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. 

  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 \(A_{1}^{*} B=B A_{2}^*\) .
  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. 

  1. If \(A\) commutes with \(A^{*} A\) , does it follow that \(A\) is normal?
  2. If \(A^{*} A\) commutes with \(A A^{*}\) , does it follow that \(A\) is normal?

Exercise 3. 

  1. A linear transformation \(A\) is normal if and only if there exists a polynomial \(p\) such that \(A^{*}=p(A)\) .
  2. If \(A\) is normal and commutes with \(B\) , then \(A\) commutes with \(B^{*}\) .
  3. 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. 

  1. 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 \(B=p(A)\) .
  2. 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.