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.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Sequences of contractions on L1-spaces

Let T_n, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L¹ for which T_ng → g weakly in L¹ for g?M. If T_n1 → 1 strongly, then T_nf → f strongly for all f in the closed vector sublattice in L¹ generated by M.This result can be applied to the determination of Korovkin sets and shadows in L¹. Given a set G ? L¹, its shadow S(G) is the set of all f?L¹ with the property that T_nf → f strongly for any sequence of contractions T_n, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L¹. For instance, if 1 ?G, then, where M is the linear hull of G and $\text{B}$_M is the sub-σ-algebra of $\text{B}$ generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements.     

A new topological degree theory for perturbations of the sum of two maximal monotone operators

Let X be an infinite dimensional real reflexive Banach space with dual space X^∗ and G⊂X, open and bounded. Assume that X and X^∗ are locally uniformly convex. Let T:X⊃D(T)→2^X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X^∗ maximal monotone, and C:X⊃D(C)→X^∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized (S₊)-condition w.r.t. S. Assume that D(S)=L⊂D(T)∩D(C), where L is a dense subspace of X, and 0∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+C, with degree mapping d(T+S+C,G,0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S+C, as above.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

Operators whose dual has non-separable range

Let X and Y be separable Banach spaces and T:X→Y be a bounded linear operator. We characterize the non-separability of T^?(Y^?) by means of fixing properties of the operator T.     

Best simultaneous approximation in L∞(μ, X)

Let Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L_∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ₀ is a sub‐σ ‐algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in ?ⁿ . We prove that L_∞(μ, Y) is N ‐simultaneously proximinal in L_∞(μ,X), and that if X is reflexive then L_∞(μ₀, X) is N ‐simultaneously proximinal in L_∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Class of Operators on Lh 2

Let D be the open unit disc in ? and let L_h ² be the space of quadratic integrable harmonic functions defined on D. Let \(\varphi: {\bar D}\rightarrow {\rm C}\) be a function in L^∞(D) with the property that φ(b) = lim_x→b,x?Dφ(x) for all b ? ?D. Define the operator C_φ in L_h ² as follows: C_φf = Q(φ·f),f ? L_h ², where Q is the orthogonal projection from L² (D) on L_h ². The following results are proved. If φ¦_?D ≡ 0, then C_φ is a compact linear operator and if φ¦_?D vanishes nowhere, then C_φ is a Fredholm operator.     

A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

Isometry Groups of Proper Hyperbolic Spaces

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H²_cb(G, L^p(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X). Partially supported by Sonderforschungsbereich 611.     

Translation-invariant operators continuous in measure

Let G be a compact abelian group and let L(G) be the space of measurable functions on G, equipped with the topology of convergence in measure. The only continuous translation-invariant linear operators on L(G) are the finite linear combinations of the translations themselves.     

On the Similarity of Perturbed Multiplication Operators

Let S be the multiplication operator by an independent variable x in L ₂(0,1), and let V be an integral operator of Volterra type. In this note, we find sufficient conditions for the similarity of the operators T := S + V and S and discuss some generalizations to an abstract setting of the results obtained.     

Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor

Let T be a free ergodic measure-preserving action of an abelian group G on (X,μ). The crossed product algebra R_T=L^∞(X,μ)? G has two distinguished masas, the image C_T of L^∞(X,μ) and the algebra S_T generated by the image of G. We conjecture that conjugacy of the singular masas S_T⁽¹⁾ and S_T⁽²⁾ for weakly mixing actions T⁽¹⁾ and T⁽²⁾ of different groups implies that the groups are isomorphic and the actions are conjugate with respect to this isomorphism. Our main result supporting this conjecture is that the conclusion is true under the additional assumption that the isomorphism γ : R_T⁽¹⁾→R_T⁽²⁾ such that γ(S_T⁽¹⁾)=S_T⁽²⁾ has the property that the Cartan subalgebras γ(C_T⁽¹⁾) and C_T⁽²⁾ of R_T⁽²⁾ are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group Aut(R_T,S_T), and a weaker one about entropy as a conjugacy invariant. We study also the Pukanszky and some related invariants of S_T, and show that they have a simple interpretation in terms of the spectral theory of the action T. It follows that essentially all values of the Pukanszky invariant are realized by the masas S_T, and there exist non-conjugate singular masas with the same Pukanszky invariant.     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

Browder-Weyl theorems, tensor products and multiplications

A Banach space operator T∈B(X) satisfies Browder's theorem if the complement of the Weyl spectrum σw(T) of T in σ(T) equals the set of Riesz points of T; T is polaroid if the isolated points of σ(T) are poles (no restriction on rank) of the resolvent of T. Let Φ(T) denote the set of Fredholm points of T. Browder's theorem transfers from A,B∈B(X) to S=LARB (resp., S=A⊗B) if and only if A and B^∗ (resp., A and B) have SVEP at points μ∈Φ(A) and ν∈Φ(B) for which λ=μν∉σw(S). If A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to LARB; again, restricting ourselves to the completion of X⊗Y in the projective topology, if A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to A⊗B.     

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.     

Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere S_X and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)