Linear Algebra Done Right · Problem 7D.7

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Give an example of two self-adjoint operators

T_1, T_2 \in \mathcal{L}(\mathbf{F}^4)

such that the eigenvalues of both operators are

2, 5, 7

but there does not exist a unitary operator

S \in \mathcal{L}(\mathbf{F}^4)

such that

T_1 = S^* T_2 S

. Be sure to explain why there is no unitary operator with the required property.