A note on the dilation of a certain family of tetrablock contractions
Tirthankar Bhattacharyya, Mainak Bhowmik

TL;DR
This paper constructs explicit tetrablock isometric dilations for a family of tetrablock contractions, clarifying conditions under which such dilations exist, and addresses a previously claimed non-dilatable example.
Contribution
It provides explicit dilation constructions for a family of tetrablock contractions and establishes necessary and sufficient conditions for their dilations, resolving a debated example.
Findings
Explicit dilation for the family of tetrablock contractions.
Construction of dilations for the adjoint contractions.
Necessary and sufficient conditions for tetrablock isometric dilations.
Abstract
We find an explicit tetrablock isometric dilation for every member of a family of tetrablock contractions indexed by a parameter in the closed unit disc (only the first operator of the tetrablock contraction depends on the parameter). The dilation space is the same for any member of the family. Explicit dilation for the adjoint tetrablock contraction for every member of the family mentioned above is constructed as well. This example is important because it has been claimed in the literature that this example does not have a dilation. Taking cue from this construction and using Toeplitz operators on , we obtain necessary and sufficient conditions for a tetrablock contraction to have a certain type of tetrablock isometric dilation.
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Taxonomy
TopicsHolomorphic and Operator Theory · Geometry and complex manifolds · Analytic and geometric function theory
