In this paper, we introduce new classes of operators related to the class of polynomially normal operators which are described as follows: (ⅰ) w-quasi polynomially normal operators includes polynomially normal operators recently studied in [7, 6]. A bounded linear operator 5 on a complex Hilbert space H is said to be m-quasi polynomially normal operator if there exists a nontrivial polynomial P = Σ_(0≤k≤n)b_kz~k ∈C[z] for which S~(*m)(P(S)S* - S*P(S))S~m = 0 (? Σ_(0≤k≤n)b_k~S~(*m)(S~kS* - S*S~k)S~n = 0),where m is a natural number, (ⅱ) Polynomially C-normal operators includes C-normal operators studied in [13, 23, 25]. An operator S is called polynomially C-normal operator if there exists a nontrivial polynomial P ∈ C[z] and a conjugation operator C on H for which CP(S)S* - S*P(S)C = 0(?Σ_(0≤k≤n)b_k(CS~kS* - S*S~kC) = 0). A detailed study of certain properties of some members of the first class has been presented. However, an initiation to the study of the second class has been given.
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