Abstract: The concept of a special normal operator, self-conjugate operator, in Hilbert space was extended to a polynomial conjugate operator. Based on the properties and spectrum feature of a normal operator in Hilbert space, the spectrum of the extended operator—polynomial conjugate operatorTwas investigated using the analogy method and the Banach algebra technique. The result shows that the spectrum of the polynomial conjugate operator is a finite eigenvalue spectrum. The conclusion was proved by using an example