New York Journal of Mathematics
Volume 21 (2015) 1347-1369


Haripada Sau

A note on tetrablock contractions

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Published: December 17, 2015
Keywords: Tetrablock, tetrablock contraction, spectral set, Beurling-Lax-Halmos theorem, functional model, fundamental operator.
Subject: 47A15, 47A20, 47A25, 47A45.

A commuting triple of operators (A,B,P) on a Hilbert space H is called a tetrablock contraction if the closure of the set
E={(a11,a22,detA): A=\begin{pmatrix} a11 & a12 a21 & a22 \end{pmatrix} with ||A|| <1}
is a spectral set. In this paper, we construct a functional model and produce a set of complete unitary invariants for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations
A-B*P = DPX1DP and B-A*P=DPX2DP,
where X1,X2B(DP) play a pivotal role. As a result of the functional model, we show that every pure tetrablock isometry (A,B,P) on an abstract Hilbert space H is unitarily equivalent to the tetrablock contraction
(MG1*+G2z, MG2*+G1z,Mz) on H2DP*(D),
where G1 and G2 are the fundamental operators of (A*,B*,P*). We prove a Beurling-Lax-Halmos type theorem for a triple of operators
where E is a Hilbert space and F1,F2B(E). We also deal with a natural example of tetrablock contraction on a functions space to find out its fundamental operators.


The author's research is supported by University Grants Commission Center for Advanced Studies.

Author information

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India