Spectrum of a linear operator on a Banach space

[Zoe-b]

Homework Statement

I have a number of problems, to be completed in the next day or so (!) that I am pretty stuck with where to begin. They involve calculating the spectra of various different linear operators.

Homework Equations

The first was:
Let X be the space of complex-valued continuous functions on Ω a closed bounded subset of ℂ, with supremum norm. Define for x $\in$ X, t $\in$ Ω
(Tx)(t) = tx(t)

I have found the spectrum of this by showing ker(λI - T) is zero and λI - T is onto for all λ not in Ω. So the spectrum of T is Ω.

I am pretty happy with this- I later found the same example in a book online and they agree with my answer.

I now have various ones involving sequences and I'm a lot more confused with these:
i) Let X be space of cts functions converging to zero with sup norm, & define:
T((a_j)) = ((j+1)^-1a_j+1)
I think norm of T is 1/2 and this gives a radius bound for possible λ but have no idea really where to go from here- I think the kernel is trivial for all non zero λ but not getting very far with showing whether its surjective or not.

ii) Let S be the bounded linear operator on l¹ defined by
T((a_j)) = (a_j - 2a_j+1 + a_j+2)

Show the spectrum of T is a cardioid.
Again I can find a general bound for λ using the norm of T but get stuck after this.

The Attempt at a Solution

Above.. any ideas on what to try next, or advice in general on methods of calculating spectra would be very welcome. Thanks in advance!
Zoe

 

[mighty2000]

Dear Zoe,

Thank you for reaching out for help with your problems involving calculating the spectra of linear operators. I understand that these problems are due soon and you are feeling stuck on where to begin. I would be happy to offer some advice and guidance on how to approach these types of problems.

Firstly, it is important to have a clear understanding of the definitions and concepts related to spectra. The spectrum of a linear operator is the set of all complex numbers λ such that λI - T is not invertible. In other words, the spectrum is the set of all values for which the linear operator T does not have an inverse. This can also be thought of as the set of values for which the equation λx - Tx = 0 does not have a unique solution.

In the first problem you mentioned, you were able to find the spectrum of the linear operator T by showing that the kernel of λI - T is zero and that λI - T is onto for all values of λ not in Ω. This is a good approach to take in general when trying to find the spectrum of a linear operator. You can also use this approach for the other problems you mentioned.

For the second problem, you have correctly identified that the norm of T is 1/2, which gives a radius bound for possible λ values. To show whether or not λI - T is onto for all non-zero values of λ, you can try to find the inverse of λI - T. If you are able to find an inverse, then λI - T is onto and the value of λ is not in the spectrum. If you are not able to find an inverse, then λI - T is not onto and the value of λ is in the spectrum. Keep in mind that the inverse may not always exist, in which case you will need to try a different approach.

For the third problem, you can again use the approach of finding the inverse of λI - T to determine the spectrum. However, since this operator is defined on the space of sequences, you may need to use some properties of sequences to help you with your calculations.

In general, when calculating spectra of linear operators, it is important to have a good understanding of the properties and definitions involved, and to try different approaches until you are able to find the spectrum. It may also be helpful to look at examples in textbooks or online to get a better understanding of the concept.

I hope this helps and good