Assume that $$T(f) = \int\_D K(z,w) f(w) du dv+\int\_D H(z,w) \overline{f(w)} du dv$$ is an operator. It seems to me that its conjugate operator is $$T^\*(f) = \int\_D \overline{K(w,z)}f(w) du dv+\int\_D \overline{H(w,z)} \cdot \overline{f(w)} du dv$$ however something is wrong. Here $D$ is for example the unit disk in the complex plane. What is wrong?



Conjugate operator
Let me define by $C$ the operator of complex conjugation, then $T= K+HC$; I presume by the conjugate operator $T^\ast$ you mean the Hermitian adjoint of $T$; as discussed [here](https://physics.stackexchange.com/questions/510383/is-complex-conjugation-operator-hermitian), $C^\ast=C$, so $T^\ast=K^\ast+CH^\ast=K^\ast+H^{\rm t}C$, with $H^t$ the transpose. Hence
$$T^\*(f)[z] = \int\_D \overline{K(w,z)}f(w) \,dw+\int\_D H(w,z) \overline{f(w)}\, dw.$$


