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Linear Algebra, Optimisation and Numerical Methods / Adjoints, Self-Adjoint, and Normal Operators

Normal Operators

The previous note studied operators satisfying T∗=TT^*=T. This note widens the lens to normal operators — those satisfying the weaker condition TT∗=T∗TTT^*=T^*T — the class properly containing both self-adjoint operators and unitary/orthogonal operators (the next note). Normal operators are exactly the class for which the (complex) spectral theorem holds in its cleanest form — they always have an orthonormal eigenbasis, even though (unlike self-adjoint operators) their eigenvalues need not be real.
Learning objectives. By the end of this note you should be able to:
  • verify whether an operator is normal;
  • prove that self-adjoint operators are always normal;
  • prove the key norm identity characterising normality;
  • explain why normality, not self-adjointness, is the natural hypothesis for the complex spectral theorem.

In this lesson

  1. Definition and the containing hierarchy
  2. The characterising norm identity

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