The group reduction for bounded cosine functions on UMD spaces

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(-A)^{1/2} generates a bounded C_0-group. The proof uses a transference principle for cosine functions.Comment: 16 pages, research articl

A transference principle for general groups and functional calculus on UMD spaces

We prove a transference principle for general (i.e., not necessarily bounded) strongly continuous groups on Banach spaces. If the Banach space has the UMD property, the transference principle leads to estimates for the functional calculus of the group generator. In the Hilbert space case, the results cover classical theorems of McIntosh and Boyadzhiev-de Laubenfels; in the UMD case they are analogues of classical results by Hieber and Pruess. By using functional calculus methods, consequences for sectorial operators are derived. For instance it is proved, that every generator of a cosine function on a UMD space has bounded H-infinity calculus on sectors.Comment: 17 pages, no figures. To be published in Mathematische Annale

Transference Principles for Semigroups and a Theorem of Peller

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows to recover the classical transference results of Calder\'on, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, culminating in a new proof of classical results by Peller from 1982. The method allows a generalization of his results away from Hilbert spaces to \Ell{p}-spaces and --- involving the concept of $\gamma$-boundedness --- to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups

Form Inequalities for Symmetric Contraction Semigroups

Consider --- for the generator ${-}A$ of a symmetric contraction semigroup over some measure space $\mathrm{X}$, $1\le p < \infty$, $q$ the dual exponent and given measurable functions $F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ --- the statement: $$\mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0$$ {\em for all $\mathbb{C}^d$-valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in \mathrm{L}^q(\mathrm{X})$ for all $j$.} It is shown that this statement is valid in general if it is valid for $\mathrm{X}$ being a two-point Bernoulli $(\frac{1}{2}, \frac{1}{2})$-space and $A$ being of a special form. As a consequence we obtain a new proof for the optimal angle of $\mathrm{L}^{p}$-analyticity for such semigroups, which is essentially the same as in the well-known sub-Markovian case. The proof of the main theorem is a combination of well-known reduction techniques and some representation results about operators on $\mathrm{C}(K)$-spaces. One focus of the paper lies on presenting these auxiliary techniques and results in great detail.Comment: 29 pages; submitted to: Proceedings of the IWOTA, Amsterdam, July 2014. For this updated version, the term "complete contraction" has been exchanged for "absolute contraction" in order to avoid confusion with terminology used in operator space theory. Some small misprints and errors have been corrected, and a reference has been added. The proof of Theorem 4.11 was incomplete and has been amende

On systems with quasi-discrete spectrum

In this paper we re-examine the theory of systems with quasi-discrete spectrum initiated in the 1960's by Abramov, Hahn, and Parry. In the first part, we give a simpler proof of the Hahn--Parry theorem stating that each minimal topological system with quasi-discrete spectrum is isomorphic to a certain affine automorphism system on some compact Abelian group. Next, we show that a suitable application of Gelfand's theorem renders Abramov's theorem --- the analogue of the Hahn-Parry theorem for measure-preserving systems --- a straightforward corollary of the Hahn-Parry result. In the second part, independent of the first, we present a shortened proof of the fact that each factor of a totally ergodic system with quasi-discrete spectrum (a "QDS-system") has again quasi-discrete spectrum and that such systems have zero entropy. Moreover, we obtain a complete algebraic classification of the factors of a QDS-system. In the third part, we apply the results of the second to the (still open) question whether a Markov quasi-factor of a QDS-system is already a factor of it. We show that this is true when the system satisfies some algebraic constraint on the group of quasi-eigenvalues, which is satisfied, e.g., in the case of the skew shift.Comment: 25 pages. Accepted for publication in Studia Mathematic

Square Function Estimates and Functional Calculi

In this paper the notion of an abstract square function (estimate) is introduced as an operator X to gamma (H; Y), where X, Y are Banach spaces, H is a Hilbert space, and gamma(H; Y) is the space of gamma-radonifying operators. By the seminal work of Kalton and Weis, this definition is a coherent generalisation of the classical notion of square function appearing in the theory of singular integrals. Given an abstract functional calculus (E, F, Phi) on a Banach space X, where F (O) is an algebra of scalar-valued functions on a set O, we define a square function Phi_gamma(f) for certain H-valued functions f on O. The assignment f to Phi_gamma(f) then becomes a vectorial functional calculus, and a "square function estimate" for f simply means the boundedness of Phi_gamma(f). In this view, all results linking square function estimates with the boundedness of a certain (usually the H-infinity) functional calculus simply assert that certain square function estimates imply other square function estimates. In the present paper several results of this type are proved in an abstract setting, based on the principles of subordination, integral representation, and a new boundedness concept for subsets of Hilbert spaces, the so-called ell-1 -frame-boundedness. These abstract results are then applied to the H-infinity calculus for sectorial and strip type operators. For example, it is proved that any strip type operator with bounded scalar H-infinity calculus on a strip over a Banach space with finite cotype has a bounded vectorial H-infinity calculus on every larger strip.Comment: 49