On m-Isometric and m-Symmetric Operators of Elementary Operators

Given Hilbert space operators A,B and X, let △A,B and δA,B denote, respectively, the elementary operators △A,B(X) = I − AXB and the generalised derivation δA,B(X) = AX − XB. This paper considers the structure of operators Dm d1,d2 (I) = 0 and Dm d1,d2 compact, where m is a positive integer, D =△ or δ, d1 =△A∗,B∗ or δA∗,B∗ and d2 = △A,B or δA,B. This is a continuation of the work done by C. Gu for the case △m δA∗,B∗, δA,B (I) = 0, and the author with I.H. Kim for the cases △m δA∗,B∗,δA,B (I) = 0 or △m δA∗,B∗,δA,B is compact, and δm △A∗,B∗,△A,B (I) = 0 or δm △A∗,B∗,δA,B is compact. Operators Dm d1,d2 (I) = 0 are examples of operators with finite spectrum, indeed the operators A,B have at most a two point spectrum, and if Dm d1,d2 is compact, then (the non-nilpotent operators) A, B are algebraic. Dm d1,d2 (I) = 0 implies Dn d1,d2 (I) = 0 for integers n ≥ m: the reverse implication, however, fails. It is proved that Dm d1,d2 (I) = 0 implies Dd1,d2 (I) = 0 if and only if of A and B (are normal, hence) satisfy a Putnam-Fuglede commutativity property.