Kunze Linear Algebra · Problem 12

Operators on Inner Product Spaces

Mathematics — Methods · self-adjoint-algebras-spectral

Problem

Let

\mathfrak{F}

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

V

and

\mathfrak{a}_0

the self-adjoint algebra generated by

\mathfrak{F}

. Let

\mathfrak{a}

be the self-adjoint algebra generated by

\mathfrak{F}

and the identity operator

I

. Show that (a)

\mathfrak{a}

is the set of all operators on

V

of the form

cI + T

where

c

is a scalar and

T

an operator in

\mathfrak{a}_0

. (b) There is at most one root

r

\mathfrak{a}

such that

r(T) = 0

for all

T

\mathfrak{a}_0

. (c) If one of the roots of

\mathfrak{a}

0

\mathfrak{a}_0

, the projections

P_1, \ldots, P_k

in the resolution of the identity defined by

\mathfrak{F}

may be indexed in such a way that

\mathfrak{a}_0

consists of all operators on

V

of the form

T = \sum_{j=2}^{k} c_j P_j

where

c_2, \ldots, c_k

are arbitrary scalars. (d)

\mathfrak{a} = \mathfrak{a}_0

if and only if for each root

r

\mathfrak{a}

there exists an operator

T

\mathfrak{a}_0

such that

r(T) \neq 0