posted this. Of course, a bilateral shift with constant nonzero weights is invertible, so the its spectrum cannot be a disk centered at zero. I've edited my post to reflect as much. The result comes from the paper I cited. It should be noted that this paper deals with the case of bilateral weighted shifts as well. – J. Loreaux Jun 7 '12 at 0:56

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