toSelfAdjoint — Mathlib

English

For a star-normal operator T, the equation adjoint T x = 0 is equivalent to T x = 0 for any x.

Русский

Для оператора нормального по звёздочке, равенство adjoint T x = 0 эквивалентно T x = 0.

LaTeX

$$$adjoint(T)\\,x = 0 \\iff T\\,x = 0$$$

Lean4

/-- The **Hellinger--Toeplitz theorem**: Construct a self-adjoint operator from an everywhere
  defined symmetric operator. -/
def toSelfAdjoint (hT : IsSymmetric T) : selfAdjoint (E →L[𝕜] E) :=
  ⟨⟨T, hT.continuous⟩, ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr hT⟩

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