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  "body": "\n## Summary\n\nA subnormal operator is a bounded operator on a Hilbert space that is the restriction of a normal operator to an invariant subspace.\nEvery normal operator is subnormal, and every subnormal operator is [[hyponormal|35XKRHEC3Q]].\nThe unilateral shift is the basic example, and multiplication by $z$ on spaces of analytic functions gives many more.\nScott Brown showed that subnormal operators have nontrivial invariant subspaces.\n\n## Explanations\n\n### L1 human\n\nA subnormal operator is a normal operator seen only on part of the space that it maps into itself.\nMultiplication by $z$ on square-integrable functions on the unit circle is normal.\nOn the smaller space of functions that extend analytically into the disc it is no longer normal, but it is subnormal.\nIn finite dimensions a subnormal operator is already normal, so the notion only matters in infinite dimensions.\n",
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