Structure of Hilbert Space Operators

On commutativity of the commutant of strongly irreducible operator

“…Obviously, strong irreduciblity of an operator is invariant under similarity sense. Moreover, it is shown in [4] and [5] that the class of such operators is a suitable replacement of the Jordan blocks in B(H).…”

“…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).…”

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

“…(2) Extensions of Hilbert modules by Carlson, Clark, Foias, Guo, Didas and Eschmeier (see [DiEs06], [CaCl95], [CaCl97], [Gu99]). (3) K 0 -group and similarity classification by Jiang, Wang, Ji, Guo (see [JJ07], [JWa06], [JGuJ05]. (4) Classification programme of homogeneous operators by Clark, Bagchi, Misra, Sastry and Koranyi (see [MiS90], [BaMi03], [KMi11] and [KMi09]).…”

The approximate Jordan forms of operators on Σ e 1 type Banach spaces

“…It is easy to prove that every strongly irreducible operator on a finite dimensional space admits a Jordan block representation with respect to some basis. As far as we know, Gilfeather [4] and Jiang [17] gave the concept of strongly irreducible operators, respectively. Jiang further pointed out that the strongly irreducible operators can be considered as an approximate replacement of Jordan blocks on infinite dimensional spaces and he hoped that a theorem similar to the Jordan Standard Theorem can be set up with this replacement on infinite-dimensional spaces.…”

“…The work of Herrero, Power and Jiang [13,14,17] has showed that the strongly irreducible operators can be viewed as an approximate replacement of Jordan blocks on infinite-dimensional separable Hilbert spaces and they have answered a number of questions about operator structure of separable Hilbert spaces raised by Herrero and Jiang. For instance, on a complex separable infinite-dimensional Hilbert space, given an operator T with a connected spectrum, they proved that there exists a compact operator K with arbitrary small norm such that T + K is strongly irreducible. Therefore, the set of operators with connected spectra is the norm-closure of the set of strongly irreducible operators [14].…”