Operator realizations of non-commutative analytic functions - Augat, Méric L.; Martin, Robert T. W.; Shamovich, Eli (Forum of Mathematics, Sigma, 2025)
A realization is a triple, $(A,b,c)$ , consisting of a $d-$ tuple, $A= (A_1, … , A_d )$ , $d\in \mathbb {N}$ , of bounded linear operators on a separable, complex Hilbert space, $\mathcal {H}$ , and vectors $b,c \in \mathcal {H}$ . Any such realization defines an analytic non-commutative (NC) function in an open neighbourhood of the origin, $0:= (0, … , 0)$ , of the NC universe of $d-$ tuples of square matrices of any fixed size. For example, a univariate realization, i.e., where A is a single bounded linear operator, defines a holomorphic function of a single complex variable, z, in an open neighbourhood of the origin via the realization formula $b^{*} (I-zA)^{-1} c$ . It is well known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at $0$ . Such finite realizations contain valuable information about the NC rational functions they generate. By extending to infinite-dimensional realizations, we construct, study and characterize more general classes of analytic NC functions. In particular, we show that an NC function is (uniformly) entire if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that a formal Taylor series extends globally to an entire or meromorphic function in the complex plane, $\mathbb {C}$ , if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This motivates our definition of the field of global (uniformly) meromorphic NC functions as the field of fractions generated by NC rational expressions in the ring of NC functions with jointly compact realizations. This definition recovers the field of meromorphic functions in $\mathbb {C}$ when restricted to one variable.