Measure, Integration & Real Analysis, 2021a

Section 10C Compact Operators 325
18 Suppose T ∈B(l 2 ) is defined by T(a 1 , a 2 , a 3 ,...)=(a 2 , a 3 , a 4 ,...). Suppose
also that α ∈ F and |α| < 1.
(a) Show that the geometric multiplicity of α as an eigenvalue of T equals 1.
(b) Show that the algebraic multiplicity of α as an eigenvalue of T equals ∞.
19 Prove that the geometric multiplicity of an eigenvalue of a normal operator on a
Hilbert space equals the algebraic multiplicity of that eigenvalue.
20 Prove that every nonzero eigenvalue of a compact operator on a Hilbert space
has finite algebraic multiplicity.
21 Prove that if T is a compact operator on a Hilbert space and α is a nonzero
eigenvalue of T, then the algebraic multiplicity of α as an eigenvalue of T equals
the algebraic multiplicity of α as an eigenvalue of T ∗ .
22 Prove that if V is a separable Hilbert space, then the Banach space C(V) is
separable.
Measure, Integration & Real Analysis, by Sheldon Axler