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.