Characterizations of logA≥logB and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB ^r X^1–r A ^r XB ^r holds for r [0, 1]. In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logAlogB) by a norm inequality.Secondly, we shall study the condition under which , where is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=T) via Aluthge transformation.     

A new proof of the Komlós-Révész-theorem

The Komlós-Révész theorem states: For r.v.s.X _n with X _{n 1}M there exists a subsequenceX _{k n} and a r.v.X with X₁M such that

Numerical stability of the Chebyshev method for the solution of large linear systems

Summary This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systemsAx+g=0 whereA=A ^* is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x _k} approximates the solution such that x _k – is of order AA ^–1 where is the relative computer precision.We also point out that in general the Chebyshev method is not well-behaved, which means that the computed residualsr _k=Ax _k+g are of order A²A ^–1.This work was supported in part by the Office of Naval Research under Contract N0014-67-0314-0010, NR 044-422 and by the National Science Foundation under Grant GJ32111     

Round-off error analysis of iterations for large linear systems

Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA _x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A ^*>0 andA has PropertyA. This means that the computed resultx _k approximates the exact solution with relative error of order A·A ^–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax _k –b is of order A² A ^–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

A fixed point theorem for local pseudo-contractions in uniformly convex spaces

It is proved that if D is a bounded open subset of a uniformly convex Banach space X and is a continuous mapping which is a local pseudo-contraction (e.g., locally nonexpansive) on D, then T has a fixed point in D if there exists xD such that z–Tz相似文献   

Small into isomorphisms onL ∞ spaces

IfT is an isomorphism ofL (A, ) intoL (B, ) which satisfies the condition T T ^–11+, where (A, ) is a -finite measure space, thenT/T is close to an isometry with an error less than 4.     

On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

A Note on the Optimal L 2-Estimate of the Finite Volume Element Method

In this note, the optimal L ²-error estimate of the finite volume element method (FVE) for elliptic boundary value problem is discussed. It is shown that u–u _{h 0}Ch ²|ln h|^1/2f_1,1 and u–u _{h 0}Ch ²f_1,p , p>1, where u is the solution of the variational problem of the second order elliptic partial differential equation, u _h is the solution of the FVE scheme for solving the problem, and f is the given function in the right-hand side of the equation.     

Upper bounds for theL p -norms of martingales

Summary Let (f _n ) be a martingale. We establish a relationship between exponential bounds for the probabilities of the typeP(|f _n |>·T(f _n )) and the size of the constantC _p appearing in the inequality f ^* _p C _p T ^*(f) _p , for some quasi-linear operators acting on martingales.This research was supported in part by NSF Grant, no. DMS-8902418On leave from Academy of Physical Education, Warsaw, Poland     

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.     

Convergence of random products of compact contractions in Hilbert space

Let {T₁, ..., T_N} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. Roughly, the S_n converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): x_n1, Tx_n1 implies (I-T)x_n0 weakly, (S): x_n1, Tx_n1 implies (I-T)x_n0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x²K(x²–Tx²).We have three main results in the event that the T_i's are compact contractions. First, if r assumes each value infinitely often, then S_n converges uniformly to the projection Q on the subspace _i= ₁ ^N [x|T_ix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T₁, ..., T_N) denotes the algebraic semigroup generated by the T_i's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.     

Generalizations of the projection method with applications to SOR theory for hermitian positive semidefinite linear systems

Summary Ann×n complex matrixB is calledparacontracting if B₂1 and 0x[N(I-B)]Bx₂<x₂. We show that a productB=B _k B _k–1 ...B ₁ ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB _i 's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.The work of this author was supported in part by NSF Research Grant No. MCS-8400879     

Extreme values and the law of the iterated logarithm

Summary IfX takes values in a Banach spaceB and is in the domain of attraction of a Gaussian law onB, thenX satisfies the compact law of the iterated logarithm (LIL) with respect to a regular normalizing sequence { _n } iffX satisfies a certain integrability condition. The integrability condition is equivalent to the fact that the maximal term of the sample {X ₁, X ₂,..., X _n} does not dominate the partial sums {S _n}, and here we examine the precise influence of these maximal terms and its relation to the compactLIL. In particular, it is shown that if one deletes enough of the maximal terms there is always a compactLIL with non-trivial limit set.Supported in part by NSF Grant MCS-8219742Work done while visiting the University of Wisconsin, Madison, with partial support by NSF Grant MCS-8219742     

An extremal property of outer functions

Our main result is the following: iff (z) is in the space H², and F(z) is its outer part, then F⁽ⁿ⁾_H2F⁽ⁿ⁾_H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971.     

Stability inL 1 of some integral operators

A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL ¹, i.e. lim _n P ⁿ f=0 forfL ¹ and f(x) dx=0.     

A superfast algorithm for multi-dimensional Padé systems

For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n ₀,...,n _k ) inO(n · log²n) block-matrix operations, where n=n ₀+...+n _k . Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(n²). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.Supported in part by NSERC grant No. A8035.Partially supported by NSERC operating grant No. 6194.     

On the Global Solvability of the Cauchy Problem for Singular Functional Differential Equations

Sufficient conditions are found for the global solvability of the weighted Cauchy problem where f : C([a, b]; R ⁿ) L _loc([a, b]; R ⁿ) is a singular Volterra operator, c ₀ R ⁿ, h : [a, b] [0, +] is a function continuous and positive on [a, b], and · is the norm in R ⁿ.     

Discretely normed Abelian groups

A discrete norm on an Abelian groupA is a non-negative function · A which satisfies the triangle inequality, is homogenous with respect to scaling ofA by and is bounded away from 0 onA/{0}.A countable Abelian group is discretely normed if and only if the group is free.     

Perron-frobenius theory for Banach spaces with a hyperbolic cone

If X is a real Banach space, then the inequality x defines so-called hyperbolic cone in E=X. We develop a relevant version of Perron-Frobenius-Krein-Rutman theory.