Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

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.     

A generalization of Fermat's little theorem

Fermat's Little Theorem states that if p is a prime number and gcd (x,p) = 1, then x^p?1 ≡ 1 (modp) If the requirement that gcd (x,p) = 1 is dropped, we can say x^p ≡ x(modp)for any integer x. Euler generalized Fermat's Theorem in the following way: if gcd (x,n) = 1 then x^φ(n) ≡ 1(modn), where φ is the Euler phi-function. It is clear that Euler's result cannot be extended to all integers x in the same way Fermat's Theorem can; that is, the congruence x^φ(n)+1 ≡ x(modn)is not always valid. In this note we show exactly when the congruence x^φ(n)+1 ≡ x(modn) is valid.     

Faber-hypercyclic operators

Let Ω be a bounded domain of the complex plane whose boundary is a closed Jordan curve and (F _n ) _n≥0 the sequence of Faber polynomials of Ω. We say that a bounded linear operator T on a separable Banach space X is Ω-hypercyclic if there exists a vector x of X such that {F _n (T)x: n ≥ 0} is dense in X. We show that many of the results in the spectral theory of hypercyclic operators involving the unit disk or its boundary have Ω-hypercyclic counterparts which involve the domain Ω or its boundary. The influence of the geometry of Ω or the smoothness of its boundary on Faber-hypercyclicity is also discussed.     

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 Theories Having Three Countable Models

A theory T is called almost ??₀-categorical if for any pure types p₁(x₁),…,p_n(x_n) there are only finitely many pure types which extend p₁(x₁) ∪…∪p_n(x_n). It is shown that if T is an almost ??₀-categorical theory with I(??₀,T) = 3, then a dense linear ordering is interpretable in T.     

Hamilton cycles,avoiding prescribed arcs,in close-to-regular tournaments

The local irregularity of a digraph D is defined as i_l(D) = max {|d⁺ (x) − d⁻ (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V₁, V₂, …, V_c} be a partition of V(T) such that |V₁| ≥ |V₂| ≥ … ≥ |V_c|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2i_l(T) + 2|V₁| + 2|V₂| − 2, i_l(T) + 3|V₁| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., i_l(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999     

Almost Injective Transformations of an Infinite Set

Let V be an infinite-dimensional vector space, let n be a cardinal such that ?₀ ≤ n ≤ dim V, and let AM(V, n) denote the semigroup consisting of all linear transformations of V whose nullity is less than n. In recent work, Mendes-Gonçalves and Sullivan studied the ideal structure of AM(V, n). Here, we do the same for a similarly-defined semigroup AM(X, q) of transformations defined on an infinite set X. Although our results are clearly comparable with those already obtained for AM(V, n), we show that the two semigroups are never isomorphic.     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

Zero product determined Jordan algebras,I

We show that the Jordan algebra 𝒮 of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {.,?.} from 𝒮?×?𝒮 into a vector space X satisfies {x, y}?=?0 whenever x?○?y?=?0, then there exists a linear map T : 𝒮?→?X such that {x,?y}?=?T(x?○?y) for all x, y?∈?𝒮 (here, x?○?y?=?xy?+?yx).     

A polynomial method for the pos/neg weighted 3-median problem on a tree

Let a connected undirected graph G  =  (V, E) be given. In the classical p-median problem we want to find a set X containing p points in G such that the sum of weighted distances from X to all vertices in V is minimized. We consider the semi-obnoxious case where every vertex has either a positive or negative weight. In this case we have two different objective functions: the sum of the minimum weighted distances from X to all vertices and the sum of the weighted minimum distances. In this paper we show that for the case p = 3 an optimal solution for the second model in a tree can be found in O(n ⁵) time. If the 3-median is restricted to vertices or if the tree is a path then the complexity can be reduced to O(n ³). This research has partially been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.     

On weak positive supercyclicity

A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT ⁿ x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ _p (T ^*) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT ⁿ x: n ∈ ℂ, r ∈ ℝ₊} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions. Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225. Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ? we say that x ∈ [0, 1] is the p-limit point of a sequence (x _n ) _n∈? ? [0, 1] (in symbols, x = p -lim _n∈? x _n ) if for every neighbourhood V of x, {n ∈ ?: x _n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f ^p : [0, 1] → [0, 1] is defined by f ^p (x) = p -lim _n∈? f ⁿ (x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f ^p . For a filter F we also define the ω ^F -limit set of f at x. We consider a question about continuity of the multivalued map x → ω _f ^F (x). We point out some connections between the Baire class of f ^p and tame dynamical systems, and give some open problems.     

On embedding trees into uniformly convex Banach spaces

We investigate the minimum value ofD =D(n) such that anyn-point tree metric space (T, ρ) can beD-embedded into a given Banach space (X, ∥·∥); that is, there exists a mappingf :T →X with 1/D ρ(x,y) ≤ ∥f(x) −f(y)∥ ≤ρ(x,y) for anyx,y εT. Bourgain showed thatD(n) grows to infinity for any superreflexiveX (and this characterized super-reflexivity), and forX =ℓ _p, 1 <p < ∞, he proved a quantitative lower bound of const·(log logn)^min(1/2,1/p). We give another, completely elementary proof of this lower bound, and we prove that it is tight (up to the value of the constant). In particular, we show that anyn-point tree metric space can beD-embedded into a Euclidean space, with no restriction on the dimension, withD =O(√log logn). This paper contains results from my thesis [Mat89] from 1989. Since the subject of bi-Lipschitz embeddings is becoming increasingly popular, in 1997 I finally decided to publish this English version. Supported by Czech Republic Grant GAČR 0194 and by Charles University grants No. 193, 194.     

Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables

Let V be a complex vector space with basis {x ₁, x ₂, . . . , x _n } and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x ₁, x ₂, . . . , x _n with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those words.     

Gromov hyperbolic cubic graphs

If X is a geodesic metric space and x ₁; x ₂; x ₃ ∈ X, a geodesic triangle T = {x ₁; x ₂; x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.     

Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C* and D* may overlap. When T: V W is linear and K V and D W are cones, these results will be applied to C = T(K) and D, giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D. The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T(X) = AX + X* A yields new and known results about the existence of block diagonal X's satisfying the Lyapunov condition: T(X) is an interior point of D. For the same V, W and D, T(X) = X – B* XB will be studied for certain cones K of entry-wise nonnegative X's.     

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).     

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)