Linear Algebra Done Right · Problem 7D.10

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

is a self-adjoint operator such that

\|Tv\| \leq \|v\|

for all

v \in V

. (a) Show that

I - T^2

is a positive operator. (b) Show that

T + i\sqrt{I - T^2}

is a unitary operator.