Superstable operators on Banach spaces

It has been shown by W. Arendt—C.J.K. Batty and Yu.I. Lyubich— V.Q. Phong that the powers of a linear contraction on a reflexive Banach space converge strongly to zero if the boundary spectrum is countable and contains no eigenvalues. In this paper we characterize the countability of the boundary spectrum through a stronger convergence property in terms of ultrapower extensions. This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

Subnormal semigroups of composition operators

In this article we describe a model for subnormal semigroups of composition operators (with linear fractional symbol) acting on the Hardy space . We also discuss cyclicity of such semigroups in the context of more general results studied by J. H. Shapiro and P. S. Bourdon.

     

Inhomogeneous semigroups of commuting operators

The concepts of the homogeneously continuable semigroup of operators, and of infinitesimal and reproducing families of a semigroup, are introduced. The class of strongly continuous homogeneously continuable semigroups of commuting linear operators is discussed. This class contains in particular the class (C₀) of homogeneous semigroups. An analog of the Hill-Yosida theorem is proved for it.     

Ultrapowers and semigroups of operators

We study two semigroups of operators between Banach spaces, related with the finite representability ofc ₀ and ℓ₁. We show that these semigroups are open, have nice duality properties and can be characterized in terms of compact perturbations, and in terms of the properties of their ultrapowers. We obtain analogous results for their associated dual semigroups. Supported in part by DGICYT Grant PB 94-1052 (Spain). Supported by a postdoctoral Grant of the Ministry of Spain for Education and Science     

Strongly continuous semigroups of semi-Fredholm operators

Properties of infinitesimal generators of C₀ semigroups of semi-Fredholm operators are investigated. In addition, perturbations of the generator which give rise to such semigroups are studied.     

Embedding operators into strongly continuous semigroups

We study linear operators T on Banach spaces for which there exists a C₀-semigroup (T(t))_t≥0 such that T = T(1). We present a necessary condition in terms of the spectral value 0 and give classes of examples for which such a C₀-semigroup does or does not exist. Received: 22 December 2008, Revised: 7 April 2009     

Scalar-type spectral operators and holomorphic semigroups

We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold.

1. A generates a uniformly bounded holomorphic semigroup {e?^zA}_Re(z)≥0.
2. If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖F_N‖} _N=1 ^∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {F_N(s)x} _N=1 ^∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation.

The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e^?zA}_Re(z)≥0 generated by A, as follows. $$\frac{1}{2}(E\{ s\} )x + (E[0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X.     

Standard triangularization of semigroups of non-negative operators

We show that, under mild conditions, a semigroup of non-negative operators on L^p(X,μ) (for 1?p<∞) of the form scalar plus compact is triangularizable via standard subspaces if and only if each operator in the semigroup is individually triangularizable via standard subspaces. Also, in the case of operators of the form identity plus trace class we show that triangularizability via standard subspaces is equivalent to the submultiplicativity of a certain function on the semigroup.     

Contraction semigroups of elliptic quadratic differential operators

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.     

Ergodic semigroups of positivity preserving self-adjoint operators

We prove that an ergodic semigroup of positivity preserving self-adjoint operators is positivity improving. We also present a new proof (using Markov techniques) of the ergodicity of semigroups generated by spatially cutoff P(?)₂ Hamiltonians.     

BMO spaces associated with semigroups of operators

We study BMO spaces associated with semigroup of operators on noncommutative function spaces (i.e. von Neumann algebras) and apply the results to boundedness of Fourier multipliers on non-abelian discrete groups. We prove an interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on noncommutative L _p spaces for all 1 < p < ∞, with optimal constants in p.     

Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces (Ω,μ). We assume that there exists a semigroup of positive operators on Lp(Ω,μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's H¹-BMO duality theory. We also get a H¹-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative Lp spaces.     

Superstable groups

We initiate a study of superstable groups, generalizing previous work of Zil'ber and Cherlin. After having introduced the various tools of stability and model theory needed for that purpose we prove a general ‘Indecomposability Theorem’ and apply them to prove:(1) the definability of many subgroups of superstable groups (which has the consequence that, for superstable groups, “to be simple” is a first-order property);(2) the existence of ‘large’ abelian subgroups of all superstable groups; this allows us for example to give a transparent proof to the theorem of Cherlin stating that superstable division rings are commutative.This study of superstable groups is continued in [4].