Linear Algebra Done Right · Problem 7C.13

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

T \in \mathcal{L}(V)

is self-adjoint and

\alpha \in \mathbf{R}

. (a) Prove that

T - \alpha I

is a positive operator if and only if

\alpha

is less than or equal to every eigenvalue of

T

. (b) Prove that

\alpha I - T

is a positive operator if and only if

\alpha

is greater than or equal to every eigenvalue of

T