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    Products of self-adjoint operators

    E-HK-83-7

    Area
    Algebra
    Topics
    Adjoint Operators, Linear Algebra
    Status
    draft

    Show that the product of two self-adjoint operators is self-adjoint if and only if the two operators commute.

    Solution

    Let \(S\) and \(T\) be self-adjoint. Then \[ S^*=S, \qquad T^*=T. \] For adjoints of products, \[ (ST)^*=T^*S^* \] [@HK71]. Therefore \[ (ST)^*=TS. \]

    1. If \(S\) and \(T\) commute, then \(ST\) is self-adjoint.

    If \(ST=TS\), then \[ (ST)^*=TS=ST, \] which is exactly the definition of self-adjointness.

    2. If \(ST\) is self-adjoint, then \(S\) and \(T\) commute.

    If \(ST\) is self-adjoint, then \[ ST=(ST)^*. \] But the displayed identity above gives \((ST)^*=TS\). Hence \[ ST=TS. \] Thus the two operators commute.

    Consequently \[ \boxed{ST\text{ is self-adjoint}\iff ST=TS.} \]

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