This is the possibly infinite dimensional version of a Hermitian matrix, since linear operators are the possibly infinite dimensional version of matrices.
There's a catch though: now we don't have explicit matrix indices here however in general, the generalized definition is shown at: en.wikipedia.org/w/index.php?title=Hermitian_adjoint&oldid=1032475701#Definition_for_bounded_operators_between_Hilbert_spaces
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A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In mathematical terms, a matrix \( A \) is Hermitian if it satisfies the condition: \[ A = A^* \] where \( A^* \) denotes the conjugate transpose of \( A \).