m-Isometric Operators with Null Symbol and Elementary Operator Entries

A pair (A,B) of Banach space operators is strict (m,X)-isometric for a Banach space operator X∈B(X) and a positive integer m if ▵A,Bm(X)=∑j=0mmjLAjRBj(X)=0 and ▵A,Bm−1(X)≠0, where LA and RB∈B(B(X)) are, respectively, the operators of left multiplication by A and right multiplication by B. Define operators EA,B and EA,B(X) by EA,B=LARB and EA,B(X)n=EA,Bn(X) for all non-negative integers n. Using little more than an algebraic argument, the following generalised version of a result relating (m,X)-isometric properties of pairs (A1,A2) and (B1,B2) to pairs (EA1,A2(S1),EB1,B2(S2)) and (EA1,A2,EB1,B2) is proved: if Ai,Bi,Si,X are operators in B(X), 1≤i≤2 and X a quasi-affinity, then the pair (EA1,A2(S1),EB1,B2(S2)) (resp., the pair (EA1,A2,EB1,B2)) is strict (m,X)-isometric for all X∈B(X) if and only if there exist positive integers mi≤m, 1≤i≤2 and m=m1+m2−1, and a non-zero scalar β such that I−EβA1,A2(S1) is (strict) m1-nilpotent and I−E1βB1,B2(S2) is (strict) m2-nilpotent (resp., (βA1,B1) is strict (m1,I)-isometric and (1βB2,A2) is strict (m2,I)-isometric).