CHARACTERISATIONS OF OPERATORS OF LOWER SEMI-FREDHOLM TYPE IN NORMED LINEAR SPACES

Abstract

Let X and Y be normed linear spaces. A linear operator T: D(T) ? X → Y is called an F-operator if its adjoint T′: D(T) ? Y′ → D(T)' is a φ+ -operator, i.e. has closed range and finite dimensional-kernel. Characterisations of an F_-operator T are obtained in the general case and in the case when T is closable. Unbounded strictly cosingular operators are defined and shown to belong to the class of F_ -admissible pertubations whenever Y is complete.     

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract

Let X and Y be normed spaces and T: D(T) ? X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily l_p. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.     

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.     

On the linearity of regression

Summary A stochastic process X={X _t :tT| is called spherically generated if for each random vector , there exist a random vector Y=(Y₁,..., Y _m) with a spherical (radially symmetric) distribution and a matrix A such that X is distributed as AY. X is said to have the linear regression property if (X ₀¦X ₁,..., X _n) is a linear function of X ₁,..., X _n whenever the X _j's are elements of the linear span of X. It is shown that providing the linear span of X has dimension larger than two, then X has the linear regression property if and only if it is spherically generated. The class of symmetric stable processes which are spherically generated is shown to coincide with the class of socalled sub-Gaussian processes, characterizing those stable processes having the linear regression property.This research was supported by a grant from the University of Wisconsin-Milwaukee     

Generating Random Vectors in (ℤ/<Emphasis Type="Italic">p</Emphasis>ℤ)<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> via an Affine Random Process

This paper considers some random processes of the form X _n+1=T X _n +B _n (mod p) where B _n and X _n are random variables over (ℤ/pℤ) ^d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B _n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)² steps suffice to make X _n close to uniformly distributed where X ₀ is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p ^b steps are needed for X _n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X ₀ is the zero vector.     

An Algebraic Characterization of Blumberg Spaces

Abstract

Let X be a topological space and let C(X) be the ring of continuous real-valued functions on X. We study T′(X) as an over-ring of C(X), where T′(X) denotes the set of all real-valued functions on X such that for each f ∈ T′(X) there exists a dense open subspace D of X such that f|D ∈ C(D). In this paper new algebraic characterizations of discrete spaces, open-hereditarily irresolvable spaces, and Blumberg spaces are obtained.     

On Certain Densely Invariant Quantities of Linear Operators

Let L(X,Y) denote the class of linear transformations T:D(T) ? X → Y where X and Y are normed spaces. A quantity f is called densely invariant if for every system L(X, Y) and every operator T ? L(X,Y) we have f(T/E)= f(T) whenever E is a core of T. The density invariance of certain well known quantities is established. In case Y is complete and T is closable, a stronger property is shown to hold for some of these quantitites, namely invariance under restriction to dense subspaces.     

MULTIPLIERS OF MATRIX SPACES

Abstract

Suppose X and Y are FK spaces in which ? the span of the coordinate vectors (e_n) is dense. Let L(X,Y) denote the space of all matrices of the form E_i(T(e_j)) as T ranges over all continuous linear operators from X into Y; here e_i represents the i^th coordinate vector and E_i represents the i^th coordinate functional. Let M(L(X, Y)) denote the space of all matrices B such that (B(i,j)A(i,j)) is in L(X,Y) whenever A is in L(X,Y). In this paper we shall show how the summability properties of X and Y determine the extent of M(L(X,Y)) and conversely how the extent of M(L(X,Y)) determines the summability properties of both X and Y.     

ON THE PROJECTION OF THE EFFICIENT SET AND POTENTIAL APPLICATIONS

Abstract

We study the efficient set X _E for a multiple objective linear program by using its projection into the linear space L spanned by the independent criteria. We show that in the orthogonally complementary space of L, the efficient points form a polyhedron, while in L an efficiency-equivalent polyhedron for the projection P(X _E ) of X _E can be constructed by algorithms of outer and inner approximation types. These algorithms can be also used for generating all extreme points of P(X _E ). Application to optimization over the efficient set for a multiple objective linear program is considered.     

Minimum and Surjectivity Moduli Preserving in Banach Space

Let m(T) and q(T) be respectively the minimum and the surjectivity moduli of T∈ℬ(X), where ℬ(X) denotes the algebra of all bounded linear operators on a complex Banach space X. If there exists a semi-invertible but non-invertible operator in ℬ(X) then, given a surjective unital linear map φ: ℬ(X)⟶ℬ(X), we prove that m(T)=m(φ(T)) for all T∈ℬ(X), if and only if, q(T)=q(φ(T)) for all T∈ℬ(X), if and only if, there exists a bijective isometry U∈ℬ(X) such that φ(T)=UTU ⁻¹ for all T∈ℬ(X).     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

DISJOINTNESS OF OPERATOR RANGES IN BANACH SPACES

Abstract

Let X be a Banach space. A linear subspace of X is called an operator range if it coincides with the range of a bounded linear operator defined on some Banach space. The paper studies disjointness and inclusion properties of various types of operator ranges in a separable infinite dimensional Banach space X. One of the main results is the following: Let E be a non-closed operator range in X. Then X contains a non-closed dense operator range R with the properties E∩= {0}, and R is decomposable, i.e. R = M + N where M,N are closed and infinite dimensional and M ∩ N = {0} (Theorem 6.2).     

Strongly Generated Banach Spaces and Measures of Noncompactness

To generalize the Hausdorff measure of noncompactness to other classes of bounded sets (like e. g. conditionally weakly compact or Asplund sets), we introduce Grothendieck classes. We deduce integral inequalities for quantities (called Grothendieck measures) related to these classes. As a by-product, we can answer a question concerning the measure of noncompactness for linear T : X → Y introduced in [14], and generalize a theorem about weak solutions of differential equations in Banach spaces.     

On the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings

Let X be a normed linear space and let S and T be multi-valued mappings of X into a family of closed, not necessarily compact subsets of X. In this paper some results on the convergence of the Ishikawa iterates associated with a pair S, T which satisfy the condition (8) below, are obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999     

Semigroups of Transformations Preserving an Equivalence Relation and a Cross-Section

Abstract

For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T(X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y)?∈?ρ then (xa, ya)?∈?ρ) and R (if r?∈?R then ra?∈?R). The semigroup T(X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T(X, ρ, R) in terms of Green's relations, describe the regular elements of T(X, ρ, R), and determine the following classes of the semigroups T(X, ρ, R): regular, abundant, inverse, and completely regular.     

Chapter 4:algebraic sets,polynomials, and other functions

Let T be a linear operator on the vector space V ofn×n matrices over a field F. We discuss two types of problems in this chapter. First, what can we say about T if we assume that T maps a given algebraic set such as the special linear group into itself? Second, let p(x) be a polynomial function (such as det) on V into F. What can we say about T if Tpreserves p(x), i.e. p(T(X)) = p(X) for all X in V?