 

Haripada Sau
A note on tetrablock contractions view print


Published: 
December 17, 2015

Keywords: 
Tetrablock, tetrablock contraction, spectral set, BeurlingLaxHalmos theorem, functional model, fundamental operator. 
Subject: 
47A15, 47A20, 47A25, 47A45. 


Abstract
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={(a_{11},a_{22},detA): A=\begin{pmatrix} a_{11} & a_{12} a_{21} & a_{22} \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
AB*P = D_{P}X_{1}D_{P} and
BA*P=D_{P}X_{2}D_{P},
where X_{1},X_{2} ∈ B(D_{P}) 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
(M_{G1*+G2z},
M_{G2*+G1z},M_{z})
on
H^{2}_{D}_{P*}(D),
where G_{1} and G_{2} are the fundamental operators of (A*,B*,P*).
We prove a BeurlingLaxHalmos type theorem for a triple of operators
(M_{F1*+F2z},M_{F2*+F1z},M_{z}),
where E is a Hilbert space and F_{1},F_{2} ∈ B(E).
We also deal with a natural example of tetrablock contraction on a functions space to find out its fundamental operators.


Acknowledgements
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
sau10@math.iisc.ernet.in

