Generalized inverses of cone preserving maps

Let K₁ and K₂ be solid cones in Rⁿ and R^m respectively, and let A be a linear operator such that AK₁?K₂. Necessary and sufficient conditions for the existence of various kinds of generalized inverses of A taking K₂ into K₁ are given. Projections which belong to π(K), the set of all linear operators that preserve a are studied. In particular, it is proved that a projection P in π(K) is a simplicial nonnegative linear combination of its rank-one operators iff Im P∩K is a simplical cone. An open question of Burns, Fiedler, and Haynsworth concerning minimal generating matrices of polyhedral cones is also extended and answered affirmatively.     

On additive maps preserving certain semi-Fredholm subsets

Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ?(X)?→??(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and ?emrl [M. Mbekhta and P. ?emrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n?≥?1, if φ preserves in both directions ? _n (X) (resp., 𝒬 _n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T)?=?UTV for all T or φ(T)?=?UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.     

Maps preserving common zeros between subspaces of vector-valued continuous functions

For metric spaces X and Y, normed spaces E and F, and certain subspaces A(X, E) and A(Y, F) of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps \({T:A(X,E)\rightarrow A(Y,F)}\) preserving common zeros, that is, maps satisfying the property

$Z(f) \cap Z(g) \neq \emptyset \Longleftrightarrow Z(Tf) \cap Z(Tg) \neq \emptyset \quad\quad\quad{\rm (P)}$

for any \({f, g \in A(X, E)}\), where \({Z(f) = \{x \in X: f(x) = 0\}}\). Moreover, we provide some examples of subspaces for which the automatic continuity of linear bijections having the property (P) is derived.

     

Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.

     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

We construct a bijective continuous area preserving map from a class of elongated dipyramids to the sphere, together with its inverse. Then we investigate for which such solid polyhedrons the area preserving map can be used for constructing a bijective continuous volume preserving map to the 3D-ball. These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and refinable grids on the elongated dipyramids. In particular, we show that HEALPix grids can be obtained from these maps. We also study the optimality of the logarithmic energy of the configurations of points obtained from these grids.     

Applications of Wigner’s theorem to positive maps preserving norm of operator products

We apply Wigner’s theorem to positive maps on standard operator algebras that preserve norm of operator products or sum of singular values of operator products. It follows that such preservers are of the form ϕ(A) = U AU* with U either a unitary or antiunitary operator. Selected from Journal of Mathematical Study, 2004, 37(1): 11–16     

Zero product preserving maps on Banach algebras of Lipschitz functions

Let (K,d) be a non-empty, compact metric space and α∈]0,1[. Let A be either lipα(K) or Lipα(K) and let B be a commutative unital Banach algebra. We show that every continuous linear map T:A→B with the property that T(f)T(g)=0 whenever f,g∈A are such that fg=0 is of the form T=wΦ for some invertible element w in B and some continuous epimorphism Φ:A→B.     

Generalized hypergeometric functions: product identities and weighted norm inequalities

We describe a method of obtaining weighted norm inequalities for generalized hypergeometric functions. This method is based upon our recent convolution theorem and some classical hypergeometric identities. In particular, it is shown that some product identities involving the divergent hypergeometric series lead to the convergent hypergeometric inequalities. A number of the new weighted norm inequalities for the Gaussian hypergeometric function, confluent hypergeometric function, and other generalized hypergeometric functions are presented.     

Linear maps preserving essential spectral functions and closeness of operator ranges

Let H be an infinite-dimensional complex Hilbert space and (H)the algebra of all bounded linear operators on H. Some characterizationsare obtained of linear maps on (H) preserving essential spectralfunctions such as the intersection of left essential spectrumand right essential spectrum of operators, preserving Fredholmoperators, and preserving semi-Fredholm operators, and preservingcloseness of operator ranges. A result of Mbekhta, Rodman andemrl (Integral Equations Operator Theory 55 (2006) 93-109) ongeneralized invertibility preserving linear maps is improved.     

Contractive Korovkin subsets in weighted spaces of continuous functions

We characterize those particular subsets of weighted spaces of continuous functions defined on locally compact Hausdorff spaces having the property that any net of positive linear contractions strongly converges to the identity operator provided it converges pointwise on them. Some variants and applications are indicated as well.     

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C₀(X) into C₀(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ_n δ ?h_n for a (possibly finite) sequence {x_n }_n of distinct points in X and a norm null sequence {h_n }_n of mutually disjoint functions in C₀(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On continuous maps between Grassmann manifolds

LetG _n,k denote the Grassmann manifold ofk-planes in ?ⁿ. We show that for any continuous mapf: G _n,k→G_n,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w₁(γ_{n, k}), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.     

Non-linear maps preserving solvability

Let M_n be the algebra of all n×n complex matrices and let L be the general linear Lie algebra gl(n,C) or the special linear Lie algebra sl(n,C). A bijective (not necessarily linear) map preserves solvability in both directions if both and ⁻¹ map every solvable Lie subalgebra of L into some solvable Lie subalgebra. If n3 then every such map is either a composition of a bijective lattice preserving map with a similarity transformation and a map [a_ij][f(a_ij)] induced by a field automorphism , or a map of this type composed with the transposition. We also describe the general form of such maps in the case when n=2. Using Lie's theorem we will reduce the proof of this statement to the problem of characterizing bijective maps on M_n preserving triangularizability of matrix pairs in both directions. As a byproduct we will characterize bijective maps on M_n that preserve inclusion for lattices of invariant subspaces in both directions.     

Linear maps preserving invariants

Let be a complex reductive group. Let denote for all . We show that, ``in general', . In case is the adjoint group of a simple Lie algebra , we show that is an order 2 extension of . We also calculate for all representations of .

     

Generalized Padé-type approximants to continuous functions

Summary We define generalized Padé-type approximants to continuous functions on a compact subset Eof Rⁿsatisfying the Markov's inequality and we show that the Fourier series expansion of a generalized Padé-type approximant to a u ∈ C ^∞(E ) matches the Fourier series expansion of uas far as possible. After studying the errors, we give integral representations and an answer to the convergence problem of a generalized Padé-type approximation sequence.     

Circuit preserving edge maps

It is proved that any one-to-one edge map f from a 3-connected graph G onto a graph G′, G and G′ possibly infinite, satisfying f(C) is a circuit in G′ whenever C is a circuit in G is induced by a vertex isomorphism. This generalizes a result of Whitney which hypothesizes f(C) is a circuit in G′ if and only if C is a circuit in G.