We develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary n-tuples of operators. The main topics are the following: Free holomorphic functions and Hausdorff derivations; Cauchy, Liouville, and Schwartz type results for free holomorphic functions; Algebras of free holomorphic functions; Free analytic functional calculus and noncommutative Cauchy transforms; Weierstrass and Montel theorems for free holomorphic functions; Free pluriharmonic functions and noncommutative Poisson transforms; Hardy spaces of free holomorphic functions.
In this paper we continue the study of free holomorphic functions on the unit ball of B(H)^n, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Several classical results from complex analysis have free analogues in our noncommutative setting.
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.
The theory of characteristic functions for row contractions is used to determine the group $Aut(B(H)^n_1)$ of all free holomorphic automorphisms of the unit ball of $B(H)^n$. We show that the noncommutative Poisson transform commutes with the action of the automorphism group $Aut(B(H)^n_1)$. This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra $C^*(S_1,..., S_n)$, which leave invariant the noncommutative disc algebra $\cA_n$. This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals n…
We propose a general method for constructing continuous Banach bundles whose fibers are algebras of holomorphic functions on subvarieties of a closed noncommutative ball. These algebras are of the form $\mathcal{A}_d/I_x$, where $\mathcal{A}_d$ is the noncommutative disk algebra introduced by G. Popescu, and $I_x$ is a graded ideal in $\mathcal{ A}_d$, which depends continuously on the point $x$ of the topological space $X$. Similarly, we construct bundles with fibers isomorphic to the algebras $\mathcal{F}_d/I_x$ of holomorphic functions on subvarieties of an open noncommutative ball. Here $\…
For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebra by $\mathcal A,$ we study the Gleason parts of the set of scalar-valued homomorphisms $\mathcal M(\mathcal A)$ on $\mathcal A.$ Following remarks on the general situation, we focus on the case $X = c_0.$
We study the spectrum $M_b(U)$ of the algebra of bounded type holomorphic functions on a complete Reinhardt domain in a symmetrically regular Banach space $E$ as an analytic manifold over the bidual of the space. In the case that $U$ is the unit ball of $\ell_p$, $1<p<\infty$, we prove that each connected component of $M_b(B_{\ell_p})$ naturally identifies with a ball of a certain radius. We also provide estimates for this radius and in many natural cases we have the precise value. As a consequence, we obtain that for connected components different from that of evaluations, these radii are str…
The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) = \int_{S^n}G_\om(z)\bs n(\om)f(\om)\,dμ(\om)$, $z\in L_n.$ The inverse map $\tilde f\mapsto f$ is constructed here using the Cauchy-Hua formula for the Lie ball following the work of M. Morimoto \cite{Mori2}.
We prove an analogue of Alexander's Theorem for holomorphic mappings of the unit ball in a complex Hilbert space: Every holomorphic mapping which takes a piece of the boundary of the unit ball into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball. We also show that it is enough to assume that the mapping is only Gateaux-holomorphic.
We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly c-algebrable for all separable Banach spaces. For specific spaces including lp or duals of Lorentz sequence spaces, we have strongly c-algebrability and spaceability even for the subalgebra of uniformly continuous holomorphic functions on the ball.
We find a lower bound for the essential norm of the difference of two composition operators acting on $H^2(B_N)$ or $A^2_s(B_N)$ ($s>-1$). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.
In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna-Pick interpolation problem for analytic functions with positive real parts on the open unit disc. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball on C^n.
Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some ch…
In this paper we introduce a hyperbolic distance $δ$ on the noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $δ$ is invariant under the action of the group $Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\cH)^n]_1$, and show that the $δ$-topology and the usual operator norm topology coincide on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric space with respect to the hype…
In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture for complex analytic subsets that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $\mathcal{T}(L^{\infty})$, which implies the essential normality of the quotient module. Combining some other techniques we actually obtain the $p$-essential normality for $p>2d$, where $d$ is the complex dimension of the analytic su…
This paper is an attempt to find large classes of noncommutative multivariable functions $g:Ω\subset [B(\cH)^n]_1^-\to B(\cH)^n$ for which a reasonable operator model theory and dilation theory can be developed for the noncommutative domain $g(Ω)$. In other words, we want to transfer the free analogue of Nagy-Foias theory from the unit ball $[B(\cH)^n]_1$ to other noncommutative domains, using appropriate maps.
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric $π$-property. Besid…
We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions…
We study mean ergodic composition operators on infinite dimensional spaces of holomorphic functions of different types when defined on the unit ball of a Banach or a Hilbert space: that of all holomorphic functions, that of holomorphic functions of bounded type and that of bounded holomorphic functions. Several examples in the different settings are given.
The main result of the paper is the construction of explicit uniformly bounded basis in the spaces of complex homogenous polynomials on the unit ball of $C^3$, extending an earlier result of the author in the $C^2$ case
Let $B_{R}$ be the ball in the euclidean space $\mathbb{R}^{n}$ with center 0 and radius $R$ and let $f$ be a complex-valued, infinitely differentiable function on $B_{R}.$ We show that the Laplace-Fourier series of $f$ has a holomorphic extension which converges compactly in the Lie ball $\hat {B_{R}}$ in the complex space $\mathbb{C}^{n}$ when one assumes a natural estimate for the Laplace-Fourier coefficients.
Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B$ and $ψ(z)$ a holomorphic function on $B$, where $B$ is the unit ball of ${\Bbbb C}^n$. Let $0<p,s<+\infty, -n-1<q<+\infty, q+s>-1$ and $α\geq 0,$ this paper gives some necessary and sufficient conditions for the weighted composition operator $\wco$ induced by $ϕ$ and $ψ$ to be bounded and compact between the space $\Fs$ and $α$-{\sl Bloch} space $β^α.$
Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B_n$ and $ψ(z)$ a holomorphic function on $B_n$, and $H(B_n)$ the class of all holomorphic functions on $B_n$, where $B_n$ is the unit ball of $C^n$, the weight composition operator $W_{ψ,ϕ}$ is defined by $W_{ψ,ϕ}=ψf(ϕ)$ for $f\in H(B_n)$. In this paper we estimate the essential norm for the weighted composition operator $W_{ψ,ϕ}$ acting from the Hardy space $H^p$ to $H^q$ ($0<p,q\leq \infty$). When $p=\infty$ and $q=2$, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necess…
Denote by $ B_X $ the unit ball of an infinite-dimensional complex Hilbert space $ X. $ Let $ψ\in H(B_X),$ the space of all holomorphic functions on the unit ball $B_X,$ $φ\in S(B_X)$ the set of holomorphic self-maps of $B_X. $ Let $C_{ψ, φ}: \mathcal B_ν(B_X)$ (and $ \mathcal B_{ν,0}(B_X)$) $\to \mathcal B_μ(B_X) $ (and $ \mathcal B_{μ,0}(B_X)$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $ C_φ$ and integral operator $T_ψ.$ In this paper, we characterize the boundedness and compactness of $ C_{ψ,φ} $ via the estimates for the restrictions of $ ψ$ …
We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $λ_{ϕ, w}$ is the Shilov boundary for algebras of holomorphic functions on $λ_{ϕ, w}$ if $ϕ$ satisfies the $δ_2$-condition.
We investigate which boundary points in the closed unit ball of the Bergman space of integrable holomorphic functions on the unit disc are strongly exposed. This requires study of the Bergman projection and its kernel, the annihilator of Bergman space. We show that all polynomials in the boundary of the unit ball are strongly exposed.
If F is an automorphism of the spectral unit ball, we show that, in a neighborhood of any cyclic (i.e. non-derogatory) matrix of the ball, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.
We introduce Hausdorff operators over the unit disc and give conditions for boundedness of such operator in Bloch, Bergman, and Hardy spaces on the disc. Identity approximation by Hausdorff operators is also considered.
We characterise continuity of composition operators on weighted spaces of holomorphic functions $H_{v}(B_{X})$, where $B_{X}$ is the open unit ball of a Banach space which is homogeneous, that is, a $JB^{\ast}-$triple.
This paper focuses on the embedding problem of linear fractional maps which explains when a linear fractional self-map of $B_{N}$ can be a member of a semigroup of holomorphic self-maps on the unit ball $B_{N}$ of the complex $N$-dimensional Euclidean space $\mathbb{C}^{N}$.
We prove the plasticity of the unit ball of $c$. That is, we show that every non-expansive bijection from the unit ball of $c$ onto itself is an isometry. We also demonstrate a slightly weaker property for the unit ball of $c_0$ -- we prove that a non-expansive bijection is an isometry, provided that it has a continuous inverse.
We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator exists. Using such 'trace theorems' it is possible to construct discrete increments to spectra of some unitary representation and to catch singular unitary representations in the spectra. We also discuss spectral problems related to Berezin type kernels on riemann symmetric spaces.
Let (M,g) be a simply connected complete Kahler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in C^n, where dim M = n.
Let $g$ be a holomorphic function of the unit ball $B$ in the $n$-dimensional space, and denote by $T_g$ and $I_g$ the induced extended Ces$\acute{a}$ro operator and another integral operator. The boundedness and compactness of $T_g$ and $I_g$ acting on the Zygmund spaces in the unit ball are discussed and necessary and sufficient conditions are given in this paper.
We identify the exposed points of the unit ball of the dual space of the ball algebra. As a corollary, we show that the predual of $H^\infty(\mathbb{B}_d)$ has no extreme points in its unit ball.
