Rational dilation of tetrablock contractions revisited

DOI10.1016/J.JFA.2019.108275zbMATH Open1513.47020arXiv1907.10832MaRDI QIDQ6322665

Publication date: 25 July 2019

Abstract: A classical result of Sz.-Nagy asserts that a Hilbert-space contraction operator

T

can be lifted to an isometry

V

. A more general multivariable setting of recent interest for these ideas is the case where (i) the unit disk is replaced by a certain domain contained in

{m a t h b b C}^{3}

(called the {em tetrablock}), (ii) the contraction operator

T

is replaced by a commutative triple

(T_{1}, T_{2}, T)

of Hilbert-space operators having

m a t h b b E

as a spectral set (a tetrablock contraction) . The rational dilation question for this setting is whether a tetrablock contraction

(T_{1}, T_{2}, T)

can be lifted to a tetrablock isometry

(V_{1}, V_{2}, V)

(a commutative operator tuple which extends to a tetrablock-unitary tuple

(U_{1}, U_{2}, U)

---a commutative tuple of normal operators with joint spectrum contained in the distinguished boundary of the tetrablock). We discuss necessary conditions for a tetrablock contraction to have a tetrablock-isometric lift. We present an example of a tetrablock contraction which does have a tetrablock-isometric lift but violates a condition previously thought to be necessary for the existence of such a lift. Thus the question of whether a tetrablock contraction always has a tetrablock-isometric lift appears to be unresolved at this time.

Mathematics Subject Classification ID

Several-variable operator theory (spectral, Fredholm, etc.) (47A13) Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) (47A56) Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators (47A68) Dilations, extensions, compressions of linear operators (47A20) Canonical models for contractions and nonselfadjoint linear operators (47A45) Hardy spaces (30H10)

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