Abstract
In this paper, we introduce a new family of operators which is called doubly polynomially normal. An operator T is an element of L-b(H) is said to be doubly polynomially or (P, Q)-normal if there exist two polynomials P and Q such that P(z) = Sigma(0 <= k <= n) a(k)z(k)is an element of poly(1)(C) and Q(z)=Sigma(0 <= j <= m) b(j)z(j)is an element of poly(1)(C) if it satisfies
P(T)Q(T*) - Q(T*)P(T)=0,
or equivalently
Sigma(0 <= k <= n) (0 )(<= j <= m) a(k)b(j)(T-k(T*)(j) - (T*)T-j(k)) = 0.