Spectra on different spaces

Question

This is a method request: I am looking for techniques that allow me to investigate problems like this:

Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\infty,0].$ (the real part of the spectrum is not strictly positive)

Assume now that this operators has a bounded extension $T_2:\ell^2 \rightarrow \ell^2.$ Is it possible to derive meaningful conditions under which $\Re(\sigma(T_2)) \subset (-\infty,0].$

Spectral theory on different spaces is rarely treated somewhere, so I was wondering if anything in that direction exists?

Answer 1

I know it's a while since this question was asked, but I think the book "Linear Operators and their Spectra" by E. Brian Davies might contain some information about what's being asked here. In particular, the book contains a couple of results about 'consistent' operators and `compatible' Banach spaces.

I think you can say quite a lot if the essential spectra of $T_1$ and $T_2$ don't get in the way of things too much. In particular, it looks like $\sigma(T_1)=\sigma(T_2)$ if both your operators are Riesz (see Theorem 4.2.15 for the case where both operators are compact - I think the proof goes through verbatim for Riesz operators more generally).

Things are unfortunately less straightforward when the operators have different essential spectra (see, for example Example 2.2.11).