Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

An adiabatic theorem for section determinants of spectral projections

We derive an adiabatic‐type theorem that expresses the section determinants of spectral projections of a selfadjoint operator through the solution to an operator‐valued Wiener‐Hopf equation. The solution theory of this equation is developed and for a special case a concrete criterion that ensures uniqueness of the solution is presented. Furthermore, for a special class of operators a dichotomy criterion, which is used in the proof of the adiabatic theorem, is proved. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong convergence studied by a hybrid type method for monotone operators in a Banach space

In this paper, we study a strong convergence for monotone operators. We first introduce the hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function.     

Dichotomy,bisectorial operators and unbounded spectral projections

For an operator having a uniformly bounded resolvent on a strip around the imaginary axis, the existence of—possibly unbounded—spectral projections corresponding to the left and right half-plane is proved. The operator is dichotomous if these projections are bounded, and an abstract perturbation theorem for dichotomy is derived. All results apply, with certain simplifications, to bisectorial operators. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Statistical approximation properties of q-Bleimann,Butzer and Hahn operators

The main aim of this study is to introduce a new generalization of q-Bleimann, Butzer and Hahn operators and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical BBH operators. The second aim of this study is to construct a bivariate generalization of the operator and also obtain the statistical approximation properties.     

Domination problem for narrow orthogonally additive operators

The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.     

Weak convergence of inner superposition operators

The equivalence of the weak (pointwise) and strong convergence of a sequence of inner superposition operators is proved as well as the criteria for such convergence are provided. Besides, the problems of continuous weak convergence of such operators and of representation of a limit operator are studied.

     

Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.      

Equi-statistical convergence of positive linear operators

Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.     

The structure of hemispaces in Rn

We study hemispaces (i,e., convex sets with convex complements) in Rⁿ. We give several geometric characterizations of hemispaces and several ways of representing them with the aid of linear operators and lexicographical order. We obtain a metric-affine classification of hemispaces, in terms of their “rank” and “type,” and a “decomposition theorem.” We also give some characterizations of affine transformations which preserve a hemispace.     

Interpolation problems in nest algebras

The main result of this paper is a theorem which allows one to determine when a finitely generated left ideal in certain reflexive operator algebras is trivial (i.e., contains the identity). This is based on a formula which expresses the distance from such an algebra to an arbitrary operator on the underlying Hilbert space. As an application, we are able to deduce an operator-theoretic variant of the Corona theorem. Some applications of the distance formula to quasitriangular operators are given, and we present some new “inner-outer” factorization theorems along the way to the main result.     

An abstract Szeg? theorem

Given a semi-group U(t) of bounded linear operators with bounded self-adjoint generator A we estimate the logarithm of the section determinants of U(t) in terms of A. When A is subject to an additional condition, which is related to so-called Følner sequences of orthogonal projections, this estimate implies a Szeg? type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when A is a Toeplitz operator or a compact operator.     

Rate of convergence in L_{p} approximation

In the present paper we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from \(L_{p}\left[ a,b\right] \) into itself using the concept of \(\mathcal {A}\) -summation processes. We also study the rate of convergence of these operators.     

The solution existence and convergence analysis for linear and nonlinear differential-operator equations in Banach spaces within the Calogero type projection-algebraic scheme of discrete approximations

The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated solution sets, based on a Leray-Schauder type fixed point theorem, are obtained.     

Multiplicative Decompositions of Holomorphic Fredholm Functions and ψ*-Algebras

In this article we construct multiplicative decompositions of holomorphic Fredholm operator valued functions on Stein manifolds with values in various algebras of differential and pseudo differential operators which are submultiplicative ψ* - algebras, a concept introduced by the first author. For Fredholm functions T(z) satisfying an obvious topological condition we. Prove (0.1) T(z) = A(z)(I + S(z)), where A(z) is holomorphic and invertible and S(z) is holomorphic with values in an “arbitrarily small” operator ideal. This is a stronger condition on S(z) than in the authors' additive decomposition theorem for meromorphic inverses of holomorphic Fredholm functions [12], where the smallness of S(z) depends on the number of complex variables. The Multiplicative Decomposition theorem (0.1) sharpens the authors' Regularization theorem [11]; in case of the Band algebra L(X) of all bounded linear operators on a Band space, (0.1) has been proved by J. Letterer [20] for one complex variable and by M. 0. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov [26] for the Banach ideal of compact operators.     

A Generalization of Tyuriemskih's Lethargy Theorem and Some Applications

Tyuriemskih's Lethargy Theorem is generalized to provide a useful tool for establishing when a sequence of (not necessarily) linear operators that converges point wise to the identity operator actually converges arbitrarily slowly. Then this generalization is used to answer affirmatively a 2010 conjecture of ours as well as establishing that all of the classical operators of Bernstein, Hermite-Fejer, Landau, Fejer, and Jackson converge arbitrarily slowly to the identity operator (and not just almost arbitrarily slowly as we established in 2010).     

Finite-difference operators in the study of differential operators: Solution estimates

The main results of the present paper are related to the use of finite-difference operators for estimating the norms of inverses of differential operators with unbounded operator coefficients. We obtain a new proof of the Gearhart-Prüss spectral mapping theorem for operator semigroups in a Hilbert space and estimate the exponential dichotomy exponents of an operator semigroup.     

Strong convergence of viscosity approximation methods for finding zeros of accretive operators in Banach spaces

In this paper, we introduce a composite iterative scheme by viscosity approximation method for finding a zero of an accretive operator in Banach spaces. Then, we establish strong convergence theorems for the composite iterative scheme. The main theorems improve and generalize the recent corresponding results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415-424] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631-643] as well as Aoyama et al. [K. Aoyama, Y Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350-2360], Benavides et al. [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248-249 (2003) 62-71], Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m-accretive operators, Fixed Point Theory and Appl. 2006 (2006) 1-10] and Kamimura and Takahashi [S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilberts spaces, J. Approx. Theory 106 (2000) 226-240].     

Generic convergence of infinite products of positive linear operators

We present several results concerning the asymptotic behavior of (random) infinite products of generic sequences of positive linear operators on an ordered Banach space. In addition to a weak ergodic theorem we also obtain convergence to an operator of the formf(·) wheref is a continuous linear functional and is a common fixed point.     

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Abstract

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ?. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δ^α and interpret them to those of previous works. Also, by using the convergence of Δ^α-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δ^α-statistical convergence of positive linear operators.