Some new observations on interpolation in the spectral unit ball
Gautam Bharali
备注：
Added a definition (Def.1.1); 2 of the 4 results herein are minor refinements of those in the author's preprint math.CV/0608177; to appear in Integral Eqns. Operator Theory
发表信息：
Integral Equations Operator Theory 59 (2007) no. 3, 329-343
We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ω_n, n\geq 2. We begin by showing that a known necessary condition for the existence of a $\mathcal{O}(D;Ω_n)$-interpolant (D here being the unit disc in the complex plane), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem -- one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ω_n, n\geq 2.
We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In part…
We solve a three point Nevanlinna-Pick problem in the Euclidean ball. In particular, we determine a class of rational functions that interpolate this problem.
We give necessary and sufficient conditions for solving the spectral Nevanlinna--Pick lifting problem. This reduces the spectral Nevanlinna--Pick problem to a jet interpolation problem into the symmetrized polydisc.
We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is constructed from the knowledge of $f$ and its derivatives on these lines. We prove that this formula approximates any function when the number of lines increases. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.
We study the interpolation sets for the Hardy-Sobolev spaces defined on the unit ball of ${\bf C}^n$. We begin by giving a natural extension to ${\bf C}^n$ of the condition that is known to be necessay and sufficient for interpolation sets lying on the boundary of the unit disc. We show that under this condition the restriction of a function in the Hardy-Sobolev space to the set always exists, and lies in a Besov space. We then show that under the assumption that there is an holomorphic distance function for the set, there is an extension operator from these spaces to the Hardy-Sobolev ones.
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
We present sufficient conditions on a smooth uniformly flat hypersurface W in the unit ball to be an interpolation hypersurface or a sampling hypersurface for generalized Bergman spaces associated to the unit ball with its Bergman metric. The conditions are phrased in terms of a geometric Beurling-type densities, similar to the densities defined for hypersurfaces in C^n by Ortega-Cerda, Schuster and the second author. For the case of sampling, our proofs are different than those in the C^n case; we use an approach closer in spirit to the work of Berndtsson and Ortega-Cerda in the 1-dimensional…
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light o…
This paper surveys results concerning peak sets and boundary interpolation sets for the unit disc. It includes hitherto unpublished results proved by David C. Ullrich on peak sets for the Zygmund class.
We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in $\mathbb{C}^d$.
In this article, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows, generalizing a result of Ransford-White for the spectral unit ball, that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider $2$-point and $3$-point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary cond…
We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.
We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.
We provide generalisations of two Schwarz-type lemmas -- the first a result of Globevnik and the other due to Ransford and White -- for holomorphic mappings into the spectral unit ball. The first concerns mappings of the unit disc in C into the spectral unit ball, while the second concerns self- mappings. The aforementioned results apply to holomorphic mappings that map the origin to the origin. We extend these results to general holomorphic mappings into the spectral unit ball. We also show that our results are sharp.
We study the complexity of holomorphic isometries and proper maps from the complex unit ball to type IV classical domains. We investigate on degree estimates of holomorphic isometries and holomorphic maps with minimum target dimension. We also construct a real-parameter family of mutually inequivalent holomorphic isometries from the unit ball to type IV domains. We also provide examples of non-isometric proper holomorphic maps from the complex unit ball to classical domains.
In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.
We first give an exposition on holomorphic isometries from the Poincaré disk to polydisks and from the Poincaré disk to the product of the Poincaré disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.
In this paper we prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, our result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.
As outlined below, this paper is devoted to a Carleson-type-measure-based study of the holomorphic Campanato $2$-space on the open unit ball $\mathbb B_n$ of $\mathbb C^n$, comprising all Hardy $2$-functions whose oscillations in non-isotropic metric balls on the compact unit sphere $\mathbb S_n$ are proportional to some power of the radius other than the dimension $n\ge 1$.
We study the holomorphic functions with Hadamard gaps in $\mathcal{N}_p$-spaces on the unit ball of $\mathbb{C}^n$ when $0<p \le n$ and $p>n$. A corollary on analytic functions with Hadamard gaps on $\mathcal{N}_p$-spaces on the unit disk is also given.
We construct isometric holomorphic embeddings of the unit ball into higher rank symmetric domains, first discovered by Mok, in an explicit way using Jordan triple systems, and prove uniqueness results for all domains, including the exceptional domains of dimension 16 and 27.
In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings between the unit balls in complex spaces. This extends the classical Schwarz-Pick lemma and the related result proved by Pavlovic.
In this paper we extend a method of Arveson and McCullough to prove a tangential interpolation theorem for subalgebras of $H^\infty$. This tangential interpolation result implies a Toelitz corona theorem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent-Wick and Douglas-Sarkar.
We characterize the Carleson measures for an exponential Bergman space on the unit ball of $\mathbb C^n$ in terms of the ball induced by the complex Hessian of the logarithm of the weight function. The boundedness (or compactness) of integral operators, Cesàro operators and Toeplitz operators, is given using the Carleson measure (or vanishing Carleson measure) characterization.
In this article, we initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.
We prove rigidity results for holomorphic proper maps from the complex unit ball $\mathbb{B}^n$ to the Type IV bounded symmetric domain $D^{IV}_m$ where $n \geq 4, n+1\leq m \leq 2n-3$. In addition, a classification result is established when $m=n+1.$
In this paper, we prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma.
Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of $\mathbb C^n$, $n\geq 2$, consider its (random) zero variety $Z(f_L)$. We study the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ inside a pseudo-hyperbolic ball of radius $r$. We first express this variance as an integral of a positive function in the unit disk. Then we study its asymptotic behaviour as $L\to\infty$ and as $r\to 1^{-}$. Both the results and the proofs generalise to the ball those given by Jeremiah Buckley for the unit disk.
In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.
We consider inductive limits of weighted spaces of holomorphic functions in the unit ball of $\mathbb C^n$. The relationship between sets of uniqueness, weakly sufficient sets and sampling sets in these spaces is studied. In particular, the equivalence of these sets, under general conditions of the weights, is obtained.
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.
This paper deals with an extremal problem for harmonic functions in the unit ball of $\mathbf{R}^n$. We are concerned with the pointwise sharp estimates for the gradient of real--valued bounded harmonic functions. Our main result may be formulated as follows. The sharp constants in the estimates for the absolute value of the radial derivative and the modulus of the gradient of a bounded harmonic function coincide near the boundary of the unit ball. This result partially confirms a conjecture posed by D. Khavinson.
We study some properties of hyperbolic Gaussian analytic functions of intensity $L$ in the unit ball of $\mathbb C^n$. First we deal with the asymptotics of fluctuations of linear statistics as $L\to\infty$. Then we estimate the probability of large deviations (with respect to the expected value) of such linear statistics and use this estimate to prove a hole theorem.
In this paper, we prove the Khavinson conjecture for hyperbolic harmonic functions on the unit ball. This conjecture was partially solved in \cite{JKM2020}.
We prove an Alexander type theorem for the spectral unit ball $Ω_n$ showing that there are no non-trivial proper holomorphic mappings in $Ω_n$, $n\geq 2$.
We describe the growth of $p$th means, $1<p<\frac{2n-1}{2(n-1)}$, of the invariant Green potential in the unit ball in $\mathbb{C}^n$ in terms of smoothness properties of a measure. In particular, a criterion of boundedness of $p$th means of the potential is obtained, a result of M. Stoll is generalized.
Let $A$ be a sequence of points of $\mathbb{B}^n$ the unit ball in $\mathbb{C}^n.$ In terms of interpolating vectorial function (or Amar's function)[1], we give a necessary condition on $A$ to be interpolating for the weighted Bergman space $B^p_α(\mathbb{B}^n). In the particular case of Hardy space $H^p (\mathbb{B}^2)$, this condition is sufficient no optimal. In the main theorem proof, we resolve Gleason's problem (vectorial form) in $B^p_α(\mathbb{B}^n)$
The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.
We derive atomic decompositions and frames for weighted Bergman spaces of several complex variables on the unit ball in the spirit of Coifman, Rochberg, and Luecking. In contrast to our predecessors, we use group theoretic methods, in particular the representation theory of the discrete series of $SU(n,1)$ and its covering groups. One of the benefits is a much larger class of admissible
We establish weak factorizations for a weighted Bergman space $A^p_{\a}$, with $1<p<\infty$, into two weighted Bergman spaces on the unit ball of $\C^n$. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of $\C^n$.
We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$, and we give a sufficient capacity condition in order to identify non-cyclic vectors.
In this paper, a new characterization is provided for the boundedness, compactness and essential norm of the difference of two weighted composition operators on weighted-type spaces in the unit ball of $\mathbb{C}^n$.
In 2002 A.\ Hartmann and X.\ Massaneda obtained necessary and sufficient conditions for interpolation sequences for classes of analytic functions in the unit disc such that $\log M(r,f)=O((1-r)^{-ρ})$, $0<r<1$, $ρ\in (0 , +\infty)$, where $M(r,f)=\max\{ |f(z)|: |z|=r\}$. Using another method, we give an explicit construction of an interpolating function in this result. As an application we describe minimal growth of the coefficient $a$ such that the equation $f''+a(z)f=0$ possesses a solution with a prescribed sequence of zeros.
We give here some precisions and improvements about the validity of the explicit reconstruction of any holomorphic function on a ball of $\mathbb{C}^2$ from its restrictions on a family of complex lines. Such validity depends on the mutual distribution of the lines. This condition can be geometrically described and is equivalent to a stronger stability of the reconstruction formula in terms of permutations and subfamilies of these lines. The motivation of this problem comes from possible applications in mathematical economics and medical imaging.
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.
The boundedness and compactness of weighted composition operators from $H^\infty$ to the Bloch space in the unit ball of Cn are investigated in this paper. In particular, some new characterizations for the boundedness and the essential norm of weighted composition operators are given.
