Scalar perturbations of the Sz.-Nagy—Foias characteristic function

Bagchi, Bhaskar; Misra, Gadadhar

doi:10.1007/978-3-0348-8374-0_5

Cited by 3 publications

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“…. Then T = S −1 (T 0 ⊕T 0 )S. We know that similar transformations do not change the spectrum, then σ(T ) = σ(T 0 ⊕ T 0 ) = σ(T 0 ) = D, since the spectrum of a weakly homogeneous operator is T or D by using the spectral mapping theorem (see [3]).…”

Section: Strong Irreducible Decompositions and Weak Homogeneity Letmentioning

confidence: 98%

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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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Section: Strong Irreducible Decompositions and Weak Homogeneity Letmentioning

confidence: 98%

“…Operator T ∈ L(H) is said to be homogeneous if σ(T ) ⊂ D, the closed unit disc and φ(T ) ∼ u T for every φ in M öb (see [3,35,37]).…”

Section: Homogeneous Operators Let M öB Denote the M öBious Group Of ...mentioning

confidence: 99%