A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]^–1T[(P)]<-12 max {T, P^–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]^–1T[(N)]<-12c² max {tT, N^–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.     

Resolvent of the Toeplitz operator may increase arbitrarily fast

It is proved that for every sequence of points _n from the unit circle, _n1, and for an arbitrary sequence of positive numbers A_n, A_n, there exists a continuous real function u, such that for the Toeplitz operator T (acting in the Hardy space H²) with the symbol =e ^iu we have the estimates (T–_nI)^–1>A_n, n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 157, pp. 175–177, 1987.     

On the stability of semi-implicit methods for ordinary differential equations

The aim of this paper is to analyze the stability properties of semi-implicit methods (such as Rosenbrock methods,W-methods, and semi-implicit extrapolation methods) for nonlinear stiff systems of differential equations. First it is shown that the numerical solution satisfies y ₁ (h)y ₀, if the method is applied with stepsizeh to the systemy =Ay ( denotes the logarithmic norm ofA). Properties of the function(x) are studied. Further, conditions for the parameters of a semi-implicit method are given, which imply that the method produces contractive numerical solutions over a large class of nonlinear problems for sufficiently smallh. The restriction on the stepsize, however, does not depend on the stiffness of the differential equation. Finally, the presented theory is applied to the extrapolation method based on the semi-implicit mid-point rule.     

Hilbertraum-Approximation und Jacobi-Matrizen

Zusammenfassung SeiH ein Hilbertraum mit Norm und Skalarprodukt (,). Seien _n ,n=0, 1, 2, ... undf Elemente H, s, i natürliche Zahlen und >0 derart dass gilt: 1) _n =1,2)( _f ), _k )=0 für j–k >s, 3) ( _f , _n ) fürj k, 4) (f, _f =0 fürj>i. Seienf _n undf ^* die Projektionen vonf auf die durch ₀,..., _n bezw. ₀,₁,₂,... aufgespannten abgeschlossenen Unterräume. Das Hauptresultat der Arbeit besagt dass für hinreichend kleines KonstantenC>0 und 0<q<1 existieren mit: f _n -f ^*Cq ⁿ . Dieses Resultat steht in enger Beziehung zu gewissen unendlichen MatrizenA=(a _jk ), die charakterisiert sind durch: (*) es existiertm>0 so dassa _jk =0 für |j-k|>m. Das Hauptresultat wird auf unendliche lineare GleichungssystemeAf=0,Af=g angewandt, woA eine Matrix mit der Eigenschaft (*) ist, deren Diagonalen gewissen Wachstumsbedingungen genügen.

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LetH be a Hilbert space with norm and scalar product (,). Let _n ,n=0, 1, 2, ... andf be elements H, i, s integers and >0 such that: 1) _n =1,2)( _f ), _k )=0 for j–k >s, 3) ( _f , _n ) forj k, 4) (f, _f =0 forj>i. Letf _n andf ^* be the projections off onto the closed subspaces spanned by ₀,..., _n and ₀,₁,₂ .... respectively. The main result says that for sufficiently small there are constantsC>0 and 0<q<1 with f _n -f ^*Cq ⁿ . This result is closely related to certain infinite matricesA=(a _jk ) with the property: (*) there exists anm>0 such thata _jk =0 for |j-k|>m. The result is applied to infinite systems of linear equationsAf=0 andAf=g, whereA is a matrix with property (*), whose diagonals satisfy certain growth conditions.

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Dem Andenken von Professor Eduard Stiefel gewidmet     

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     

Comparison theorems for nonlinear evolution equations and their application to the investigation of the dependence of the solutions of the equation ut=δϕ(u) ON ϕ and on the boundary conditions

One obtains estimates of the form, whereu. are generalized solutions of the equationsdu/dt=Au, du/dt=Bu whileA, B are non-linear,m-dissipative operators in a Banach space, and there exists an operatorP:D(A)D(B), such thatPw · W+BPw –Aw, uniformly on some setw. These results are applied to the investigation of the dependence of the solutions of the Cauchy, Dirichlet problems and of the problem with the boundary condition –du/dn=(u) for the equation u₁=(u) on the continuous nondecreasing functions and.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 183–198, 1984.The author is sincerely grateful to O. A. Ladyzhenskaya and N. N. Ural'tseva for their interest in this paper and for useful discussions.     

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     

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.     

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     

Attracting properties for one dimensional flows of a general barotropic viscous fluid. Periodic flows

Summary We consider the motion of a barotropic compressible fluid in a one dimensional bounded region with impermeable boundary, see equation (1.1). Here, u(t, q) denotes the velocity and v(t, q) the specific volume. The quantity log v(t, q) measures the displacement of v(t, q) with respect to the equilibrium v 1. For the sake of brevity we denote here different norms by the simbol . We show that there is a positive constant r₀=r₀(), a small ball B₁ (r) (with radius R₁ (r), ), and a large ball B(r) (with radius R(r), ) such that the following holds, for each r [0, r₀ [(i) If f(t) < r for all t 0, and if (u(0), log v(0))R(r) (i.e. (u(0), log v(0)) B(r)) then, for sufficiently large values of t, (u(t), log v(t))R₁ (r); (ii) The solutions starting at time t=0 from the large ball B(r) have all the same asymptotic behaviour (see (1.11)); (iii) If f is T-periodic then there is a (unique) T-periodic solution (u(t), log v(t)) inside the small ball B₁ (r). This periodic solution atracts all solutions which intersect the large ball B(r). Periodic solutions had been previously studied only for very specific pressure laws, namely p(v)-log v and p(v)-v^–1.     

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.     

Approximation for the measures of ergodic transformations of the real line

Summary Let be the Frobenius-Perron operator corresponding to a nonsingular point-transformation of the real lineR into itself and let for each natural numbern, P _n be the discrete analogue ofP _f. It is shown that under fairly weak restrictions on, the equationf=P _f f has an unique solutionf ₀ such thatf ₀>0 (a.e.), ¦f ₀¦=1, and that this solution can be approximated inL ¹ (R) in two different ways: (1) by the sequence wheref0, ¦f¦=1, and (2) by the sequence {s _On } of simple functions such thats _n=P_n(s _On ).     

Spectral properties of certain operators on Hardy spaces of planar regions

If G is any region in the complex plane and H^P (G), 1p, denotes the Hardy space of analytic functions on G, then the principal aim of this paper is to present the spectral properties of the multiplication operators M defined on H^P(G) by Mf=f, where is any bounded analytic function on G. In order to do this, the speccial case in which G is bounded and (z)=z is studied in detail. In addition an operator called the truncation operator that is not a multilication operator and acts on the Hardy spaces of regions of the form /K for a compact set K is studied.Dedicated to the memory of my brotherThis research was partially supported by National Science Foundation grant MCS 83-204-26.     

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.     

The functional equation of multiplicative derivation is superstable on standard operator algebras

LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let : _{+ +} be a function with the property lim _t (t)t ^–1=0. Assume that a mappingD:A B(X) satisfies D(AB)–AD(B)–D(A)B<(A B) for all operatorsA, B D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT–TA for someTB(X).This work was supported by the Research Council of Slovenia     

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.     

Wave operators and similarity of certain non-normal one-parameter groups

This paper starts from a self-adjoint Schrödinger operator H(0) for three particles. If the interaction is dilation-analytic, H(0) has an analytic continuation H() (>0). G(t,) (–(±,a,) defined as strong limits, when t±, of t-dependent operators. The wave operators establish transformations under which the subgroups are similar to unitary groups. The scattering matrix determined by G(t,) is diagonal with respect to a.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

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   

The decomposition theorem for functions satisfying the law of large numbers

LetB be a Banach space with the Radon-Nikodym property and (S, , ) a probability space. Then anf: SB satisfies the strong law of large numbers if and only if there exists a Bochner integrable functionf ₁ and a Pettis integrable functionf ₂,f ₂f ₂=0 in the Glivenko-Cantelli norm, such thatf=f ₁+f ₂. The composition is unique.     

The possibility of integral formulas in R n

We consider the integral We solve the problem of determination of necessary and sufficient conditions in order that (u) be independent of the values of u(x) inside a bounded domain . These conditions are written in the form of a set of differential equations for the functions f(x,u,¯p,T^ij) on the set m{x; u+¯p+ T^ij<}. For such functions (u) is represented in the form of a boundary integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 52, pp. 35–51, 1975.