Berezin transforms on noncommutative polydomains

Popescu, Gelu

doi:10.1090/tran/6466

Berezin transforms on noncommutative polydomains
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Trans. Amer. Math. Soc. 368 (2016), 4357-4416 Request permission

Abstract:

This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ in $B(\mathcal {H})^n$. An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain. These transforms are used to prove that each such polydomain has a universal model $\mathbf {W}=\{\mathbf {W}_{i,j}\}$ consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra $F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})$ as the weakly closed algebra generated by $\{\mathbf {W}_{i,j}\}$ and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$. It is shown that the Berezin transform is a completely isometric isomorphism between $F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})$ and the algebra of bounded free holomorphic functions on the radial part of $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$. A characterization of the Beurling type joint invariant subspaces under $\{\mathbf {W}_{i,j}\}$ is also provided.

It has been an open problem for quite some time to find significant classes of elements in the commutative polydisc for which a theory of characteristic functions and model theory can be developed along the lines of the Sz.-Nagy–Foias theory of contractions. We give a positive answer to this question, in our more general setting, providing a characterization for the class of tuples of operators in $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ which admit characteristic functions. The characteristic function is constructed explicitly as an artifact of the noncommutative Berezin kernel associated with the polydomain, and it is proved to be a complete unitary invariant for the class of completely non-coisometric tuples. Using noncommutative Berezin transforms and $C^*$-algebra techniques, we develop a dilation theory on the noncommutative polydomain $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$.

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