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1. CJM 2004 (vol 56 pp. 742)
Similarity Classification of Cowen-Douglas Operators Let $\cal H$ be a complex separable Hilbert space
and ${\cal L}({\cal H})$ denote the collection of
bounded 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-Douglas
operator, if there exists $\Omega$, a connected open subset of
$C$, and $n$, a positive integer, such that
(a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$
(b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$;
(c) $\bigvee_{z{\in}{\Omega}}$\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 strongly
irreducible Cowen-Douglas operators by using the $K_0$-group of
the commutant algebra as an invariant.
Categories:47A15, 47C15, 13E05, 13F05 |