Measure, Integration & Real Analysis, 2021a

296 Chapter 10 Linear Maps on Hilbert Spaces
10.36 spectrum is closed
The spectrum of a bounded operator on a Banach space is a closed subset of F.
Proof Suppose T is a bounded operator on a Banach space V. Suppose α 1 , α 2 ,...
is a sequence in sp(T) that converges to some α ∈ F. Thus each T − α n I is not
invertible and
lim
n→∞ (T − α nI) =T − αI.
The set of noninvertible elements of B(V) is a closed subset of B(V) (by 10.25).
Hence the equation above implies that T − αI is not invertible. In other words,
α ∈ sp(T), which implies that sp(T) is closed.
Our next result provides the key tool used in proving that the spectrum of a
bounded operator on a nonzero complex Hilbert space is nonempty (see 10.38). The
statement of the next result and the proofs of the next two results use a bit of basic
complex analysis. Because sp(T) is a closed subset of C (by 10.36), C \ sp(T) is
an open subset of C and thus it makes sense to ask whether the function in the result
below is analytic.
To keep things simple, the next two results are stated for complex Hilbert spaces.
See Exercise 6 for the analogous results for complex Banach spaces.
10.37 analyticity of (T − αI) −1
Suppose T is a bounded operator on a complex Hilbert space V. Then the function
α ↦→ 〈 (T − αI) −1 f , g 〉
is analytic on C \ sp(T) for every f , g ∈ V.
Proof Suppose β ∈ C \ sp(T). Then for α ∈ C with |α − β| < 1/‖(T − βI) −1 ‖,
we see from 10.22 that I − (α − β)(T − βI) −1 is invertible and
(
I − (α − β)(T − βI)
−1 ) ∞
−1 = ∑ (α − β) k( (T − βI) −1) k .
k=0
Multiplying both sides of the equation above by (T − βI) −1 and using the equation
A −1 B −1 =(BA) −1 for invertible operators A and B,weget
(T − αI) −1 =
Thus for f , g ∈ V,wehave
〈
(T − αI) −1 f , g 〉 =
∞
∑
k=0
∞
∑
k=0
(α − β) k( (T − βI) −1) k+1 .
〈 ((T − βI)
−1 ) 〉
k+1 f , g (α − β) k .
The equation above shows that the function α ↦→ 〈 (T − αI) −1 f , g 〉 has a power series
expansion as powers of α − β for α near β. Thus this function is analytic near β.
Measure, Integration & Real Analysis, by Sheldon Axler