Spectral Theory in Hilbert Space

56 9. EXTENSION THEORY
definitions of Sλ and Dλ and identical proofs. We now define
Dλ = {(u, λu) ∈ T ∗ } = {(u, λu) | u ∈ Dλ}
Eλ = {(u, λu + v) ∈ T ∗ | v ∈ D λ } .
As before it is clear that Eλ for non-real λ is the direct sum of Dλ and
Dλ since if a = i we have (u, λu+v) = a(v, λv)+(u−av, λ(u−av))
2 Im λ
and that this direct sum is topological (i.e., the projections from Eλ
onto Dλ and Dλ are bounded).
Theorem 9.16. For any non-real λ holds T ∗ = T ˙+Eλ as a topological
direct sum.
Corollary 9.17. If Im λ > 0 then dim Dλ = n+, dim D λ = n−.
The proofs of Theorems 9.16 and 9.17 is the same as for Theorems
9.6 and 9.7 respectively, and are left as exercises.