Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer     

Multi-dimensional coupled diffusion process

In this paper, the formal differential expression is considered, whereQ ₁,Q ₂ are general elliptic operators with bounded coefficientsa _ij (x), b _i (x) and bounded absorptionC ₁₁ (x),C ₂₂ (x),xR ⁿ . It is proved thatL generates semigroup the coupled diffusion semigroup acting on functions with zero limit at infinity. The associated Markov process is constructed and shown to have nice trajectory properties.     

On tame operators between non-archimedean power series spaces

Let p ∈ {1, ∞}. We show that any continuous linear operator T from A1 (a) to Ap (b) is tame, i.e., there exists a positive integer c such that sup x||Tx||k/|x|ck ∞ for every k ∈ N. Next we prove that a similar result holds for operators from A∞(a) to Ap(b) if and only if the set Mb,a of all finite limit points of the double sequence (bi /aj ) i,j∈N is bounded. Finally we show that the range of every tame operator from A∞(a) to A∞(b) has a Schauder basis.     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

Kolmogorov equations for measures

We consider a semigroup of operators in the Banach space C _b (H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.      

Arens Regularity of Semigroup Algebras with Certain Locally Convex Topologies

The authors have recently introduced and studied a locally convex topology β¹(S) on the semigroup algebra M_a(S) of a locally compact semigroup S; as the main result, they showed that the strong dual of (M_a(S),β¹(S)) can be identified with the Banach space L^∞₀(S,M_a(S)) for a large class of locally compact semigroups S. Here, an application of this result is made to define and investigate an Arens multiplication on the second dual of (M_a(S),β¹(S)).     

Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds

We prove that if M is a three-manifold with scalar curvature greater than or equal to ?2 and Σ?M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of Σ is greater than or equal to 4π(g(Σ)?1), where g(Σ) denotes the genus of Σ. In the equality case, we prove that the induced metric on Σ has constant Gauss curvature equal to ?1 and locally M splits along Σ. We also obtain a rigidity result for cylinders (I×Σ,dt ²+g _Σ), where I=[a,b]?? and g _Σ is a Riemannian metric on Σ with constant Gauss curvature equal to ?1.     

Set-valued contractions and fixed points

A common fixed point theorem is proved for a family of set-valued contraction mappings in gauge spaces. This result is related to a recent result of Frigon for ‘generalized contractions’ and it includes a method for approximating the fixed point. The remainder of the paper is devoted to results for families of set-valued contraction mappings in hyperconvex spaces. It is proved, for example, that if M is a hyperconvex metric space and f_α is a family of set-valued contractions indexed over a directed set Λ and taking values in the space of all nonempty admissible subsets of M endowed with the Hausdorff metric, then the condition f_β(x)⊆f_α(x) for all x∈M and β⩾α implies that the set of points x∈M for which x∈⋂_α∈Λf_β(x) is nonempty and hyperconvex.     

Bicommutants and ranges of derivations

Let V be a vector space, V* its dual space and L(V) the algebra of all linear operators on V. For an operator a?∈?L(V) let a* be its adjoint acting on V*, and for a subset R of L(V) let R″ be its bicommutant. If R is a left noetherian subalgebra of L(V), then {a*: a?∈?R}″?=?{a*: a?∈?R″}. When R is singly generated R″ is described precisely. Further, for any two operators a, b?∈?L(V), b?∈?(a)″ if and only if the derivations d _a and d _b satisfy d _b (F(V))???d _a (F(V)), where F(V) is the set of all finite rank operators on V. In this case the inclusion d _b (L(V))???d _a (L(V)) also holds.     

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on H_b(V), V absolutely convex open set.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

QUASI-FROBENIUS BIMODULES OF FUNCTIONS ON A SEMIGROUP

Let _AM_B be a QF-bimodule, A a left Artinian ring, B a right Artinian ring, G a semigroup with a unit element (a monoid). Let M^G be the set of all functions on G with values in M. Consider M^G as an (AG, BG)-bimodule over the semigroup rings AG and BG. It is proved that the annihilator maps I → r_M^G (I) and R → l_AG (R) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated left ideals I ? AG and the set of right BG-submodules R ? M^G finitely generated over B. The maps J → l_M^G (J) and L → r_AG (L) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated right ideals J ? AG and the set of left AG-submodules L ? M^G finitely generated over A. This result also makes it possible, starting from a given QF-bimodule _A M_B , to construct new QF-bimodules _AG/IS_BG/J as bimodules of functions on a semigroup with values in M.     

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B _{( −∞, ∞ )}¹, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}. The author would like to thank to the referee for useful suggestions.     

A method of constructing Krall's polynomials

We propose a method of constructing orthogonal polynomials P_n(x) (Krall's polynomials) that are eigenfunctions of higher-order differential operators. Using this method we show that recurrence coefficients of Krall's polynomials P_n(x) are rational functions of n. Let P_n^(a,b;M)(x) be polynomials obtained from the Jacobi polynomials P_n^(a,b)(x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We find necessary conditions for the parameters a,b in order for the polynomials P_n^(a,b;M)(x) to obey a higher-order differential equation. The main result of the paper is the following. Let a be a positive integer and b⩾−1/2 an arbitrary real parameter. Then the polynomials P_n^(a,b;M)(x) are Krall's polynomials satisfying a differential equation of order 2a+4.     

On uniformly continuous Nemytskij operators generated by set-valued functions

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all Hölder functions ${\varphi : I \to C}Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all H?lder functions j: I ? C{\varphi : I \to C} into the set H _β (I, clb(Z)) of all H?lder set-valued functions f: I ? clb(Z){\phi : I \to clb(Z)} and is uniformly continuous, then

F(x,y)=A(x,y) \text+* B(x),       x ? I, y ? CF(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C