WEYL'S THEOREMS FOR POSINORMAL OPERATORS
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Abstract
An operator T belonging to the algebra B(H) of bound-ed linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator P 2 B(H)such that TT¤ = T¤PT. A posinormal operator T is said to be condi-tionally totally posinormal (resp., totally posinormal), shortened to T 2 CTP (resp., T 2 TP), if to each complex number ¸ there corre-sponds a positive operator P¸ such that j(T¡¸I)¤j2 = jP 1/2¸ (T¡¸I)j2(resp., if there exists a positive operator P such that j(T ¡¸I)¤j2 =jP1/2 (T ¡ ¸I)j2 for all ¸). This paper proves Weyl's theorem type results for TP and CTP operators. If A 2 TP, if B¤ 2 CTP is isoloid and if dAB 2 B(B(H)) denotes either of the elementary op-erators ±AB(X) = AX ¡XB and 4AB(X) = AXB ¡X, then it is proved that dAB satis¯esWeyl's theorem and d¤AB satis¯es a-Weyl's theorem.
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