Measure, Integration & Real Analysis · Problem 10C.11

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · compact-operators-fredholm

Problem

Suppose

T

is a compact operator on a Hilbert space and

\alpha \in \mathbf{F} \setminus \{0\}

. ((a)) Prove that

\mathrm{range}(T - \alpha I)^{m-1} = \mathrm{range}(T - \alpha I)^m

for some

m \in \mathbf{Z}^+

. ((b)) Prove that

\mathrm{null}(T - \alpha I)^{n-1} = \mathrm{null}(T - \alpha I)^n

for some

n \in \mathbf{Z}^+

. ((c)) Show that the smallest positive integer

m

that works in (a) equals the smallest positive integer

n

that works in (b).