Yu. I. Lyubich

Splitting-off of the boundary spectrum for almost-periodic operators and representations of semigroups

Results, communicated in Ukr. Mat. Zh.,36, No. 5, 1984, are presented here; they refer to almost periodic operators and representations of semigroups in an arbitrary Banach space and are concentrated around the so-called boundary spectrum splitting-off theorem.

On the normal solvability of cohomological equations on compact topological spaces

The functional equation ϕ(Fx) - ϕ(x) = γ(x) in continuous functions on a compact topological space X is considered, F: X → X is a continuous mapping. It is proved that the equation is normally solvable in C(X) if and only if F is preperiodic, i.e. F p+l = F l for some p ≥ 1, l ≥ 0. The solvability problem in measurable functions is also investigated.

Perron-frobenius theory for Banach spaces with a hyperbolic cone

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

A spectral criterion of asymptotic almost periodicity for uniformly continuous representations of abelian semigroups

Assume that the unitary asymptotic spectrum of a representation T is at most countable. Then for asymptotic almost periodicity it is necessary and sufficient that the eigensubspaces of the representations T, T*, corresponding to the unitary weights, should be in duality.

Separation of roots of matrix and operator polynomials

In a complex Hilbert spaceX for an arbitrary operator polynomialL(λ) (λ ∈ C) of degreem the following theorem is proved. If the equation (L(λ)x, x)=0 hasm distinct roots at every pointx ∈X, ‖x‖=1, then there existm pairwise disjoint connected sets in C such that each set contains a root at everyx. The minimal distance between the roots is separated from zero under the same assumption on the discriminant and the leading coefficient of that equation.

Exponential numbers of linear operators in normed spaces

Abstract Let X be a real or complex normed space, A be a linear operator in the space X, and x ϵ X. We put E(X, A, x) = min{l : l>0, ∥Al x∥ ≠ ∥x∥}, or 0 if ∥Ak x∥ = ∥x∥ for all integer k>0. Then let E(X, A) = supx E(X, A, x) and E(X) = supA E(X, A). If dim X ≥ 2 then E(X) ≥ dim X + 1. A space X is called E-finite if E(X) The main results are following. If X is polynomially normed of a degree p, then it is E-finite; moreover, E(X) ≤ Cpn+p−1 (over R), and E(X) ≤ (C p 2 n+p 2−1 ) 2 (over C). If X is Euclidean complex, then n2 − n + 2 ≤ E(X) ≤ n2 − 1 for n ≥ 3; in particular, E(X) = 8 if n = 3. Also, E(X) = 4 if n = 2. If X is Euclidean real, then [ n 2 ] 2 − [ n 2 ] + 2 ≤ E(X) ≤ n(n + 1) 2 , and E(X) = 3 if n = 2. Much more detailed information on E-numbers of individual operators in the complex Euclidean space is obtained. If A is not nilpotent, then E(X, A) ≤ 2ns − s2, where s is the number of nonzero eigenvalues. For any operator A we prove that E(X, A) ≤ n2 − n + t, where t is the number of distinct moduli of nonzero nonunitary eigenvalues. In some cases E-numbers are “small” and can be found exactly. For instance, E(X, A) ≤ 2 if A is normal, and this bound is achieved. The topic is closely connected with some problems related to the number-theoretic trigonometric sums.

Perron-Frobenius theory for finite-dimensional spaces with a hyperbolic cone

Abstract If X is a real n-dimensional space provided with a subnorm π, then the inequality ξ ⩾ π(x) defines a so-called hyperbolic cone inE =R⊕ X. In this case the Perron-Frobenius theory admits some special features. A relevant characterization of nonnegative operators in a matrix form is given first. Auxiliary information from spectral theory and from the geometry of subnormed spaces is collected as preparation.

The cohomological equations in nonsmooth categories

The functional equation

Lower bounds for projective designs, cubature formulas and related isometric embeddings

\varphi(Fx) - \varphi(x) = \gamma(x)

Upper bound for isometric embeddings ℓ₂^{}→ℓ_{}ⁿ

is considered in topological, measurable and related categories from the point of view of functional analysis and general theory of dynamical systems. The material is presented in the form of a self-contained survey.