16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.5.10

Operators on Inner Product Spaces

Mathematics — Linear Algebra · roots-self-adjoint-algebra

Problem

Let

\mathfrak{F}

be any family of linear operators on a finite-dimensional inner product space

V

and

\mathfrak{A}

the self-adjoint algebra generated by

\mathfrak{F}

. Show that (a) each root of

\mathfrak{A}

defines a root of

\mathfrak{F}

; (b) each root

r

\mathfrak{A}

is a multiplicative linear function on

A

, i.e.,

r(TU) = r(T)r(U)

r(cT + U) = cr(T) + r(U)

for all

T

and

U

\mathfrak{A}

and all scalars

c