Essential spectra of operators in the class ℬ_{𝓃}(Ω)

Abstract

For a connected open subset Ω \Omega of the plane and n n a positive integer, let B n ( Ω ) {\mathcal {B}_n}(\Omega ) be the space introduced by Cowen and Douglas in their paper Complex Geometry and Operator Theory. Our paper deals with characterizing the essential spectrum of an operator T T in B n ( Ω ) {\mathcal {B}_n}(\Omega ) for which σ ( T ) = Ω ¯ \sigma (T) = \bar \Omega and the point spectrum of T ∗ {T^ * } is empty. This class of operators forms an important part of B n ( Ω ) {\mathcal {B}_n}(\Omega ) denoted by B n ′ ( Ω ) {\mathcal {B}’_n}(\Omega ) . We use this characterization to give another proof of the result of Axler, Conway and McDonald on determining the essential spectrum of the Bergman operator. Let A n ( G ) = { S : T = S ∗ is in B ′ n ( G ∗ ) } {A_n}(G) = \left \{ {S:T = {S^ * }{\text {is}}\;{\text {in}}{{\mathcal {B}’}_n}({G^ * })} \right \} . We also characterize the weighted shifts in A 1 ( G ) {A_1}(G) .