The solution of characteristic value-vector problems by Newton's method

LetT(λ) be a bounded linear operator in a Banach spaceX for eachλ in the scalar fieldS. The characteristic value-vector problemT(λ)x = 0 with a normalization conditionφ x = 1, whereφ ε X ^*, is formulated as a nonlinear problem inX xS:P(y) ≡ (T(λ)x, φ x - 1) = 0,y= (X, A). Newton's method and the Kantorovič theorem are applied. For this purpose, representations and criteria for existence ofP′(y)⁻¹ are obtained. The continuous dependence onT of characteristic values and vectors is investigated. A numerical example withT(λ) =A +λB +λ ² C is presented. Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.     

A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

Optimization of convex functions on w*-compact sets

We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw ^*-lower semicontinuous function ϕ defined on aw ^*-compact convex setC in a dual Banach spaceX ^* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw ^*−H _σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

The minimal cardinality where the Reznichenko property fails

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood ofx. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a setX of real numbers such thatC _p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space^Nℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC _p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ. The author is partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

On Characterization of Best Approximation with Certain Constraints

SupposeKis the intersection of a finite number of closed half-spaces {K_i} in a Hilbert spaceX, andx∈X\K. Dykstra's cyclic projections algorithm is a known method to determine an approximate solution of the best approximation ofxfromK, which is denoted byP_K(x). Dykstra's algorithm reduces the problem to an iterative scheme which involves computing the best approximation from the individualK_i. It is known that the sequence {x_j} generated by Dykstra's method converges to the best approximationP_K(x). But since it is difficult to find the definite value of an upper bound of the error ‖x_j−P_K(x)‖, the applicability of the algorithm is restrictive. This paper introduces a new method, called thesuccessive approximate algorithm, by which one can generate a finite sequencex₀, x₁, …, x_kwithx_k=P_K(x). In addition, the error ‖x_j−P_K(x)‖ is monotone decreasing and has a definite upper bound easily to be determined. So the new algorithm is very applicable in practice.     

A Chebyshev Set and its Distance Function

We prove that in a Banach space X with rotund dual X^* a Chebyshev set C is convex iff the distance function d_C is regular on X\C iff d_C admits the strict and Gâteaux derivatives on X\C which are determined by the subdifferential ∂x−

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x" height="11" width="10"> for each xX\C and

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x" height="11" width="10">P_C(x)\{cC:x−c=d_C(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff P_C is continuous. If the norms of X and X^* are Fréchet differentiable then C is convex iff d_C is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂⁻d_C(x) (∂_Fd_C(x)) is nonempty on X\C.     

Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-δ-algebra of and X a Banach space. Let multifunction t → Γ(t), t T, have a (X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε E_p x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = , then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.     

C-Cosine Functions and the Abstract Cauchy Problem, I

IfAis the generator of an exponentially boundedC-cosine function on a Banach spaceX, then the abstract Cauchy problem (ACP) forAhas a unique solution for every pair (x, y) of initial values from (λ − A)⁻¹C(X). The main result is a characterization of the generator of aC-cosine function, which may not be exponentially bounded and may have a nondensely defined generator, in terms of the associated ACP.     

Convergence of best approximations in a smooth Banach space

Let X be a reflexive, strictly convex Banach space such that both X and X^* have Fréchet differentiable norms, and let {C_n} be a sequence of non-empty closed convex subsets of X. We prove that the sequence of best approximations {p(x ¦ C_n)} of any x ε X converges if and only if lim C_n exists and is not empty. We also discuss measurability of closed convex set valued functions.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces

Let X be a connected finite CW complex and d _X : K ₀(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K ₀(A) = ℚ⊕ kerd _x and K ₁(A) = K ₁(C(X)), then A is isomorphic to an inductive limit of M _{n !}(C(X)). Received April 19, 2001, Accepted April 27, 2001.     

Simplex pivots on the set packing polytope

This paper considers the set packing problem max{wx: Ax b, x 0 and integral}, whereA is anm × n 0–1 matrix,w is a 1 ×n weight vector of real numbers andb is anm × 1 vector of ones. In equality form, its linear programming relaxation is max{wx: (x, y) P(A)} whereP(A) = {(x, y):Ax +I _m y =b, x0,y0}. Letx ¹ be any feasible solution to the set packing problem that is not optimal and lety ¹ =b – Ax ¹; then (x ¹,y ¹) is an integral extreme point ofP(A). We show that there exists a sequence of simplex pivots from (x ¹,y ¹) to (x*,y*), wherex* is an optimal solution to the set packing problem andy* =b – Ax*, that satisfies the following properties. Each pivot column has positive reduced weight and each pivot element equals plus one. The number of pivots equals the number of components ofx* that are nonbasic in (x ¹,y ¹).This research was supported by NSF Grants ECS-8005360 and ECS-8307473 to Cornell University.     

Separabilite vague dans l’espace des mesures sur un compact

Using the continuum hypothesis we construct a compact spaceK such that the spaceM (K)) of measures onK is vaguely separable, i.e., thatC (K) is injected intol ^∞, but thatC (K) is not isomorphic to a subspace ofl ^∞. It is shown that ifC(K) is isomorphic to a subspace ofC (K) is positively isometric to a subspace ofl ^∞(⌈). Nevertheless, under the continuum hypothesis one can construct a compact spaceL such that the spaceM ₁ ⁺ (L) of probabilities onL is vaguely separable, butL cannot be the support of a measureμ withL ¹(μ) separable in the norm.      

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Limiting -subgradient characterizations of constrained best approximation

In this paper, we show that the strong conical hull intersection property (CHIP) completely characterizes the best approximation to any x in a Hilbert space X from the set

K:=C∩{xX:-g(x)S},