Abstract: The concept of a self-conjugate operator in Hilbert space was extended by introducing the definition of a conjugate square-operator. The properties of the operator and the necessary and sufficient conditions for the regular value to exist were studied using the concept and properties of normal operators in Hilbert space, the spectrum mapping principle and analogy. The results obtained show that, whenT*=T2,the spectrum of conjugate square-operatorTis a finite spectrum.