Measure, Integration & Real Analysis, 2021a

294 Chapter 10 Linear Maps on Hilbert Spaces
10B
Spectrum
Spectrum of an Operator
The following definitions play key roles in operator theory.
10.32 Definition eigenvalue; eigenvector; spectrum; sp(T)
Suppose T is a bounded operator on a Banach space V.
• A number α ∈ F is called an eigenvalue of T if T − αI is not injective.
• A nonzero vector f ∈ V is called an eigenvector of T corresponding to an
eigenvalue α ∈ F if
Tf = α f .
• The spectrum of T is denoted sp(T) and is defined by
sp(T) ={α ∈ F : T − αI is not invertible}.
If T − αI is not injective, then T − αI is not invertible. Thus the set of eigenvalues
of a bounded operator T is contained in the spectrum of T. IfV is a finite-dimensional
Banach space and T ∈B(V), then T − αI is not injective if and only if T − αI is
not invertible. Thus if T is an operator on a finite-dimensional Banach space, then
the spectrum of T equals the set of eigenvalues of T.
However, on infinite-dimensional Banach spaces, the spectrum of an operator does
not necessarily equal the set of eigenvalues, as shown in the next example.
10.33 Example eigenvalues and spectrum
Verifying all the assertions in this example should help solidify your understanding
of the definition of the spectrum.
• Suppose b 1 , b 2 ,...is a bounded sequence in F. Define a bounded linear map
T : l 2 → l 2 by
T(a 1 , a 2 ,...)=(a 1 b 1 , a 2 b 2 ,...).
Then the set of eigenvalues of T equals {b k : k ∈ Z + } and the spectrum of T
equals the closure of {b k : k ∈ Z + }.
• Suppose h ∈L ∞ (R). Define a bounded linear map M h : L 2 (R) → L 2 (R) by
M h f = fh.
Then α ∈ F is an eigenvalue of M h if and only if |{t ∈ R : h(t) =α}| > 0.
Also, α ∈ sp(M h ) if and only if |{t ∈ R : |h(t) − α| < ε}| > 0 for all ε > 0.
• Define the right shift T : l 2 → l 2 and the left shift S : l 2 → l 2 by
T(a 1 , a 2 , a 3 ,...)=(0, a 1 , a 2 , a 3 ,...) and S(a 1 , a 2 , a 3 ,...)=(a 2 , a 3 , a 4 ,...).
Then T has no eigenvalues, and sp(T) ={α ∈ F : |α| ≤1}. Also, the set of
eigenvalues of S is the open set {α ∈ F : |α| < 1}, and the spectrum of S is the
closed set {α ∈ F : |α| ≤1}.
Measure, Integration & Real Analysis, by Sheldon Axler