The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid in that the unitary equivalence class of the operator and the flag structure determine each other. We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.2010 Mathematics Subject Classification. 47B32, 47B35. Key words and phrases. The Cowen-Douglas class, strongly irreducible operator, homogeneous operator, curvature, second fundamental form. 2.A new class of operators in B 2 (Ω) 2.1. Definitions. If T is an operator in B 2 (Ω), then there exists a pair of operators T 0 and T 1 in B 1 (Ω) and a bounded operator S such that T = T 0 S 0 T 1 . This is Theorem 1.49 of [8, page 48]. We show, the other way round, that two operators T 0 and T 1 from B 1 (Ω) combine with the aid of an arbitrary bounded linear operator S to produce an operator in B 2 (Ω).Proposition 2.1. Let T be a bounded linear operator of the form T 0 S 0 T 1 . Suppose that the two operators T 0 , T 1 are in B 1 (Ω). Then the operator T is in B 2 (Ω).Proof. Suppose T 0 and T 1 are defined on the Hilbert spaces H 0 and H 1 , respectively. Elementary considerations from index theory of Fredholm operators shows that the operator T is Fredholm and ind(T ) = ind(T 0 ) + ind(T 1 ) (cf. [2, page 360]). Therefore, to complete the proof that T is in B 2 (Ω), all we have to do is prove that the vectors in the kernel ker(T − w), w ∈ Ω, span the Hilbert space H = H 0 ⊕ H 1 .Let γ 0 and t 1 be non-vanishing holomorphic sections for the two line bundles E 0 and E 1 corresponding to the operators T 0 and T 1 , respectively. For each w ∈ Ω, the operator T 0 − w 0

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Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist , a connected open subset of complex plane C, and n, a positive integer, such thatIn the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; ଁ

Abstract. Let A be an AI algebra; that is, A is the C * -algebra inductive limit of a sequence