The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

Weakly sequentially continuous differentiable mappings

It is well known that a (linear) operator between Banach spaces is completely continuous if and only if its adjoint takes bounded subsets of Y^* into uniformly completely continuous subsets, often called (L)-subsets, of X^*. We give similar results for differentiable mappings. More precisely, if UX is an open convex subset, let be a differentiable mapping whose derivative is uniformly continuous on U-bounded subsets. We prove that f takes weak Cauchy U-bounded sequences into convergent sequences if and only if f^′ takes Rosenthal U-bounded subsets of U into uniformly completely continuous subsets of . As a consequence, we extend a result of P. Hájek and answer a question raised by R. Deville and E. Matheron. We derive differentiable characterizations of Banach spaces not containing ℓ₁ and of Banach spaces without the Schur property containing a copy of ℓ₁. Analogous results are given for differentiable mappings taking weakly convergent U-bounded sequences into convergent sequences. Finally, we show that if X has the hereditary Dunford–Pettis property, then every differentiable function as above is locally weakly sequentially continuous.     

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].     

Dynamics of composite functions meromorphic outside a small set

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E=E(f) and the cluster set of f at any a∈E with respect to is equal to . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f○g and g○f. Denote by F(f) and J(f) the Fatou and Julia sets of f. Let U be a component of F(f○g) and V be a component of F(g○f) which contains g(U). We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.     

Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and be nonexpansive mappings satisfying the weakly inward condition and F(T)≠∅, and be a fixed contractive mapping. The implicit iterative sequence {xt} is defined by for t∈(0,1)

xt=P(tf(xt)+(1−t)Txt).     

An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness

Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform from L¹(R) into C(E) is onto. Helson proved that then, for μ∈M(E), is possible only if μ=0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness.     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces

Let D be nonempty open convex subset of a real Banach space E. Let be a continuous pseudocontractive mapping satisfying the weakly inward condition and let be fixed. Then for each t∈(0,1) there exists satisfying yt∈tTyt+(1−t)u. If, in addition, E is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of has fixed point property for nonexpansive self-mappings, then T has a fixed point if and only if {yt} remains bounded as t→1; in this case, {yt} converges strongly to a fixed point of T as t→¹⁻. Moreover, an explicit iteration process which converges strongly to a fixed point of T is constructed in the case that T is also Lipschitzian.     

Convexity properties of harmonic measures

It is shown that any convex combination of harmonic measures , where U₁,…,Uk are relatively compact open neighborhoods of a given point x∈Rd, d?2, can be approximated by a sequence of harmonic measures such that each Wn is an open neighborhood of x in U₁∪?∪Uk.This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures.This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner.These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hörmander's condition, for example, sub-Laplacians on stratified Lie algebras).     

Dissipative sequences in infinite ergodic theory

Let T be an invertible, ergodic, measure-preserving transformation on a nonatomic, infinite, σ-finite measure space . A set is weakly wandering under the sequence of integers W={n_i}_i=1^∞ if the sets T^n_iA are disjoint. If, for a given sequence W, there exists a weakly wandering set of positive measure, then we call W a weakly wandering sequence. A sequence {k_i}_i=1^∞ is dissipative if, for every function , we have , almost everywhere. Every weakly wandering sequence is dissipative. In 1978, Hajian and Ito constructed a dissipative sequence which is not weakly wandering. An example of a dissipative sequence which is not a finite union of weakly wandering sequences is presented. Also, some further properties of dissipative sequences are studied.     

Product of diagonal entries of the unitary orbit of a 3-by-3 normal matrix

Let N be a 3×3 normal matrix. We investigate the sets where U(3) is the group of 3×3 unitary matrices and 1?k?3. Geometric properties of these sets are studied, namely, star-shapedness and simple connectedness are investigated. A method for the numerical estimation of is also provided for normal matrices of size 3.     

On harmonic typically real mappings

In this paper, we prove that a univalent orientation-preserving harmonic mapping defined on the unit disk U with the normalization f(0)=0, , is a typically real mapping, if f(U) is a starlike domain with respect to the origin or f(U) is convex in one direction.     

Radon-Nikodym property for the projective tensor product of Köthe function spaces

Let E be an order continuous Köthe function space (or an order continuous Banach lattice) and X be a dual Banach space. Then , the projective tensor product of E and X, has the Radon-Nikodym property if and only if both E and X do.     

Sampling sets and sufficient sets for A

We give new characterizations of the subsets S of the unit disc of the complex plane such that the topology of the space A^−∞ of holomorphic functions of polynomial growth on coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A^−∞. Our approach is based on a study of the so-called (p,q)-sampling sets which generalize the A^−p-sampling sets of Seip. A characterization of (p,q)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khôi and Thomas.     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.     

Lim's theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀x∈E, ∀α∈[0,1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .