Spectral inclusion relations for 𝑇,𝑇|𝑌 and 𝑇/𝑌

Abstract

Let X X be a complex Banach space and Y Y be an invariant subspace of a bounded linear operator T T in X X . Then it is easy to prove that (1) σ ( T ) ⊂ σ ( T | Y ) ∪ σ ( T / Y ) \sigma (T) \subset \sigma (T|Y) \cup \sigma (T/Y) , (2) σ ( T | Y ) ⊂ σ ( T ) ∪ σ ( T / Y ) \sigma (T|Y) \subset \sigma (T) \cup \sigma (T/Y) , and (3) σ ( T | Y ) ⊂ σ ( T ) ∪ σ ( T / Y ) \sigma (T|Y) \subset \sigma (T) \cup \sigma (T/Y) . In this paper, we first study these relations for unbounded linear operators: While (3) holds without any conditions, (1) and (2) do not hold in general. We shall make clear conditions on T T that guarantee (1) and (2). Next we introduce the notion of extended spectrum for unbounded linear operators and prove similar results for the extended spectrum.