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=== Normal operators ===
Every normal operator is subnormal by definition, but the converse is not true in general. A simple class of examples can be obtained by weakening the properties of [[unitary operator]]s. A unitary operator is an isometry with [[dense set|dense]] [[range (mathematics)|range]]. Consider now an isometry ''A'' whose range is not necessarily dense. A concrete example of such is the [[unilateral shift]], which is not normal. But ''A'' is subnormal and this can be shown explicitly. Define an operator ''U'' on
:<math>H \oplus H</math>
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