On derivable mappings

A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C^∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.     

Best proximity points of non-self mappings

Let A,B be nonempty subsets of a Banach space X and let T:A→B be a non-self mapping. Under appropriate conditions, we study the existence of solutions for the minimization problem min _x∈A ∥x?Tx∥.     

Hereditary invertible linear surjections and splitting problems for selections

Let A+B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h:X→A+B splits into a sum h=f+g of continuous mappings f:X→A and g:X→B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.     

Linear mappings derivable at some nontrivial elements

Suppose that A is an algebra and M is an A-bimodule. Let A be any element in A. A linear mapping δ from A into M is said to be derivable at A if δ(ST)=δ(S)T+Sδ(T) for any S,T in A with ST=A. Given an algebra A, such as a non-abelian von Neumann algebra or an irreducible CDCSL algebra on a Hilbert space H with dimH?2, we show that there exists a nontrivial idempotent P in A such that for any Q∈PAP which is invertible in PAP, every linear mapping derivable at Q from A into some unital A-bimodule (for example, A or B(H)) is derivation.     

Simple image set of linear mappings in a max-min algebra

For a given linear mapping, determined by a square matrix A in a max-min algebra, the set S_A consisting of all vectors with a unique pre-image (in short: the simple image set of A) is considered. It is shown that if the matrix A is generally trapezoidal, then the closure of S_A is a subset of the set of all eigenvectors of A. In the general case, there is a permutation π, such that the closure of S_A is a subset of the set of all eigenvectors permuted by π. The simple image set of the matrix square and the topological aspects of the problem are also described.     

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

Factorization of delta-monotone linear mappings

For a delta-monotone linear mapping we prove that the factors in the polar decomposition are delta-monotone. Also, we prove that every delta-monotone linear mapping can be factored into a product of (1-ε)-monotone mappings for any ε∈(0,1). As an application in nonlinear case, we give a new proof of the following fact: the quasiconformality constant K(δ,n) of a δ-monotone mapping can be chosen such that K(δ,n) tends to 1 as δ tends to 1.     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.     

A remark on weakly convex continuous mappings in topological linear spaces

Let C be a compact convex subset of a Hausdorff topological linear space and T:C→C a continuous mapping. We characterize those mappings T for which T(C) is convexly totally bounded.     

Approximately intertwining mappings

Let A be a Banach algebra, and let E be a weak Banach A-bimodule. An approximately intertwining mapping corresponding to a functional equation E(f)=0 is a mapping with f(0)=0 such that

‖E(f)‖?ε,     

Convex vector lattices and l-algebras

For A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit e_A, we have the Yosida representation  as a Riesz space in D(X_A), the lattice of extended real valued functions on the space of e_A-maximal ideas. This note is about those A for which  is a convex subset of D(X_A); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity e_A. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large).     

Qualitative stability of linear systems

This paper provides a finitely computable graph-theoretic answer to the following question concerning linear dynamical systems: When, given only the signs of entries (+, -, or 0) in a real square matrix A, can one be certain that all positive trajectories of the system ẋ = Ax are bounded? Matrices having such sign-patterns are called sign-quasistable. With “bounded” replaced by “convergent to the origin,” the matrices are called sign-stable and were fully described in earlier papers. However, when A's digraph has several strong components, so that the system is actually a hierarchy of subsystems, and when some of those subsystems fail to be sign-stable, the recognition of sign-quasistability is a very delicate matter. By means of certain graph color tests, it is possible to identify the system variables that are capable (for some choice of matrix-entry magnitudes and initial conditions) of emitting nonzero constant     

Product formula for imaginary resolvents with application to a modified Feynman integral

An abstract product formula for imaginary resolvents is proved for a pair of self-adjoint operators A, B of a complex Hilbert space. Here, A is assumed to be nonnegative and the positive part of B is arbitrary while its negative part is small with respect to A in the sense of quadratic forms. The proof is somewhat simpler than the author's original one which required both A and B to be nonnegative. When specialized, this theorem establishes the convergence of the “modified Feynman integral”—recently introduced by the author—in the most general case for which the Schrödinger equation can be solved without ambiguity.     

Remarks on pseudo-contractive mappings

Let X be a Banach space, B a closed ball centered at the origin in X, and T: B → X a pseudo-contractive mapping (i.e., (λ ? 1) ∥x ? y∥ ? ∥(λI ? T)(x) ? (λI ? T) (y)∥ for all x, y?B and λ > 1). It is shown here that the antipodal boundary condition: T(x) = ?T(?x) for all x?δB assures existence of a fixed point of T in B provided that the ball B has the fixed point property with respect to non-expansive self-mappings. Also included are some fixed point theorems which involve the Leray-Schauder condition.     

Regular abelian Banach algebras of linear maps of operator algebras

With H a complex Hilbert space we study regular abelian Banach subalgebras of the Banach algebra of bounded linear maps of B(H) into itself. If a ? b denotes the map x → axb, a, b, x ? B(H), it is shown that normalized positive maps in algebras of the form A ? A with A an abelian C¹-algebra, can be described by a generalized Bochner theorem.     

Dimensionality of biinfinite systems

The problem of determining the uniqueness of the coefficient of interpolation of M compactly supported real functions, with a biinfinite sequence of interpolation points, leads to the study of the kernel Z of the biinfinite block Toeplitz matrix

$\text{D=}\text{?}\text{?}\text{A}\text{B}\text{A}\text{B}\text{?}\text{?}$

. The dimension of Z is found by considering the “maximal solvable subspace” V (relative to A and B). Further results are obtained using the Kronecker canonical form of the matrix pencil A+λB and “restricted generalized inverses” of A (and B).     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

On single input controllability for infinite dimensional linear systems

Given a controllable linear system {A, B} where A is a Volterra operator, there exists a vector b in the range of B such that {A, b} is controllable. The case where A is a convolution operator on L²(0, ∞) is discussed and an example is given where a controllable system is not replaceable by a single input controllable system.