Operators in the Cowen-Douglas Class and Related Topics

Abstract

Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although, the study of self adjoint operators goes back a few decades, the non-self adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like A (Ω), where Ω ⊆ C m is a bounded domain, consisting of complex valued functions which are holomorphic on some open set U containing Ω, the closure of Ω. The book [29] showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The books [31] and [14] provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.such that the vectors {γ 1 (w ), . . . , γ n (w )} are linearly independent, w ∈ U . They also show that such an operator T defines a holomorphic Hermitian vector bundle E T :This means, for any fixed but arbitrary point w 0 ∈ Ω, there exists a holomorphic map γ T of the form γ T (w ) = (γ 1 (w ), . . ., γ n (w )), (T −w )γ i (w ) = 0 in some open neighbourhood U of w 0 . It is called a holomorphic frame for the operator T . Finally, Cowen and Douglas also assume that the linear span of {γ 1 (w ), . . ., γ n (w ) : w ∈ Ω} is dense in H . Let B n (Ω) denote this class of operators.One of the striking results of Cowen and Douglas says that there is a one to one correspondence between the unitary equivalence class of the operators T and the (local) equivalence classes of the holomorphic Hermitian vector bundles E T determined by them. As a result of this correspondence set up by the Cowen-Douglas theorem, the invariants the vector bundle E T like the curvature, the second fundamental form, etc.