Asuman Güven Aksoy

Nonstandard Methods in Fixed Point Theory

A unified account of the major new developments inspired by Maureys application of Banach space ultra-products to the fixed point theory for non-expansive mappings is given in this text. The first third of the book is devoted to laying a foundation for the actual fixed point theoretic results which follow. Set theoretic and Banach space ultra-products constructions are studied in detail in the second part of the book, while the remainder of the book gives an introduction to the classical fixed point theory in addition to a discussion of normal structure.

A selection theorem in metric trees

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping T* of a metric tree M with convex values has a selection T: M → M for which d(T(x),T(y)) < d H (T*(x),T*(y)) for each x,y ∈ M. Here by d H we mean the Hausdroff distance. Many applications of this result are given.

On the numerical index of vector-valued function spaces

Let X be a Banach space and μ a positive measure. In this article, we show that . Also, we investigate the positivity of the numerical index of lp -spaces.

Mixed Partial Derivatives and Fubini's Theorem

(2002). Mixed Partial Derivatives and Fubinis Theorem. The College Mathematics Journal: Vol. 33, No. 2, pp. 126-130.

Best Approximation in Numerical Radius

Let X be a reflexive Banach space. In this article, we give a necessary and sufficient condition for an operator T ∈ 𝒦(X) to have the best approximation in numerical radius from the convex subset 𝒰 ⊂ 𝒦(X), where 𝒦(X) denotes the set of all linear, compact operators from X into X. We also present an application to minimal extensions with respect to the numerical radius. In particular, some results on best approximation in norm are generalized to the case of the numerical radius.

Minimal numerical-radius extensions of operators

In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases L P , p = 1,∞, and in the case of self-adjoint extensions in L 2 , the two extensions and their norms are equal. We also show that, in the case of L P , 1 < p < ∞, and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is 1. An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from l p 3 onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for p ≠ 1, 2, ∞, and we determine the minimal numerical-radius projection in this same situation.

Bernstein's Lethargy Theorem in Fréchet spaces

In this paper we consider Bernsteins Lethargy Theorem (BLT) in the context of Frechet spaces. Let X be an infinite-dimensional Frechet space and let V = { V n } be a nested sequence of subspaces of X such that V n ź ź V n + 1 for any n ź N . Let e n be a decreasing sequence of positive numbers tending to 0. Under one additional but necessary condition on sup { dist ( x , V n ) } , we prove that there exist x ź X and n o ź N such that e n 3 ź dist ( x , V n ) ź 3 e n for any n ź n o . By using the above theorem, as a corollary we obtain classical Shapiros (1964) and Tyuriemskihs (1967) theorems for Banach spaces. Also we prove versions of both Shapiros (1964) and Tyuriemskihs (1967) theorems for Frechet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Frechet spaces will be discussed. We also give a theorem improving Konyagins (2014) result for Banach spaces. Finally, we present some applications of the above mentioned result concerning particular classes of Frechet spaces, such as Orlicz spaces generated by s -convex functions and locally bounded Frechet spaces.

A Generalization of N-Widths

This paper is a study of the n-widths defined by Kolmogorov. In section I we give definitions of n-widths of a set in a Banach space and n-widths of an operator acting between Banach spaces. Several important well known results about this concepts are also included in section I. In section II, we introduce a refined concept of an approximation scheme with respect to which a refined concept of n-widths can be defined. Theorems about generalized n-widths illustrate the fact that this is a genuine generalization. We finish by the question of finding concept of n-widths in the context of Orlicz modular spaces.

Minimal Projections with respect to Numerical Radius

In this paper we survey some results on minimality of projections with respect to numerical radius. We note that in the cases Lp, p = 1, 2, ∞, there is no difference between the minimality of projections measured either with respect to operator norm or with respect to numerical radius. However, we give an example of a projection from lp 3 onto a two-dimensional subspace which is minimal with respect to norm, but not with respect to numerical radius for p ≠ 1, 2,∞. Furthermore, utilizing a theorem of Rudin and motivated by Fourier projections, we give a criterion for minimal projections, measured in numerical radius. Additionally, some results concerning strong unicity of minimal projections with respect to numerical radius are given.

Sequences and Series of Functions

• We say that a sequence of functions {f n : D → ℝ} defined on a subset D ⊆ ℝ converges pointwise on D if for each x ∈ D the sequence of numbers {f n(x)} converge. If {f n} converges pointwise on D, then we define f : D → ℝ with \(f\left( x \right)\, = \,\mathop {\lim }\limits_{n \to \infty } \,f_n \left( x \right)\) for each x ∈ D. We denote this symbolically by f n → f on D.