K-group and similarity classification of operators

Abstract

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; ଁ

Similarity Classification and Properties of Some Extended Holomorphic Curves

“…They later proved the congruence theorem for the tuples of elements of a C * -algebra in the CowenDouglas class. Since 2002, Chunlan Jiang, Junsheng Fang and many other people obtained a similarity classification for Cowen-Douglas operators by using the K 0 group of the commutant of single operators as an invariant [5][6][7]. In the spirit of the above work, our aim is to characterize the similarity of extended holomorphic curves.…”

The commutant and similarity invariant of analytic Toeplitz operators on Bergman space

“…In this section, we will characterize the similarity invariant of some analytic Toeplitz operators using Jiang's similarity classification theorem (see [16,17]) for Cowen-Douglas operators. Here we briefly recall some preparation knowledges.…”

“…Moreover, A (T )/radA (T ) is commutative for each T in B n (Ω) ∩ (SI). [17] . (2) the isomorphism h from ∨(A (M F1 ⊕M F2 )) to N (k) sends [I] to (2n 1 , 2n 2 , .…”

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

“…(2) Extensions of Hilbert modules by Carlson, Clark, Foias, Guo, Didas and Eschmeier (see [DiEs06], [CaCl95], [CaCl97], [Gu99]). (3) K 0 -group and similarity classification by Jiang, Wang, Ji, Guo (see [JJ07], [JWa06], [JGuJ05]. (4) Classification programme of homogeneous operators by Clark, Bagchi, Misra, Sastry and Koranyi (see [MiS90], [BaMi03], [KMi11] and [KMi09]).…”