The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A^-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T^h of T, which contains the single-valued metric generalised inverse T^M and the continuous linear projector generalised inverse T⁺. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by G_H(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T^h of T is just the Moore-Penrose metric generalised inverse T^M. In the case, X and Y are finite dimension spaces Rⁿ and R^m, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.     

Copies of c0(Г) and ?∞(Г)/c0(Г) in quotients of Banach spaces with applications to Orlicz and Marcinkiewicz spaces

Let X be a Banach space, let Y be its subspace, and let Г be an infinite set. We study the consequences of the assumption that an operator T embeds ?_221E;(Г) into X isomorphically with T(c₀(Г)) ⊂ Y. Under additional assumptions on T we prove the existence of isomorphic copies of c₀(Г^ℵ0) in X/Y, and complemented copies ?_∞(Г) in X/Y. In concrete cases we obtain a new information about the structure of X/Y. In particular, L∞[O,1]/C[O,1] contains a complemented copy of ?_∞/c₀, and some natural (lattice) quotients of real Orlicz and Marcinkiewicz spaces contain lattice-isometric and positively I-complemented copies of(real) ?_∞/c₀.     

Some mapping properties of p-summing adjoint operators

Let X and Y be Banach spaces and u be a continuous linear operator from X to Y. We prove that if u^*, the adjoint operator of u, is p-summing for some p?1, then for any q?2, u takes (almost) unconditionally summable sequences in X into members of , the projective tensor product of ?q and Y.     

Unbounded strictly singular operators

Let D(T)⊂X→Y be an unbounded linear operator where X and Y are normed spaces. It is shown that if Y is complete then T is strictly singular if and only if T is the sum of a continuous strictly singular operator and an unbounded finite rank operator. A counterexample is constructed for the case in which Y is not complete.     

Minimal distance upper bounds for the perturbation of least squares problems in Hilbert spaces

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. In this paper, we present an optimal perturbation result on the least squares solutions to the operator equation Tx = y under the most general condition.     

G-narrow operators and G-rich subspaces

Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ? X is called G-rich if the quotient map q: X → X/Z is G-narrow.     

A note on abundance of certain semigroups of transformations with restricted range

Given a set X and a nonempty Y?X, we denote by T(X,Y) the subsemigroup of the full transformation semigroup on X consisting of all transformations whose range is contained in Y. We show that the semigroup T(X,Y) is right abundant but not left abundant whenever Y is a proper non-singleton subset of X.     

Lipschitz stability of operators in Banach spaces

We consider approximations of an arbitrarymap F: X → Y between Banach spaces X and Y by an affine operator A: X → Y in the Lipschitz metric: the difference F — A has to be Lipschitz continuous with a small constant ? > 0. In the case Y = ? we show that if F can be affinely ?-approximated on any straight line in X, then it can be globally 2?-approximated by an affine operator on X. The constant 2? is sharp. Generalizations of this result to arbitrary dual Banach spaces Y are proved, and optimality of the conditions is shown in examples. As a corollary we obtain a solution to the problem stated by Zs. Páles in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.     

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

The regular part of a semigroup of transformations with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$ is the largest regular subsemigroup of T(X,Y) and determines Green??s relations on T(X,Y). In this paper, we show that F(X,Y)?T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S ¹,S). Then we describe Green??s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.     

On σ-algebras related to the measurability of compositions

Given a measurable space (T, F), a set X, and a map ?: T → X, the σ-algebras N _Ф = ?_?∈Φ N _?, and M _Φ = ?_?∈Φ N _?, where G _?(t) = (t, ?(t)) and Φ ? X ^T , are considered. These σ-algebras are used to characterize the (F, B, ?)-measurability of the compositions g ○ ? and f о G _?, where g: X → Y, f: T × X → Y, and (Y, ?) is a measurable space. Their elements are described without using the operations ? ^?1 and G _? ^?1 .     

Ranks of submatrices in (skew-)Hermitian solutions to a quaternion matrix equation

Assume that X and Y are Hermitian solutions to quaternion matrix equation AXA ^?+BYB ^?=C, which are partitioned into 2×2 block forms. We in this paper give the maximal and minimal ranks of submatrices in the Hermitian solutions X=X ^?, Y=Y ^?, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. The findings of this paper widely extend the known results in the literature.     

A generalization of doubly stochastic matrices over a field

Let X and Y be m×n matrices over a field F such that Y^TX is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and $\text{Y}{}^{\mathrm{T}}\text{A =}\text{K}\text{?}\text{Y}{}^{\mathrm{T}}$ where K?Λ and $\text{K}\text{?}\text{? Λ′}$ are considered. It is shown that many properties of doubly stochastic matrices over a field have a natural generalization in terms of the set Δ(Λ,Λ′) of all such solutions.     

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

Perturbations of maximal monotone random operators

Let X be a Banach space, X¹ its dual, and Ω a measurable space. We study the solvability of nonlinear random equations involving operators of the form L + T, where L is a maximal monotone random operator from Ω × X into X¹ and T : Ω × X → X¹ a random operator of monotone type.     

More on convexity and smoothness of operators

Let X and Y be Banach spaces and T:Y→X be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T-modulus δT of convexity and the T-modulus ρT of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a Figiel-Pisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators.