Kunze Linear Algebra · Problem 4

Operators on Inner Product Spaces

Mathematics — Methods · normal-operators-characterizations

Problem

Let

T

be a linear operator on the finite-dimensional complex inner product space

V

. Prove that the following ten statements about

T

are equivalent. (a)

T

is normal. (b)

\|T\alpha\| = \|T^*\alpha\|

for every

\alpha

V

. (c)

T = T_1 + iT_2

, where

T_1

and

T_2

are self-adjoint and

T_1T_2 = T_2T_1

. (d) If

\alpha

is a vector and

c

a scalar such that

T\alpha = c\alpha

, then

T^*\alpha = \bar{c}\alpha

. (e) There is an orthonormal basis for

V

consisting of characteristic vectors for

T

. (f) There is an orthonormal basis

\mathfrak{B}

such that

[T]_{\mathfrak{B}}

is diagonal. (g) There is a polynomial

g

with complex coefficients such that

T^* = g(T)

. (h) Every subspace which is invariant under

T

is also invariant under

T^*

. (i)

T = NU

, where

N

is non-negative,

U

is unitary, and

N

commutes with

U

. (j)

T = c_1E_1 + \cdots + c_kE_k

, where

I = E_1 + \cdots + E_k

E_iE_j = 0

for

i \neq j

, and

E_i^2 = E_i = E_i^*