Free Holomorphic Functions Research Articles

Characteristic functions and functional models on noncommutative varieties

In this paper we develop a Sz.-Nagy-Foiaş type theory of characteristic functions and functional models on noncommutative domains and varieties in \(B(\mathcal{H})^n\), where \(B(\mathcal{H})\) is the algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\), associated with admissible free holomorphic functions for operator model theory. This includes, in particular, a large class of commutative domains in \(B(\mathcal{H})^n\). In the commutative case, the characteristic functions are multipliers of reproducing kernel Hilbert spaces on domains in \(\mathbb{C}^n\). We also show that the universal operator model associated with any admissible noncommutative variety has the quasi-wandering property.

Operator valued analogues of multidimensional Bohr’s inequality

AbstractLet $\mathcal {B}(\mathcal {H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal {H}$ . In this paper, we first establish several sharp improved and refined versions of the Bohr’s inequality for the functions in the class $H^{\infty }(\mathbb {D},\mathcal {B}(\mathcal {H}))$ of bounded analytic functions from the unit disk $\mathbb {D}:=\{z \in \mathbb {C}:|z|<1\}$ into $\mathcal {B}(\mathcal {H})$ . For the complete circular domain $Q \subset \mathbb {C}^{n}$ , we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002–1053] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr’s inequalities for operator valued functions in Q.

A refined nc Oka–Weil theorem

AbstractThis short note refines a noncommutative (nc) Oka–Weil theorem by using a characterization of free compact nc sets based on the notion of dilation hulls. A consequence of it is that any free holomorphic function can be represented as a free polynomial on each free compact nc set.

The Smirnov classes for the Fock space and complete Pick spaces

For a Hilbert function space $\mathcal H$ the Smirnov class $\mathcal N^+(\mathcal H)$ is defined to be the set of functions expressible as a ratio of bounded multipliers of $\mathcal H$, whose denominator is cyclic for the action of $Mult(\mathcal H)$. It is known that for spaces $\mathcal H$ with complete Nevanlinna-Pick (CNP) kernel, the inclusion $\mathcal H\subset \mathcal N^+(\mathcal H)$ holds. We give a new proof of this fact, which includes the new conclusion that every $h\in\mathcal H$ can be expressed as a ratio $b/a\in\mathcal N^+(\mathcal H)$ with $1/a$ already belonging to $\mathcal H$. The proof for CNP kernels is based on another Smirnov-type result of independent interest. We consider the Fock space $\mathfrak F^2_d$ of free (non-commutative) holomorphic functions and its algebra of bounded (left) multipliers $\mathfrak F^\infty_d$. We introduce the (left) {\em free Smirnov class} $\mathcal N^+_{left}$ and show that every $H \in \mathfrak F^2_d$ belongs to it. The proof of the Smirnov theorem for CNP kernels is then obtained by lifting holomorphic functions on the ball to free holomorphic functions, and applying the free Smirnov theorem.

An entire free holomorphic function which is unbounded on the row ball

We give an entire free holomorphic function f which is unbounded on the row ball. That is, we give a holomorphic free noncommutative function which is continuous in the free topology developed by Agler and McCarthy but is unbounded on the set of row contractions.

Algebras of free holomorphic functions and localizations

We consider the algebras of holomorphic functions on a free polydisc , and the algebra of holomorphic functions on a free ball . We show that the algebra is a localization of a free algebra and, moreover, is a free analytic algebra with generators (in the sense of J. Taylor), while the algebra is not a localization of a free algebra. In addition we prove that the class of localizations of free algebras and the class of free analytic algebras are closed under the operation of the Arens-Michael free product. Bibliography: 21 titles.

Algebras of free holomorphic functions and localizations

Рассматриваются алгебры голоморфных функций на свободном полидиске $\mathscr{F}^T(\mathbb{D}_R^n)$, $\mathscr{F}(\mathbb{D}_R^n)$ и алгебра голоморфных функций на свободном шаре $\mathscr{F}(\mathbb{B}_r^n)$. Показано, что алгебра $\mathscr{F}(\mathbb{D}_R^n)$ является локализацией свободной алгебры и, более того, свободной аналитической алгеброй с $n$ образующими (в смысле Дж. Л. Тейлора), а алгебра $\mathscr{F}(\mathbb{B}_r^n)$ не является локализацией свободной алгебры. Кроме того, доказано, что класс локализаций свободных алгебр и класс свободных аналитических алгебр замкнуты относительно операции свободного произведения Аренса-Майкла. Библиография: 21 название.

Operators affiliated to the free shift on the Free Hardy Space

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over Cd. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions in a non-commutative open unit ball in several matrix-variables, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space acts as right multiplication by a non-commutative function in the right free Smirnov class (and analogously, replacing ‘right’ with ‘left’).

Noncommutative hyperballs, wandering subspaces, and inner functions

A noncommutative domain Dfm(H), m∈N, of n-tuples of bounded linear operators on a Hilbert space H is associated with any positive regular free holomorphic function f. If Q⊂C〈Z1,…,Zn〉 is a set of polynomials in noncommutative indeterminates such that q(0)=0 for all Q∈Q, then the noncommutative varietyVf,Qm(H):={X=(X1,…,Xn)∈Dfm(H):q(X)=0 for all q∈Q} admits a universal modelB=(B1,…,Bn) acting on a model space Nf,Qm, which is a subspace of the full Fock space with n generators. If M is an arbitrary joint invariant subspace under the operators B1⊗IG,…,Bn⊗IG and the variety Vf,Qm(H) has the unique minimal dilation property, we provide an explicit description of the wandering subspaceW:=M⊖∑i=1n(Bi⊗IG)M and obtain a characterization of the inner multi-analytic operators Θ:Nf,Q1⊗G⁎→Nf,Qm⊗G, i.e. Θ|G⁎ is an isometry andΘ(G⁎)⊥(Bα⊗IG)Θ(G⁎),α∈Fn+,|α|≥1, where Fn+ is the free semigroup with n generators. We show that the noncommutative m-hyperball Dm(H) (which corresponds to f=Z1+⋯+Zn) has the unique minimal dilation property. When S=(S1,…,Sn) is the universal model for the unit ball D1(H) and W=(W1,…,Wn) is the universal model of the m-hyperball Dm(H), we provide a transfer function type characterization of the (S,W)-inner operators on Fock spaces. This extends to our noncommutative setting the corresponding results obtained by Olofsson (when n=1) and by Eschmeier in the multivariable commutative case.

Bohr inequalities for free holomorphic functions on polyballs

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball Bn, n=(n1,…,nk)∈Nk, which is a noncommutative analogue of the scalar polyball (Cn1)1×⋯×(Cnk)1. The Bohr radius Kmh(Bn) (resp. Kh(Bn)) associated with the multi-homogeneous (resp. homogeneous) power series expansions of the free holomorphic functions are the main objects of study in this paper. We extend a theorem of Bombieri and Bourgain for the disc D:={z∈C:|z|<1} to the polyball, and obtain the estimations 13k<Kmh(Bn)<2log⁡kk if k>1, extending Boas-Khavinson result for the scalar polydisc Dk.With respect to the homogeneous power series expansion, we prove that Kh(Bn)=1/3, extending the classical result, and obtain analogues of Carathéodory, Fejér, and Egerváry-Szász inequalities for free holomorphic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.When specialized to the regular polydisc Dk (which corresponds to the case n1=⋯=nk=1), we obtain new results concerning Bohr, Landau, Fejér, and Harnack inequalities for operator-valued holomorphic functions and k-pluriharmonic functions on the scalar polydisc Dk. The results of the paper can be used to obtain Bohr type inequalities for the noncommutative ball algebra An, the Hardy algebra Fn∞, and the C⁎-algebra C⁎(S), generated by the universal model S of the polyball Bn.

Bohr Inequalities on Noncommutative Polydomains

The goal of this paper is to study the Bohr phenomenon in the setting of free holomorphic functions on noncommutative regular polydomains $$\mathbf{D_f^m}$$ , $$\mathbf{f}=(f_1,\ldots , f_k)$$ , generated by positive regular free holomorphic functions. These polydomains are noncommutative analogues of the scalar polydomains $$\begin{aligned} {{\mathcal {D}}}_{f_1} ({{\mathbb {C}}})\times \cdots \times {{\mathcal {D}}}_{f_k}({{\mathbb {C}}}), \end{aligned}$$ where each $${{\mathcal {D}}}_{f_i}({{\mathbb {C}}})\subset {{\mathbb {C}}}^{n_i}$$ is a certain Reinhardt domain generated by $$f_i$$ . We characterize the free holomorphic functions on $$\mathbf{D_f^m}$$ in terms of the universal model of the polydomain and extend several classical results from complex analysis to our noncommutative setting. It is shown that the free holomorphic functions admit multi-homogeneous and homogeneous expansions as power series in several variables. With respect to these expansions, we introduce the Bohr radii $$K_{mh}(\mathbf{D_f^m})$$ and $$K_{h}(\mathbf{D_f^m})$$ for the noncommutative Hardy space $$H^\infty (\mathbf{D_{f,\text { rad}}^m})$$ of all bounded free holomorphic functions on the radial part of $$\mathbf{D_f^m}$$ . Several well-known results concerning the Bohr radius associated with classes of bounded holomorphic functions are extended to our noncommutative multivariable setting.

Free Holomorphic Functions on the Noncommutative Polydomains and Universal Models

In this paper, we study free holomorphic functions on the noncommutative polydomains \(\mathbf{D_f^m}(\mathcal {H})\), which were recently introduced by Popescu (J Funct Anal 265(10):2500–2552, 2013; Adv Math 279:104–158, 2015; Trans Am Math 368:4357–4416, 2016). We obtain some characterizations of free holomorphic functions by noncommutative Berezin transforms and the universal models associated with the abstract noncommutative polydomains \(\mathbf{D_f^m}\). As consequences, we provide noncommutative multivariable analogues of several results from the classical holomorphic function theory such as: Wiener inequality, Weierstrass theorem, Montel theorem, Vitali theorem. Moreover, we also prove that the universal models are essentially unique.

Free Holomorphic Functions on the Regular Polyball

In this paper, we study free holomorphic functions on the regular polyball, which was recently introduced by Popescu (Adv Math 279:104–158, 2015, Trans Am Math Soc 368:4357–4416, 2016). The purpose of this paper is to continue the line of Popescu to develop a theory of free holomorphic functions. Some results from the classical holomorphic function theory have free analogues in this noncommutative setting. In particular, we prove the maximum principle, Weierstrass and Montel type theorems for free holomorphic functions. As an application, we construct a metric on $$\mathrm{Hol}(\mathbf{B_n}(\mathcal {H}))$$ such that it becomes a complete space.

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs Bn(H) in B(H)n1+⋯+nk, where n=(n1,…,nk)∈Nk and B(H) is the algebra of all bounded linear operators on a Hilbert space H. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on Bn(H), study their properties, and obtain hyperbolic versions of Schwarz–Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the regular polydisk Dk(H) (which corresponds to the case n1=⋯=nk=1), our hyperbolic metric δH is complete and invariant under the group Aut(Dk) of all free holomorphic automorphisms of Dk(H), and the δH-topology induced on Dk(H) is the usual operator norm topology. The restriction of δH to the scalar polydisk Dk is equivalent to the Kobayashi distance on Dk. Most of the results of this paper are presented in the more general setting of Harnack (resp. Poisson) parts of the closed polyball Bn(H)−.

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of automorphisms of the Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball algebra A_n; the group of unitarily implemented automorphisms of the polyball algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact, locally compact topological group with respect to the topology induced by an appropriate metric. Finally, we obtain a concrete unitary projective representation of the topological group Aut(B_n)in terms of noncommutative Berezin kernels associated with regular polyballs.

Free pluriharmonic functions on noncommutative polyballs

In this paper, we study free k-pluriharmonic functions on noncommutative regular polyballs. These regular polyballs have universal operator models consisting of left creation operators acting on tensor products of full Fock spaces. We introduce and determine the class of kmulti- Toeplitz operators acting on these tensor products and show that the bounded free k-pluriharmonic functions on regular polyballs are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in a certain class, which we determine. We provide a Naimark type dilation theorem for direct products of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball, with operator-valued coefficients. We define the noncommutative the Berezin (resp. Poisson) transform of a completely bounded linear map on the C*-algebra generated by the universal operator model and give necessary an sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz-Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as Korannyi-Pukanszky version in scalar polydisks.

Pick Interpolation for free holomorphic functions

We give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron. As an application, we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function.

Berezin transforms on noncommutative polydomains

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 D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) in B ( H ) n 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 W = { W i , j } \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 ∞ ( D q m ) F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}) as the weakly closed algebra generated by { W i , j } \{\mathbf {W}_{i,j}\} and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) . It is shown that the Berezin transform is a completely isometric isomorphism between F ∞ ( D q m ) F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}) and the algebra of bounded free holomorphic functions on the radial part of D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) . A characterization of the Beurling type joint invariant subspaces under { W i , j } \{\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 D q m ( H ) \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 ∗ C^* -algebra techniques, we develop a dilation theory on the noncommutative polydomain D q m ( H ) \mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H}) .

Global Holomorphic Functions in Several Noncommuting Variables

AbstractWe define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains $${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$$ generated by positive regular free holomorphic functions f and certain classes of n-tuples $${\varphi = (\varphi_1, \ldots, \varphi_n)}$$ of formal power series in noncommutative indeterminates Z 1, . . . , Z n . Noncommutative Poisson transforms are employed to show that each abstract domain $${\mathbb{D}_f^\varphi}$$ has a universal model consisting of multiplication operators (M Z1, . . . , M Z n ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M Z1, . . . , M Z n and show that all pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ are compressions of $${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$$ to their coinvariant subspaces. We show that the eigenvectors of $${M_{Z_1}^*, \ldots, M_{Z_n}^*}$$ are precisely the noncommutative Poisson kernels $${\Gamma_\lambda}$$ associated with the elements $${\lambda}$$ of the scalar domain $${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra $${H^\infty(\mathbb{D}_f^\varphi)}$$ . We introduce the characteristic function of an n-tuple $${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$$ , present a model for pure n-tuples of operators in the noncommutative domain $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ .