Similarity Classification of Cowen-Douglas Operators

[摘要] Let $cal H$ be a complex separable Hilbert spaceand ${cal L}({cal H})$ denote the collection ofbounded linear operators on ${cal H}$.An operator $A$ in ${cal L}({cal H})$is said to be strongly irreducible, if${cal A}^{prime}(T)$, the commutant of $A$, has no non-trivial idempotent.An operator $A$ in ${cal L}({cal H})$ is said to a Cowen-Douglasoperator, if there exists $Omega$, a connected open subset of$C$, and $n$, a positive integer, such that(a) ${Omega}{subset}{sigma}(A)={z{in}C; A-z {ext {not invertible}}};$(b) $an(A-z)={cal H}$, for $z$ in $Omega$;(c) $igvee_{z{in}{Omega}}$ker$(A-z)={cal H}$ and(d) $dim ker(A-z)=n$ for $z$ in $Omega$.In the paper, we give a similarity classification of stronglyirreducible Cowen-Douglas operators by using the $K_0$-group ofthe commutant algebra as an invariant.

[学科分类] 数学（综合）