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Popescu, Gelu

Noncommutative multivariable operator theory. (English) Zbl 1277.47017

Integral Equations Oper. Theory 75, No. 1, 87-133 (2013).
Given a Hilbert space \(\mathcal{H}\), the author studies noncommutative domains \(\mathbb{D}_f^\phi(\mathcal{H})\) in \(B(\mathcal{H})^n\), generated by a positive regular free holomorphic function \(f\) and some classes of \(n\)-tuples \(\phi=(\phi_1,\dots,\phi_n)\) of formal power series in (noncommutative) indeterminates \(Z_1,\dots,Z_n\). Such a domain \(\mathbb{D}_f^\phi=\mathbb{D}_f^\phi(\mathcal{H})\) has a universal model associated to the multiplication operators \((M_{Z_1},\dots,M_{Z_n})\), described via noncommutative Poisson transforms. Among several properties, all joint invariant subspaces under \(M_{Z_1},\dots,M_{Z_n}\) are given a Beurling type characterization, and the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra \(H^\infty(\mathbb{D}_f^\phi)\) is solved.
Reviewer: Florian-Horia Vasilescu (Villeneuve d’Ascq)

Cited in 4 Documents

MSC:

47A45 Canonical models for contractions and nonselfadjoint linear operators
46L52 Noncommutative function spaces
47A20 Dilations, extensions, compressions of linear operators
46T25 Holomorphic maps in nonlinear functional analysis
46L07 Operator spaces and completely bounded maps

Keywords:

multivariable operator theory; noncommutative Reinhardt domain; free biholomorphic function; weighted Fock space; invariant subspace; noncommutative Poisson transform; characteristic function; operator model theory; commutant lifting; Nevanlinna-Pick interpolation
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References:

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