Proper holomorphic self-maps of symmetric powers of balls

We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.     

Isometers in the Cartesian product ofn unit open Hilbert balls with a hyperbolic metric

Summary In this paper we give a characterization of isometries in the Cartesian Products of n unit Hilbert balls with hyperbolic metrics and of their fixed point sets.     

Iteration of holomorphic maps on Lie balls

We investigate iterations of fixed-point free holomorphic self-maps on a Lie ball of any dimension, where a Lie ball is a bounded symmetric domain and the open unit ball of a spin factor which can be infinite dimensional. We describe the invariant domains of a holomorphic self-map f on a Lie ball D when f   is fixed-point free and compact, and show that each limit function of the iterates (fⁿ)$(f^n)$ has values in a one-dimensional disc on the boundary of D  . We show that the Möbius transformation _ga$g_a$ induced by a nonzero element a in D may fail the Denjoy–Wolff-type theorem, even in finite dimension. We determine those which satisfy the theorem.     

Iteration of compact holomorphic maps on a Hilbert ball

Given a compact holomorphic fixed-point-free self-map, , of the open unit ball of a Hilbert space, we show that the sequence of iterates, , converges locally uniformly to a constant map with . This extends results of Denjoy (1926), Wolff (1926), Hervé (1963) and MacCluer (1983). The result is false without the compactness assumption, nor is it true for all open balls of -algebras.

     

Random convex hulls in a product of balls

Summary The convex hull of a set of points sampled independently and uniformly from the Cartesian product of balls of various dimensions is investigated. Bounds on the asymptotic behavior of the expected combinatorial complexity volume, and mean width are derived when the distribution is held fixed and the sample size approaches infinity. The expected combinational complexity and volume are found to depend (up to constant factors) only on the greatest dimension of any factor ball and the number of balls of that dimension. On the other hand, the expected mean width depends only on the number of balls and the dimensions of the product.Supported by the National Science Foundation under Grants CCR-8658139 and CCR-8908782     

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 Fréchet derivatives of a holomorphic function f of a Hilbert ball into the right half-plane is obtained.     

Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator CΦ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for CΦ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in Cn (respectively Rn) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.     

Iteration of holomorphic maps in Hilbert spaces

Previous work (Gong-ning Chen, J. Math. Anal. Appl.98 (1984), 305–313) on iteration of holomorphic maps of Cⁿ is continued. The purpose of this note is to extend results given in the above mentioned reference to the case of complex Hilbert spaces. Other comments are appended.     

A projected Newton method in a Cartesian product of balls

We formulate a locally superlinearly convergent projected Newton method for constrained minimization in a Cartesian product of balls. For discrete-time,N-stage, input-constrained optimal control problems with Bolza objective functions, we then show how the required scaled tangential component of the objective function gradient can be approximated efficiently with a differential dynamic programming scheme; the computational cost and the storage requirements for the resulting modified projected Newton algorithm increase linearly with the number of stages. In calculations performed for a specific control problem with 10 stages, the modified projected Newton algorithm is shown to be one to two orders of magnitude more efficient than a standard unscaled projected gradient method.This work was supported by the National Science Foundation, Grant No. DMS-85-03746.     

非扩展非自映像不动点的迭代构造研究

运用Banach极限的技巧将收敛控制条件进一步放宽,去掉了∑∞n=1αn 1-αn<∞条件,在相对弱的条件Txn 1-Txn→w0,n→∞下证明了一个强收敛定理,改进了Wittmann的结果.     

On Weak Convergence of an Iteration Process with Minimal Norm in a Hilbert Space

We show that in a Hilbert space with a measure and a minimal norm on a chosen sequence of nodes, the interpolation process converges to the interpolated polynomial operator.     

Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space

We consider short asymptotic expansions for the probability of a sum of i.i.d. random elements to hit a ball in a Hilbert space H. The error bound for the expansion is of order O(n^-1). It depends on the first 12 eigenvalues of the covariance operator only. Moreover, the bound is non-uniform, i.e. the accuracy of the approximation becomes better as the distance between a boundary of the ball and the origin in H grows.     

Type I product systems of Hilbert modules

Christensen and Evans showed that, in the language of Hilbert modules, a bounded derivation on a von Neumann algebra with values in a two-sided von Neumann module (i.e. a sufficiently closed two-sided Hilbert module) are inner. Then they use this result to show that the generator of a normal uniformly continuous completely positive (CP-) semigroup on a von Neumann algebra decomposes into a (suitably normalized) CP-part and a derivation like part. The backwards implication is left open.In these notes we show that both statements are equivalent among themselves and equivalent to a third one, namely, that any type I tensor product system of von Neumann modules has a unital central unit. From existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered Fock modules. We, thus, find the analogue of Arveson's result that type I product systems of Hilbert spaces are symmetric Fock spaces.On the way to our results we have to develop a number of tools interesting on their own right. Inspired by a very similar notion due to Accardi and Kozyrev, we introduce the notion of semigroups of completely positive definite kernels (CPD-semigroups), being a generalization of both CP-semigroups and Schur semigroups of positive definite -valued kernels. The structure of a type I system is determined completely by its associated CPD-semigroup and the generator of the CPD-semigroup replaces Arveson's covariance function. As another tool we give a complete characterization of morphisms among product systems of time ordered Fock modules. In particular, the concrete form of the projection endomorphisms allows us to show that subsystems of time ordered systems are again time ordered systems and to find a necessary and sufficient criterion when a given set of units generates the whole system. As a byproduct we find a couple of characterizations of other subclasses of morphisms. We show that the set of contractive positive endomorphisms are order isomorphic to the set of CPD-semigroups dominated by the CPD-semigroup associated with type I system.     

A group of continuous self-maps on a topological groupoid

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every \(x\in G\), is denoted by \(S_G\). We show that \(S_G\) by a natural binary operation is a monoid. \(S_G(\alpha )\), the group of units in \(S_G\) precisely consists of those \(f\in S_G\) such that the map \(x\mapsto xf(x)\) is a bijection on G. Similar to the group of bisections, \(S_G(\alpha )\) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that \(S_G(\alpha )\) with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of \(G^2\) is isomorphic to the group \(S_G(\alpha )\) and the group of transitive bisections of G, \(Bis_T(G)\), is embedded in \(S_G(\alpha )\), where \(G^2\) is the groupoid of all composable pairs.     

Hilbert space representations of cross product algebras

Hilbert space representations of cross product *-algebras of the Hopf *-algebras with the coordinate algebras and of quantum vector spaces, and of with the coordinate algebras and of the corresponding quantum spheres, are investigated and classified. Invariant states on the coordinate *-algebras are described by two variants of the quantum trace.     

Rate of Metastability for Bruck'S Iteration of Pseudocontractive Mappings in Hilbert Space

We give a quantitative version of the strong convergence of Bruck's iteration schema for Lipschitzian pseudocontractions in Hilbert space.     

关于赋范空间上连续自映射的回归点

在赋范空间中讨论回归点的性质,主要得到了结果：（1）如果,是序列紧赋范空间X上的连续双射,x是f的任一回归点,则对于任意整数N〉0都存在f的回归点x0∈X使得f^n（x0）=x;（2）序列紧赋范空间上连续自映射的回归点集是f的强不变子集;（3）如果f是局部连通赋范空间X上的连续自映射,则f的每一个回归点或是类周期点或是类周期点的聚点．作为推论,在实直线段上得到了类似的结论．     

The Iteration Formulae of the Maslov-Type Index Theory in Weak Symplectic Hilbert Space

The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.     

A Gradient Iteration Method for Functional Linear Regression in Reproducing Kernel Hilbert Spaces

We consider a gradient iteration algorithm for prediction of functional linear regression under the framework of reproducing kernel Hilbert spaces.In the algorithm, we use an early stopping technique, instead of the classical Tikhonov regularization, to prevent the iteration from an overfitting function.Under mild conditions, we obtain upper bounds, essentially matching the known minimax lower bounds, for excess prediction risk. An almost sure convergence is also established for the proposed alg...