Weyl's theorems for posinormal operators

An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator P is an element of B(H) such that TT* = T* PT. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to T is an element of CTP (resp., T is an element of TP), if to each complex number lambda there corresponds a positive operator P-lambda such that vertical bar(T-lambda I)*vertical bar(2) = vertical bar P-lambda(1/2) (T-lambda I)vertical bar(2) (resp., if there exists a positive operator P such that vertical bar(T -lambda I)*vertical bar(2) = vertical bar P-1/2(T-lambda I)vertical bar(2) for all lambda). This paper proves Weyl's theorem type results for TP and CTP operators. If A is an element of TP, if B* is an element of CTP is isoloid and if d(AB) is an element of B(B(H)) denotes either of the elementary operators delta(AB)(X) = AX - XB and Delta(AB)(X) = AXB - X, then it is proved that d(AB) satisfies Weyl's theorem and d(AB)* satisfies alpha-Weyl's theorem.