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

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · surjective-not-injective-null-spaces

Problem

Show that if

T

is a surjective but not injective operator on a vector space

V

, then

\mathrm{null}\, T \subsetneq \mathrm{null}\, T^2 \subsetneq \mathrm{null}\, T^3 \subsetneq \cdots.