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

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · injective-surjective-invertibility

Problem

(a) Prove or give a counterexample: If

T

is a bounded operator on a Hilbert space such that

T

and

T^*

are both injective, then

T

is invertible. (b) Prove or give a counterexample: If

T

is a bounded operator on a Hilbert space such that

T

and

T^*

are both surjective, then

T

is invertible.