Lyapunov functions versus exponential contractions

For nonautonomous linear equations in a Banach space admitting a nonuniform version of exponential contraction, we give an optimal characterization of the exponential behavior in terms of strict Lyapunov sequences. In particular, we construct explicitly strict Lyapunov sequences for each exponential contraction. We also consider the particular case of quadratic Lyapunov functions, and we use the corresponding characterization of the exponential behavior in terms of these functions to show that the stability of an exponential contraction persists under sufficiently small perturbations.     

On estimate for numerical radius of some contractions

For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality , where n ≥ 2 is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1335–1339, October, 2006.     

Inner functions in Cn

In this paper we prove that if f is a holomorphic function in a strictly pseudoconvex region D in Cⁿ, n > 1, with radial limit equal to 1 in modulus at each point of some nonempty open subset S of the boundary of D, then f const in D.Translated from Matematicheskii Zametki, Vol. 19, No. 1, pp. 63–66, January, 1976.I wish to thank B. V. Shabat for a discussion of the result and for critical remarks.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Inner functions as improving multipliers

If X⊂Y are two classes of analytic functions in the unit disk D and θ is an inner function, θ is said to be (X,Y)-improving, if every function f∈X satisfying fθ∈Y must actually satisfy fθ∈X. This notion has been recently introduced by K.M. Dyakonov. In this paper we study the (X,Y)-improving inner functions for several pairs of spaces (X,Y). In particular, we prove that for any p∈(0,1) the (Qp,BMOA)-improving inner functions and the (Qp,B)-improving inner functions are precisely the inner functions which belong to the space Qp. Here, B is the Bloch space. We also improve some results of Dyakonov on the subject regarding Lipschitz spaces and Besov spaces.     

Inner functions in weak Besov spaces

It is shown that inner functions in weak Besov spaces are precisely the exponential Blaschke products.     

Inner functions on spaces of homogeneous type

One shows that the methods of M. Hakim, N. Sibony, and E. Løw, used by them for the construction of inner functions in a sphere, can be applied also in a more general situation. The fundamental result of the paper is: For any positive continuous function H on the unit sphere S of the space ^d, there exists a real function u, harmonic in the unit ball, such that the function u is bounded in B and ¦u¦ =H almost everywhere on S.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 7–14, 1983.     

Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

In this paper we prove that for an arbitrary pair {T ₁, T ₀} of contractions on Hilbert space with trace class difference, there exists a function ξ in L ¹(T) (called a spectral shift function for the pair {T ₁, T ₀}) such that the trace formula trace(f(T ₁) ? f(T ₀)) = ∫_T f′(ζ)ξ(ζ)dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk.     

Moment-free numerical integration of highly oscillatory functions

^** Email: s.olver{at}damtp.cam.ac.uk The aim of this paper is to derive new methods for numericallyapproximating the integral of a highly oscillatory function.We begin with a review of the asymptotic and Filon-type methodsdeveloped by Iserles and Nørsett. Using a method developedby Levin as a point of departure, we construct a new methodthat utilizes the same information as a Filon-type method, andobtains the same asymptotic order, while not requiring the computationof moments. We also show that a special case of this methodhas the property that the asymptotic order increases with theaddition of sample points within the interval of integration,unlike all the preceding methods whose orders depend only onthe endpoints.     

Inner functions in the unit ball of Cn

Let B be the open unit ball of $\text{C}$ⁿ, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have $\text{¦u¦ = 1}$ a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let $\text{¦X¦}$ be the norm closure, in L^∞ = L^∞(S), of X. Some results: set I is dense in the unit ball of H^∞(B) in the compact-open topology. On $\text{S,}\text{Q}\text{?}\text{Q}$ is weak¹-dense in $\text{L}{}^{\mathrm{\infty }}\text{, ¦}\text{Q}\text{?}\text{Q¦}$ does not contain $\text{H}{}^{\mathrm{\infty }}\text{, C(S) ?¦}\text{Q}\text{?}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ ¦}\text{H}\text{?}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ L}{}^{\mathrm{\infty }}$. (When $\text{n = 1, ¦Q¦ = H}{}^{\mathrm{\infty }}\text{and}\text{¦}\text{Q}\text{?}\text{Q¦ = L}{}^{\mathrm{\infty }}$.) Every unimodular $\text{?}\text{?}\text{L}{}^{\mathrm{\infty }}$ is a pointwise limit a.e. of products $\text{u}\text{v}\text{?}\text{, u}\text{?}\text{I, ν}\text{?}\text{I}$. The zeros of every $\text{? ? 0}$ in the ball algebra (but not of every H^∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if $\text{∥?∥}{}_{\mathrm{\infty }}\text{< 1}$. However, $\text{?}\text{u}$ cannot be bounded in B if u ? I is non-constant.