$\Gamma$-Unitaries, Dilation and a Natural Example

Tirthankar Bhattacharyya; Haripada Sau

doi:10.4171/prims/53-2-2

JournalsprimsVol. 53, No. 2pp. 261–285

$Γ$ -Unitaries, Dilation and a Natural Example

Tirthankar Bhattacharyya
Indian Institute of Science, Bangalore, India
Haripada Sau
Indian Institute of Science, Bangalore, India

$$\Gamma$-Unitaries, Dilation and a Natural Example cover$

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Abstract

This note constructs an explicit normal boundary dilation for a commuting pair $(S, P)$ of bounded operators with the symmetrized bidisk

Γ = {(z_{1} + z_{2}, z_{1} z_{2}) : ∣ z_{1} ∣, ∣ z_{2} ∣ \leq 1}

as a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk [11], the unit bidisk [3] and in the tetrablock [6]. The dilation is minimal and unique under a suitable condition. This paper also contains a natural example of a $Γ$ -isometry. We compute its associated fundamental operator.

Cite this article

Tirthankar Bhattacharyya, Haripada Sau, $Γ$ -Unitaries, Dilation and a Natural Example. Publ. Res. Inst. Math. Sci. 53 (2017), no. 2, pp. 261–285

DOI 10.4171/PRIMS/53-2-2