Commutativity

The spectral theorem for self-adjoint and for normal operators and the functional calculus may also be used to solve certain problems concerning commutativity. This is a deep and extensive subject; more to illustrate some methods than for the actual results we discuss two theorems from it.

Theorem 1. Two self-adjoint transformations and on a finite-dimensional inner product space are commutative if and only if there exists a self-adjoint transformation and there exist two real-valued functions and of a real variable so that and . If such a exists, then we may even choose in the form , where is a suitable real-valued function of two real variables.

Proof. The sufficiency of the condition is clear; we prove only the necessity.

Let and be the spectral forms of and ; since and commute, it follows from Section: Spectral theorem , Theorem 3, that and commute. Let be any function of two real variables such that the numbers are all distinct, and write (It is clear that may even be chosen as a polynomial, and the same is true of the functions and we are about to describe.) Let and be such that and for all and . It follows that and , and everything is proved. ◻

Theorem 2. If is a normal transformation on a finite-dimensional unitary space and if is an arbitrary transformation that commutes with , then commutes with .

Proof. Let be the spectral form of ; then . Let be such a function (polynomial) of a complex variable that for all . Since , the conclusion follows. ◻

EXERCISES

Exercise 1.

Prove the following generalization of Theorem 2: if and are normal transformations (on a finite-dimensional unitary space) and if , then .
Theorem 2 asserts that the relation of commutativity is sometimes transitive: if commutes with and if commutes with , then commutes with . Does this formulation remain true if is replaced by an arbitrary transformation ?

Exercise 2.

If commutes with , does it follow that is normal?
If commutes with , does it follow that is normal?

Exercise 3.

A linear transformation is normal if and only if there exists a polynomial such that .
If is normal and commutes with , then commutes with .
If and are normal and commutative, then is normal.

Exercise 4. If and are normal and similar, then they are unitarily equivalent.

Exercise 5.

If is Hermitian, if every proper value of has multiplicity , and if , then there exists a polynomial such that .
If is Hermitian, then a necessary and sufficient condition that there exist a polynomial such that is that commute with every linear transformation that commutes with .

Exercise 6. Show that a commutative set of normal transformations on a finite-dimensional unitary space can be simultaneously diagonalized.

EXERCISES

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