Noncommutative Multivariable Operator Theory

Popescu, Gelu

Integral Equations Operator Theory 75 (2013), no. 1, 87–133

In this paper, we study noncommutative domains Dφf(H)⊂B(H)^n generated by positive regular free holomorphic functions f and certain classes of n-tuples φ=(φ1,…,φn) of formal power series in noncommutative indeterminates Z1,…,Zn. Noncommutative Poisson transforms are employed to show that each abstract domain Dφf has a universal model consisting of multiplication operators (MZ1,…,MZn) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under MZ1,…,MZn and show that all pure n-tuples of operators in Dφf(H) are compressions of MZ1⊗I,…,MZn⊗I to their coinvariant subspaces. We show that the eigenvectors of M∗Z1,…,M∗Zn are precisely the noncommutative Poisson kernels Γλ associated with the elements λ of the scalar domain Dφf,<(C)⊂Cn. These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra H∞(Dφf). We introduce the characteristic function of an n-tuple T=(T1,…,Tn)∈Dφf(H), present a model for pure n-tuples of operators in the noncommutative domain Dφf(H) in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in Dφf(H).

Gelu Popescu

(210) 458-4761

FLN 4.01.28

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