Kunze Linear Algebra · Problem 10

Operators on Inner Product Spaces

Mathematics — Methods · ordering-of-operators

Problem

Let

V

be a finite-dimensional inner product space. If

T

and

U

are linear operators on

V

, we write

T < U

U - T

is a positive operator. Prove the following: (a)

T < U

and

U < T

is impossible. (b) If

T < U

and

U < S

, then

T < S

. (c) If

T < U

and

0 < S

, it need not be that

ST < SU