Hilbert Ball Research Articles

The Wolff hull of a compact holomorphic self-map on an infinite dimensional ball

For large classes of (finite and) infinite dimensional complex Banach spaces Z, B its open unit ball and f:B→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:B\\rightarrow B$$\\end{document} a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W(f), of f in ∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial B$$\\end{document} and prove that W(f) is proximal to the images of all subsequential limits of the sequences of iterates (fn)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f^n)_n$$\\end{document} of f. The Wolff hull generalises the concept of a Wolff point, where such a point can no longer be uniquely determined, and coincides with the Wolff point if Z is a Hilbert space. Recall that (fn)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f^n)_n$$\\end{document} does not generally converge even in finite dimensions, compactness of f (i.e. f(B) is relatively compact) is necessary for convergence in the infinite dimensional Hilbert ball and all accumulation points Γ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (f)$$\\end{document} of (fn)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f^n)_n$$\\end{document} map B into ∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial B$$\\end{document} (for any topology finer than the topology of pointwise convergence on B). The target set of f is T(f)=⋃g∈Γ(f)g(B).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T(f)=\\bigcup _{g \\in \\Gamma (f)} g(B). \\end{aligned}$$\\end{document}To locate T(f), we use a concept of closed convex holomorphic hull, Ch(x)⊂∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Ch}}(x) \\subset \\partial B$$\\end{document} for each x∈∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\in \\partial B$$\\end{document} and define a distinguished Wolff hull W(f). We show that the Wolff hull intersects all hulls from T(f), namely W(f)∩Ch(x)≠∅for allx∈T(f).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W(f) \\cap {\ ext {Ch}}(x)\ e \\emptyset \\ \\ \\hbox {for all}\\ \\ x \\in T(f). \\end{aligned}$$\\end{document}If B is the Hilbert ball, W(f) is the Wolff point, and this is the usual Denjoy–Wolff result. Our results are for all reflexive Banach spaces having a homogeneous ball (or equivalently, for all finite rank JB∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$JB^*$$\\end{document}-triples). These include many well-known operator spaces, for example, L(H, K), where either H or K is finite dimensional.

CONVEXITY OF REACHABLE SETS OF QUASILINEAR SYSTEMS

This paper investigates convexity of reachable sets for quasilinear systems under integral quadratic constraints. Drawing inspiration from B.T. Polyak's work on small Hilbert ball image under nonlinear mappings, the study extends the analysis to scenarios where a small nonlinearity exists on the system's right-hand side. At zero value of a small parameter, the quasilinear system turns into a linear system and its reachable set is convex. The investigation reveals that to maintain convexity of reachable sets of these systems, the nonlinear mapping's derivative must be Lipschitz continuous. The proof methodology follows a Polyak's scheme. The paper's structure encompasses problem formulation, exploration of parameter linear mapping and image transformation, application to quasilinear control systems, and concludes with illustrative examples.

On an interpolation sequence for a weighted Bergman space on a Hilbert unit ball

The purpose is to provide a generalization of Carleson's Theorem on interpolating sequences when dealing with a sequence in the open unit ball of a Hilbert space. Precisely, we interpolate a sequence by a function belonging to a weighted Bergman space of infinite order on a unit Hilbert ball and we furnish explicitly the upper bound corresponding to the interpolation constant.

Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces

AbstractLet be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ2. Note that the Bohr phenomenons that we have obtained are completely different from those in the case with values in and are sharp in the case with values in ℓ2.

Group of Isometries of Hilbert Ball Equipped with the Carathéodory Metric

In this article, we study the geometry of an infinite-dimensional hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carathéodory metric and learn about some special subclasses of this group. We will also find some unitary equivalence condition and compute some cardinalities.

Asymptotic Behavior of Reachable Setson Small Time Intervals

The geometric structure of small-time reachable sets plays an important role in control theory, in particular, in solving problems of local synthesis. In this paper, we consider the problem of approximate description of reachable sets on small time intervals for control-affine systems with integral quadratic constraints on the control. Using a time substitution, we replace such a set by the reachable set on a unit interval of a control system with a small parameter, which is the length of the time interval for the original system. The constraints on the control are given by a ball of small radius in the Hilbert space $$\mathbb{L}_{2}$$ . Under certain conditions imposed on the controllability Gramian of the linearized system, this reachable set turns out to be convex for sufficiently small values of the parameter. We show that in this case the shape of the reachable set in the state space is asymptotically close to an ellipsoid. The proof of this fact is based on the representation of the reachable set as the image of a Hilbert ball of small radius in $$\mathbb{L}_{2}$$ under a nonlinear mapping to $$\mathbb{R}^{n}$$ . In particular, this asymptotic representation holds for a fairly wide class of second-order nonlinear control systems with integral constraints. We give three examples of systems whose reachable sets demonstrate both the presence of the indicated asymptotic behavior and the absence of the latter if the necessary conditions are not satisfied.

Closed Range Composition Operators on the Bloch Space of Bounded Symmetric Domains

Let $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ be bounded symmetric domains realized as the unit balls of $$\hbox {JB}^*$$ -triples X and Y, respectively. In this paper, we generalize the Landau theorem to holomorphic mappings on $${\mathbb {B}}_X$$ using the Schwarz–Pick lemma for holomorphic mappings on $${\mathbb {B}}_X$$ . Next, we give a necessary condition for the composition operators $$C_{\varphi }$$ between the Bloch spaces on $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ to be bounded below by using a sampling set for the Bloch space, where $$\varphi $$ is a holomorphic mapping from $${\mathbb {B}}_X$$ to $${\mathbb {B}}_Y$$ . We also obtain other necessary conditions for the composition operators $$C_{\varphi }$$ between the Bloch spaces in the case $${\mathbb {B}}_Y$$ is a complex Hilbert ball $${\mathbb {B}}_H$$ . We give a sufficient condition for the composition operators $$C_{\varphi }$$ between the Bloch spaces on $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ to be bounded below using a sampling set for the Bloch space. In the case $$\dim X=\dim Y<\infty $$ , we also give another sufficient condition for the composition operators $$C_{\varphi }$$ between the Bloch spaces on $${\mathbb {B}}_X$$ and $${\mathbb {B}}_Y$$ to be bounded below.

A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings

In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball $${\mathbb{B}^n}$$ in ℂn, as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥Df(z0)∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.

Bloch Space of a Bounded Symmetric Domain and Composition Operators

We generalize a number of finite dimensional results on Bloch functions to infinite dimensional bounded symmetric domains. In particular, we characterize the Bloch space as well as the little Bloch space of a Hilbert ball, and give one sufficient and several necessary conditions for a composition operator on a Bloch space to be an isometry. We also answer some open questions of Allen and Colonna concerning Bloch functions and composition operators.

A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls $$\mathbb{B}$$ and $$\mathbb{B'}$$ in separable complex Hilbert spaces $$\mathcal{H}$$ and $$\mathcal{H'}$$ , respectively. It is found that if the mapping f ∈ C1+α at $${z_0} \in \partial \mathbb{B}$$ with $$f\left( {{z_0}} \right) = {w_0} \in \partial \mathbb{B}'$$ , then the Frechet derivative operator Df(z0) maps the tangent space $${T_{{z_0}}}(\partial {\mathbb{B}^n})$$ to $${T_{{w_0}}}(\partial {\mathbb{B}'})$$ , the holomorphic tangent space $$T_{{z_0}}^{(1,0)}(\partial {\mathbb{B}^n})$$ to $$T_{{w_0}}^{(1,0)}(\partial {\mathbb{B}'})$$ , respectively.

A Wolff Theorem for finite rank bounded symmetric domains

We present a Wolff Theorem for all infinite dimensional bounded symmetric domains of finite rank. Namely, if B is the open unit ball of any finite rank JB⁎-triple and f:B→B is a compact holomorphic map with no fixed point in B, we prove convex f-invariant subdomains of B (of all sizes and at all points) exist in the form of simple operator balls cλ+Tλ(B), for cλ∈B and Tλ an invertible linear map. These are exact infinite dimensional analogues of the invariant discs in Δ, the invariant ellipsoids in the Hilbert ball and invariant domains in finite dimensional triples. Results are new for rank >2, even for classical spaces such as C⁎-algebras and JB⁎-algebras.

Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains

Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.

The fixed point property and unbounded sets in spaces of negative curvature

Motivated by the well-known cases of the real Hilbert ball and complete ℝ-trees, both being particular cases of CAT(-1) spaces, we give an affirmative answer to the question of whether the geodesic boundedness property is a necessary and sufficient condition for a closed convex subset K of a complete CAT(κ) space, with κ < 0, to have the fixed point property for nonexpansive mappings.

Denjoy–Wolff theory for finite-dimensional bounded symmetric domains

Let \(B\) be a finite-dimensional bounded symmetric domain and \(f:B\rightarrow B\) be a holomorphic map having no fixed point in \(B\). For subsequential limits, \(g\), of \((f^n)\), we establish conditions, in terms of the Wolff point, \(\xi \), of \(f\), on which boundary components of \(B\) can contain \(g(B)\). We extend Herve’s 1954 theorem on the bidisc to any finite product of bounded symmetric domains, namely if \(B=B_1\times \cdots \times B_n\) and \(\xi =(\xi _1,\ldots ,\xi _n)\) then there exists \(d=(d_1,\ldots , d_n) \in \partial B\), satisfying \( \overline{K_{d_i}} \cap \overline{K_{\xi _i}}\ne \emptyset ,\) such that $$\begin{aligned} \pi _i(g(B))\subseteq d_i+B_0(d_i), \end{aligned}$$ where \(K_x\) denotes the affine boundary component of \(x\), \(\pi _i\) is projection on the \(i\)th coordinate and \(B_0(d_i)\) is a bounded symmetric subdomain of \(B_i\). This simplifies if \(\xi _i\) is extreme, and even more so if \(B_i\) is a Hilbert ball.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss the role of infinite-dimensional non-squeezing results in the study of Hamiltonian PDEs and show some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

Some Inequalities for the Horosphere Function and Hyperbolically Nonexpansive Mappings on the Hilbert Ball

Some Inequalities for the Horosphere Function and Hyperbolically Nonexpansive Mappings on the Hilbert Ball

Iteration of self-maps on a product of Hilbert balls

Let D=D1×⋯×Dp be a product of Hilbert balls, with coordinate maps πj:D¯→D¯j on the closure D¯, for j=1,…,p. Let f be a fixed-point free self-map on D, which is nonexpansive in the Kobayashi distance, and compact for p⩾2. We describe the horospheres invariant under f and show that there exist a boundary point (ξ1,…,ξp) of D and a nonempty set J⊂{1,…,p} such that each limit function h of the iterates (fn) satisfies ξj∈πj∘h(D)¯ for all j∈J and πj∘h(⋅)=ξj whenever πj∘h(D) meets the boundary of Dj. For a single Hilbert ball D1, either liminfn→∞‖f2n(0)‖<1 or (fn) converges locally uniformly to a constant map taking value at the boundary of D1.

A Generalized Version of the Earle-Hamilton Fixed Point Theorem for the Hilbert Ball

Let $D$ be a bounded domain in a complex Banach space. According to the Earle-Hamilton fixed point theorem, if a holomorphic mapping $F: D \mapsto D$ maps $D$ strictly into itself, then it has a unique fixed point and its iterates converge to this fixed point locally uniformly. Now let $\mathcal{B}$ be the open unit ball in a complex Hilbert space and let $F : \mathcal{B} \mapsto \mathcal{B}$ be holomorphic. We show that a similar conclusion holds even if the image $F(\mathcal{B})$ is not strictly inside $\mathcal{B}$, but is contained in a horosphere internally tangent to the boundary of $\mathcal{B}$. This geometric condition is equivalent to the fact that $F$ is asymptotically strongly nonexpansive with respect to the hyperbolic metric in $\mathcal{B}$.

The high order Schwarz-Pick lemma on complex Hilbert balls

In this paper we prove a high order Schwarz-Pick lemma for holomorphic mappings between unit balls in complex Hilbert spaces. In addition, a Schwarz-Pick estimate for high order Frechet derivatives of a holomorphic function f of a Hilbert ball into the right half-plane is obtained.

Convergence of a Sequence of Sets in a Hadamard Space and the Shrinking Projection Method for a Real Hilbert Ball

We propose a new concept of set convergence in a Hadamard space and obtain its equivalent condition by using the notion of metric projections. Applying this result, we also prove a convergence theorem for an iterative scheme by the shrinking projection method in a real Hilbert ball.