Similarity Classification of Cowen-Douglas Operators

Abstract

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 strongly irreducible, if A ′ (T), the commutant of A, has no non-trivial idempotent. An operator A in L(H) is said to be a Cowen-Douglas operator, if there exists Ω, a connected open subset of C, and n, a positive integer, such thatIn 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.

Algebra matrix and similarity classification of operators

“…Set S = (S ij ) n×n ∈ I, then S(i, j) = 1 R (i, i)S1 R (j, j) ∈ I, where S(i, j) is defined as (1). So I is a matrix ring of order n. Since R is a matrix ring, a simple computation shows that I i = R i or I i is an ideal of R i for 1 i n.…”

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.…”

On commutativity of the commutant of strongly irreducible operator

“…Also, Jiang [2] proved that if T is a Cowen-Douglas operator and T ∈ SI, then A (T )/rad A (T ) is commutative. Based on these conclusions, they conjectured that the answer to the above problem is positive ( [5], p. 85).…”