Norm Inequalities for Sums of Positive Operators. II

It is shown that if A and B are positive operators on a separable complex Hilbert space, then for every unitarily invariant norm. When specialized to the usual operator norm ||·|| and the Schatten p-norms ||·||_p, this inequality asserts that and These inequalities improve upon some earlier related inequalities. Other norm inequalities for sums of positive operators are also obtained.     

Inequalities for the Hadamard Weighted Geometric Mean of Positive Kernel Operators on Banach Function Spaces

Let K₁, . . . , K_n be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K₁, . . . , K_n, the operator K, satisfies the following inequalities where || · ||and r(·) denote the operator norm and the spectral radius, respectively. In the case of completely atomic measure space we show some additional results. In particular, we prove an infinite-dimensional extension of the known characterization of those functions satisfying for all non-negative matrices A₁, . . . , A_n of the same order.     

Some Geometric Aspects of Operators Acting from L 1

We study some properties of the space (L₁,X) of all continuous linear operators acting from L₁ to a Banach space X. It is proved that every operator T ∈ (L₁, X) ``almost' attains its norm at the entire positive cone of functions supported at some suitable measurable subset , μ(A) > 0. Using this fact and a new elementary technique we prove that every operator T∈ (L₁) = (L₁, L₁) is uniquely represented in the form T= R+S, R, S∈ (L₁) , where R is representable and S possess a special property (*). Moreover, this representation generates a decomposition of the space (L₁) into complemented subspaces by means of contractive projections (the fact that the subspace of all representable operators is complemented in (L₁) was proved before by Z. Liu).     

The Radon–Nikodym Property for Tensor Products of Banach Lattices

Let 1 ≤ p < ∞. We show that , the Fremlin projective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that , the Wittstock injective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from ℓ_p' to X is compact, where 1/p +1/p'= 1 and let ℓ_p' = c₀ if p = 1. The author gratefully acknowledges support from the Office of Naval Research Grant # N00014-03-1-0621     

Matrix Representations for Positive Noncommutative Polynomials

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x<sub>1</sub>, . . . , } and {h<sub>1</sub>, . . . , } for which q(X,H) is positive semidefinite whenever are tuples of selfadjoint matrices with ||X<sub>j</sub>|| ≤ 1 but H<sub>j</sub> unconstrained. The representation we obtain is a Gram representation in the variables h where P<sub>q</sub> is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose P_q such that the matrix P_q(X) is positive semidefinite for all ||X_j|| ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when g_x = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically to appear IOET July 2003] when restricted to x-domains of the type ||X_j|| ≤ 1. Partially supported by NSF, DARPA and Ford Motor Co. Partially supported by NSF grant DMS-0140112 Partially supported by NSF grant DMS-0100367     

Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals

Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of ℝ. If k belongs to class defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C¹ case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out.     

Commutativity of Almost <Emphasis Type="Italic">F</Emphasis>-algebras

Let A be an Archimedean vector lattice, let be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ ₀:A × A→ B is a positive orthosymmetric bimorphism, then there exists a positive bimorphism extension Ψ of Ψ ₀ to × in B which is orthosymmetric. This leads to a new and short proof of the commutativity of the almost f-algebras multiplications.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Exceptional times and invariance for dynamical random walks

Consider a sequence of i.i.d. random variables. Associate to each X _i (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X _i (t)} _{t ≥0} (i=1,2,···) whose invariant distribution is the law ν of X ₁(0). Benjamini et al. (2003) introduced the dynamical walk S _n (t)=X ₁(t)+···+X _n (t), and proved among other things that the LIL holds for n↦S _n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X _i (0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n↦S _n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ɛ-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that the infinite-dimensional process converges weakly in to the Ornstein–Uhlenbeck process in For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini et al. (2003) by proving that if the X _i (0)'s are lattice, mean-zero variance-one, and possess (2+ɛ) finite absolute moments for some ɛ>0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest. The research of D. Kh. is partially supported by a grant from the NSF.     

Measures with Bounded Variation with Respect to a Normed Ideal of Operators and Applications

We introduce in a natural way the notion of measure with bounded variation with respect to a normed ideal of operators and prove that for each maximal normed ideal of operators (, ), is true the following result: If U ∈ L(C(T,X), Y) with G the representing measure of U and G : Σ → ((X, Y),) has bounded variation, then U ∈ (C(T,X), Y). As an application of this result we prove that an injective tensor product of an integral operator with an operator belonging to a maximal normed ideal of operators (,) belongs also to (, ).     

Positive Solutions to Singular Semilinear Elliptic Problems

We obtain a characterization of all locally bounded functions p ≥ 0 for which the equation (E) Δu +p(x)ψ(u) = 0 has a positive solution in Ω vanishing on the boundary, where Ω is a domain of ℝ^N and ψ > 0 is a nonincreasing continuous function on ]0,∞[. In particular, for Ω = ℝ^N with N ≥ 3, it is shown that (E) has a (unique) positive solution in ℝ^N which decays to zero at infinity if and only if the set {p > 0} has positive Lebesgue measure and This condition can be replaced by if p is radial.     

Maximality of Positive Operator-valued Measures

A positive operator-valued measure is a (weak-star) countably additive set function from a σ-field Σ to the space of nonnegative bounded operators on a separable complex Hilbert space . Such functions can be written as M = V*E(·)V in which E is a spectral measure acting on a complex Hilbert space and V is a bounded operator from to such that the only closed linear subspace of , containing the range of V and reducing E (Σ), is itself. Attention is paid to an existing notion of maximality for positive operator-valued measures. The purpose of this paper is to show that M is maximal if and only if E, in the above representation of M, generates a maximal commutative von Neumann algebra.     

A fixed point theorem for a class of Sobolev maps

We prove an analog of the Brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n−1 and n/(n−1) respectively. Here Ω is a convex bounded open subset of Rⁿ.

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Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

On the exact Hausdorff dimension of the set of Liouville numbers. II

Let denote the set of Liouville numbers. For a dimension function h, we write for the h-dimensional Hausdorff measure of . In previous work, the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero has been located for various classes of dimension functions h satisfying certain rather restrictive growth conditions. In the paper, we locate the exact ``cut-point' at which the Hausdorff measure of drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function increases faster than any power function near 0, then , and if h is a dimension function for which the function increases slower than some power function near 0, then . This provides a complete characterization of all Hausdorff measures of without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if then does not have σ-finite measure. This answers another question asked by R. D. Mauldin. This work was done while Dave L. Renfro was at the Department of Mathematics at Central Michigan University.     

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .     

Hölder and Young Inequalities for the Trace of Operators

If A₁, . . . , A_m are positive semidefinite n × n matrices, and if p₁, . . . , p_m are positive real numbers such that then where |X| denotes and tr(X) denotes the trace of X. Moreover, equality holds in either of these inequalities if and only if . This result will be shown to hold as well in unital C*-algebras that have a faithful tracial state.     

An estimate on the supremum of a nice class of stochastic integrals and U-statistics

Let a sequence of iid. random variables ξ ₁, . . . ,ξ _n be given on a space with distribution μ together with a nice class of functions f(x ₁, . . . ,x _k ) of k variables on the product space For all f ∈ we consider the random integral J _n,k (f) of the function f with respect to the k-fold product of the normalized signed measure where μ _n denotes the empirical measure defined by the random variables ξ ₁, . . . ,ξ _n and investigate the probabilities for all x>0. We show that for nice classes of functions, for instance if is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too. Supported by the OTKA foundation Nr. 037886     

A Connected F-Space

We present an example of a compact connected F-space with a continuous realvalued function f for which the set is not dense. This indirectly answers a question from Abramovich and Kitover in the negative.     

Odd Degree Polynomials on Real Banach Spaces

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = expⁿ⁺¹τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ . Some of the work on this paper was done while the first author was a visitor to the Departamento de Análisis Matemático of the Universidad Complutense de Madrid, to which great thanks are given. The research of the second author was supported by grants: Institutional Research Plan AV0Z10190503, A100190502, GA ČR 201/04/0090.