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 A and B on a finite-dimensional inner product space are commutative if and only if there exists a self-adjoint transformation C and there exist two real-valued functions f and g of a real variable so that A=f(C) and B=g(C) . If such a C exists, then we may even choose C in the form C=h(A, B) , where h is a suitable real-valued function of two real variables.
Proof. The sufficiency of the condition is clear; we prove only the necessity.
Let A=\sum_{i} \alpha_{i} E_{i} and B=\sum_{j} \beta_{j} F_{j} be the spectral forms of A and B ; since A and B commute, it follows from Section: Spectral theorem , Theorem 3, that E_{i} and F_{j} commute. Let h be any function of two real variables such that the numbers h(\alpha_{i}, \beta_{j})=\gamma_{i j} are all distinct, and write C=h(A, B)=\sum_{i} \sum_{j} h(\alpha_{i}, \beta_{j}) E_{i} F_{j}. (It is clear that h may even be chosen as a polynomial, and the same is true of the functions f and g we are about to describe.) Let f and g be such that f(\gamma_{i j})=\alpha_{i} and g(\gamma_{i j})=\beta_{j} for all i and j . It follows that f(C)=A and g(C)=B , and everything is proved. ◻
Theorem 2. If A is a normal transformation on a finite-dimensional unitary space and if B is an arbitrary transformation that commutes with A , then B commutes with A^{*} .
Proof. Let A=\sum_{i} \alpha_{i} E_{i} be the spectral form of A ; then A^{*}=\sum_{i} \bar{\alpha}_{i} E_{i} . Let f be such a function (polynomial) of a complex variable that f(\alpha_{i})=\bar{\alpha}_{i} for all i . Since A^{*}=f(A) , the conclusion follows. ◻
EXERCISES
Exercise 1.
- Prove the following generalization of Theorem 2: if A_{1} and A_{2} are normal transformations (on a finite-dimensional unitary space) and if A_{1} B=B A_{2} , then A_{1}^{*} B=B A_{2}^* .
- Theorem 2 asserts that the relation of commutativity is sometimes transitive: if A^{*} commutes with A and if A commutes with B , then A^{*} commutes with B . Does this formulation remain true if A^{*} is replaced by an arbitrary transformation C ?
Exercise 2.
- If A commutes with A^{*} A , does it follow that A is normal?
- If A^{*} A commutes with A A^{*} , does it follow that A is normal?
Exercise 3.
- A linear transformation A is normal if and only if there exists a polynomial p such that A^{*}=p(A) .
- If A is normal and commutes with B , then A commutes with B^{*} .
- If A and B are normal and commutative, then A B is normal.
Exercise 4. If A and B are normal and similar, then they are unitarily equivalent.
Exercise 5.
- If A is Hermitian, if every proper value of A has multiplicity 1 , and if A B=B A , then there exists a polynomial p such that B=p(A) .
- If A is Hermitian, then a necessary and sufficient condition that there exist a polynomial p such that B=p(A) is that B commute with every linear transformation that commutes with A .
Exercise 6. Show that a commutative set of normal transformations on a finite-dimensional unitary space can be simultaneously diagonalized.
