 

Bappa Bisai and
Sourav Pal
The fundamental operator tuples associated with the symmetrized polydisc
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Published: 
February 25, 2021. 
Keywords: 
Symmetrized polydisc, fundamental operator tuple. 
Subject: 
47A13, 47A20, 47A25, 47A45. 


Abstract
A commuting tuple of operators (S_{1},..., S_{n1},P), defined on a Hilbert space H, for which the closed symmetrized polydisc is a spectral set, is called a
Γ_{n}contraction. To every Γ_{n}contraction, there is a unique operator tuple (A_{1},...,A_{n1}), defined on the closure of Ran(IP*P), such that
S_{i}  S_{ni}*P=D_{P}A_{i}D_{P},
D_{P}=(I  P*P)^{1/2}, i=1,..., n1.
This is called the fundamental operator tuple or F_{O}tuple associated with the Γ_{n}contraction. The F_{O}tuple of a Γ_{n}contraction completely determines the structure of a
Γ_{n}contraction and provides operator model and complete unitary invariant for them. In this note, we analyze the F_{O}tuples and find some intrinsic properties of them. Given a Γ_{n}contraction
(S_{1},...,S_{n1},P) and n1 operators A_{1},..., A_{n1} defined on the closure of
Ran(D_{P}), we provide a necessary and sufficient condition under which (A_{1},..., A_{n1}) becomes the F_{O}tuple of
(S_{1},...,S_{n1},P). Also for given tuples of operators
(A_{1},..., A_{n1}) and (B_{1},..., B_{n1}), defined on a Hilbert space E, we find a necessary condition and a sufficient condition under which there exist a Hilbert space H and a Γ_{n}contraction
(S_{1},..., S_{n1},P) on H such that (A_{1},..., A_{n1}) becomes the
F_{O}tuple of
(S_{1},..., S_{n1},P) and (B_{1},..., B_{n1}) becomes the
F_{O}tuple of the adjoint
(S_{1}*,..., S_{n1}*,P*). 

Acknowledgements
The first named author is supported by a Ph.D fellowship of the University Grants Commissoin (UGC). The second named author is supported by the Seed Grant of IIT Bombay with Grant No. RD/0516IRCCSH0003 (15IRCCSG032), the INSPIRE Faculty Award (Award No. DST/INSPIRE/04/2014/001462) of DST, India and the MATRICS Grant (Award No. MTR/2019/001010) of Science and Engineering Research Board (SERB) of DST, India.


Author information
Bappa Bisai:
Mathematics Department
Indian Institute of Technology Bombay
Powai, Mumbai  400076, India
bisai@math.iitb.ac.in
Sourav Pal:
Mathematics Department
Indian Institute of Technology Bombay
Powai, Mumbai  400076, India
sourav@math.iitb.ac.in

