Factorization of generalized holomorphic curve and homogeneity of operators

Hou, Yingli; Ji, Kui; Zhao, Ling

doi:10.1007/s43037-021-00127-9

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2023

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“…Similar transformations break rigidity, so there are not many geometrically similar equivalent results for such operators. Recently, Cowen-Douglas operators with certain structures have been studied [15,21,22,25,26,29,30].…”

Section: Introductionmentioning

confidence: 99%

The irreducibility and strong irreducibility of operators

Jiang¹,

Wang²

2023

Strongly Irreducible Operators on Hilbert Space

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To construct more homogeneous operators, B. Bagchi and G. Misra in [2] introduced the operator T0 T0−T1 0 T1and proved that when T 0 and T 1 are homogeneous operators with the same unitary representation U (g), it is homogeneous with associated representation U (g) ⊕ U (g). At the same time, they asked an open question, is the constructed operator irreducible? A. Korányi in [31] showed that when the (1,2)-entry of the matrix is α(T 0 − T 1 ), α ∈ C the above result is also valid, and their unitary equivalence class depends only on |α|. In this case, he and S. Hazra [20] gave a large class of irreducible homogeneous bilateral 2 × 2 block shifts, respectively, which are mutually unitarily inequivalent for α > 0. In this note, we generalize the construction to T = T0 XT1−T0X 0 T1and provide some sufficient conditions for its irreducibility. We also find that for the above-mentioned T 0 , T 1 and non-scalar operator X, T is weakly homogeneous rather than homogeneous. So the weak homogeneity problem related to T is investigated.

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