On ( m , P ) -expansive operators: products, perturbation by nilpotents, Drazin invertibility

: A generalisation of m -expansive Hilbert space operators T ∈ B ( H ) [18, 20] to Banach space operators T ∈ B ( X ) is obtained by defining that a pair of operators A , B ∈ B ( X ) is ( m , P )-expansive for some operator P ∈ B ( X ) if (cid:52) mA , B ( ) = and R B = . Unlike m -isometric and m -left invertible operators, commuting products and perturbations by commuting nilpotents of ( m , I )-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B ( H ), with Drazin inverses A d and B d , a sufficient condition proving ( A d , B d ) ∧ ( A , B ) is ( m −1, P )-isometric (resp., ( m − 1, P )-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved. operators MSC: 47A05, 47A55; Secondary47A11, 47B47

Richter [25,Lemma 1] observed that ( , I)-expansive Hilbert space operators are ( , I)-hyperexpansive (see also Athavale [18]; see Gu [26] for the Banach space analogue). This property is inherited by ( , X)expansive Hilbert space pairs (T * , T) for X ≥ . We start in this paper by proving that: if is an n-nilpotent which commutes with A and B , then (A , B ) ∈ (m, X)-left invertible implies (A + N, B + N) ∈ (m + n − )-left invertible (see [5,16,24,27]). These properties do not transfer to (m, X)-expansive operators. We prove that a su cient condition for a successful transfer of these properties to (m, X)-expansive operators is the existence of "partial expansive sequences". The existence of such sequences for left invertible pairs is a (trivial) consequence of the structure of such pairs. We prove our results, including any complementary results, in the following two sections. In addition to the notation and terminology already introduced, any further notation/terminology will be explained at the rst instance of occurence of such notation/terminology.

. (m, P)-hyperexpansive operators on Hilbert and Banach spaces.
An important, and well studied, class of (m, (Indeed, what we are saying here is that an application of powers of the elementary operator E A,B = L A R B preserves the positivity, or otherwise, of our operator Xs: it is however convenient to shorten it to "triplet (A, B, Xs) preserves order".) Order preserving triplets occur naturally.
for all positive integers t (and the triplet (A * , A, m A * ,A (P)) is an order preserving triplet). Again, if A , A and P ∈ B(H) are such that (A , A ) ∈ (m, P)-expansive for all m ≥ t for some positive integer t, then for all integers s ≥ t, for all positive integers t (so that the triplet (A , A , t A ,A (P)) preserves order). Proof. De ne the operator X ∈ B(H) by X = m A,B (P); then X ≥ . If we now let and this, since the triplet (A, B, A, B(X)) preserves order, implies Using an induction argument, it follows that for all posiitve integers n. Since X ≥ implies (L A R B ) n (X) ≥ , letting n go to in nity in the above we havẽ The hypothesis˜ and {A n˜ A,B (X)B n } n≥ is a bounded below decreasing sequence of non-negative operators. Consequently, the sequence converges to a non-negative operator, say Q.
Already we have lim In addition to generalising the result that -expansive B(H) operators are -hyperexpansive [18,25], Theorem 2.2 generalises [19, Theorem 2.1] (by dispensing with the hypothesis that (T * , T) ∈ ( , P)-expansive - [19] considers the case m = -and by proving that (T * , T) is m-hyperexpansive). Observe that for all x ∈ H. A generalisation of (m.P)-expansive operators to Banach space operators is obtained (following Bayart's [7] de nition of Banach space m-isometric operators) as follows: for some integer p ≥ and all x ∈ X [19,21]. A version of Theorem 2.2 holds for (m, P)-expansive Banach space operators.
The following theorem, generalising corresponding results from [19,21], considers the case m = of Theorem 2.2. Let T ∈ m p (P, T, x) denote T ∈ B(X) is (m, P)-expansive in the sense of (de nition) ( ).

Theorem 2.4. If an operator T ∈ B(X) is such that
for some integer p ≥ and operator P ∈ B(X), then T ∈ r p (P, T, x) for all ≤ r ≤ m (i.e., T is (m, P)hyperexpansive in the sense of (4)).
Proof. The proof below being similar to that of Theorem 2.2, we shall be economical.

follows that if T is -expansive, then T is either completely hyperexpansive,or, there exists no n-nilpotent operator, n > , commuting with T. This is absurd. (ii) Again, if T is -expansive, then T n is -expansive for all positive integers n. Suppose that T satis es the commuting product property, i.e. assume T is ( + − =) -expansive and T n is n + -expansive. Then Theorem 2.2 implies that T is (n + )-hyperexpansive. This is false.
Additional hypotheses are required for results, similar to those for (m, X)-isometries, on products and perturbation by commuting nilpotents of m-expansive operators .
Given The following theorem considers products of commuting expansive operators.

Theorem 2.6. Given operators
is an m -partial hyperexpansive sequence for (A , B ) and is an m -partial hyperexpansive sequence for (A , B ), then (A A , B B ) ∈ (m + m − , P)-expansive.
Proof. The commutativity hypotheses on A i and B i imply that L A i and R B i commute for all i = , , hence is an ( n − )-partial hyperexpansive sequence for (T * , T).
Proof. De ning the operator S as in the statement of the theorem, we have Since and N t = for all t ≥ n, S m+ n− −j = for all k ≥ n.
Assume now that k ≤ n − , and consider L m+ n− −j−k  he Drazin inverse A d of A, whenever it exists, is unique. The Drazin invertibility of A implies that is an isolated point of the spectrum of A and there exists a direct sum decomposition X = A p (X) ⊕ A −p (X) = X ⊕ X of X and a direct sum decomposition It is easily seen that The following theorem extends Theorem 3.
for some positive integers p and q, let P ∈ B(H ⊕H , H ⊕H ) have the matrix representation P = P ij i,j= . Then We prove that P ij = for all i = j ≠ , and to this end we start by considering Let r = max(p, q); then A r = B r = . Since (A, B,˜ m A,B (P)) is order preserving, multiplying m A,B (P) by A r−t on the left and by B r−t on the right, we have: Conclusion: P ≤ .
Recall that a B(H) operator P = [P ij ] i,j= ≥ if and only if P , P ≥ and P * = P = P CP for some contraction C [30]. This, since P = , implies P = P = =⇒ P = P ⊕ This completes the proof.
A particularly important case of Theorem 3.2 and 3.3 is the one in which A * = B = T: (T * , T,˜ m T * ,T (P)) then preserves order (resp., doubly preserves order if T is invertible) for all values of m. We have: In particular, if m = and T is Drazin invertible, then ||P Tx|| = ||P x|| = ||P T d x||.
where T is invertible and T q = . Furthermore Letting P ∈ B(X ⊕ X ) have the matrix representation P = [P ik ] i,k= , and letting x = ⊕ x ∈ X ⊕ X , it then follows from ( m p (P, T, x) ≤ implies) m p (P, T, T q− x) ≤ that Considering next m p (P, T, T q− x), and then repeating the process, it is seen that Since this is true for all x ∈ X , we must have