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COVARIANCE PROPERTIES Since the (n,m) abstract Pad@ approximant is an equivalence-class containing couples of abstract polynomials, we are going to represent it by one of its elements; for the sake of sfunplicity we will denote this representant also by (P,,Q,). Let the operator Pn,m (for n and m chosen) associate with the operator F the equivalence-class of the (P,,Q,). we are looking for operators @ working on F that commute more or less with the Pad@ operator ~,m: Pn,m(F)] = Pn@,m[~(F)] with n¢ and m@ depending on the considered @.

I0. I), normality is stronger than regularity. 50 Theorem I. I I. I. 810 ,. The (n,m) APA ~ . P , for F is normal if and only if 81P. ) = 8o0*+n+m+1. 1 c). 1 b) we have n_~81P . and m _< 81Q ,. 2 b) we also have 81P . _< n and 810. -< m. So n = 81P . and m = 8 1*0 . According to theorem I. 10. ) = ~o0 +n+m+] The proof goes by contraposition. 1 c) we have in any case that n _~ i and m -~ j). j+S-8oQ . Q,-P,). D s] = i. j+n+m+s÷ I. So s > i-n or s > j-re. This is in contradiction with theorem I. 5,3.

I, 81P . = 3, aiQ . ) = 8 and thus it is also a normal element in the abstract Pad@-table. § 12. ,p < ~. 54 P The space X = FI Xi, normed by one of the following Minkowski-norms i=1 P i)) 1/q IIXllq = ( ; llxill I i=l P Z IFxill(i ) i=1 tlxlt 1 = Ifxtl~ = max(llXlll (1) . . . IlxplF(p)) where IlxiH(i ) is the norm of x i in Xi and x = (x 1 . . ,xp), is a l s o a Banach space. We i n t r o d u c e the f o l l o w i n g n o t a t i o n s JY< = (x 1 . . . x,j, xj_ 1, O, xj+ 1 . . . = (x 1 .