Linear Algebra Done Right · Problem 7B.18

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · spectral-theorem

Problem

Give an example of a real inner product space

V

, an operator

T \in \mathcal{L}(V)

, and real numbers

b, c

with

b^2 < 4c

such that

T^2 + bT + cI

is not invertible. This exercise shows that the hypothesis that

T

is self-adjoint cannot be deleted in 7.26, even for real vector spaces.