16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.5.11

Operators on Inner Product Spaces

Mathematics — Linear Algebra · roots-commuting-normal-operators

Problem

Let

\mathfrak{F}

be a commuting family of diagonalizable normal operators on a finite-dimensional inner product space

V

; and let

\mathfrak{A}

be the self-adjoint algebra generated by

\mathfrak{F}

and the identity operator

I

. Show that each root of

\mathfrak{A}

is different from

0

, and that for each root

r

\mathfrak{F}

there is a unique root

s

\mathfrak{A}

such that

s(T) = r(T)

for all

T

\mathfrak{F}