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Using a recently developed $\mathcal H$-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on semigroup approximations and substantially improve some of them. We show that many of our estimates are essentially optimal, thus complementing the existing literature., Comment: 32 pages

This paper investigates when analytic Besov functions of $n$ variables act on the generators of $n$ commuting $C_0$-semigroups on a Banach space. The theory for $n=1$ has already been published, and the present paper uses a different approach to that case as well as extending to the cases when $n\ge2$. It also clarifies some spectral mapping properties and provides some operator norm estimates.

We prove a continuous-parameter version of the recent theorem of Katznelson-Tzafiri type for power-bounded operators which have a bounded calculus for analytic Besov functions. We also show that the result can be extended to some operators which have functional calculi with respect to some larger algebras., Comment: New title, to reflect the improvement in our main result. Accepted for publication in Studia Mathematica

This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that $\lim_{n\to\infty} \|T^n(I-T)\| =0$ if $T$ is a power-bounded operator on a Banach space and $\sigma(T) \cap \T \subseteq \{1\}$. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.

This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. The extend the Hille--Phillips calculus for (negative) generators $A$ of certain bounded $C_0$-semigroups, in particular for bounded semigroups on Hilbert spaces and bounded holomorphic semigroups on Banach spaces. They include functions outside the Hille-Phillips class, and they generally give sharper bounds for the norms of the resulting operators $f(A)$. The calculi are mostly based on appropriate reproducing formulas for the relevant classes of functions, and they rely on significant and interesting developments of function theory. They are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. They can also be used to derive several well-known operator norm-estimates, provide generalisations of some of them, and extend the general theory of operator semigroups. Our aim is to help readers to make use of these calculi without having to understand the details of their construction.

Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$, and assume that there is a bounded functional calculus for the operator $T$, so there is a bounded algebra homomorphism mapping functions $f \in \mathcal{A}$ to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type establish that $\lim_{n\to\infty} \|T^n f(T)\| = 0$ for functions $f \in \mathcal{A}$ whose boundary functions vanish on the unitary spectrum $\sigma(T)\cap \mathbb{T}$ of $T$, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when $\mathcal{A}$ is the Banach algebra $\mathcal{B}(\mathbb{D})$ of analytic Besov functions on $\mathbb{D}$. We prove a Katznelson-Tzafriri theorem for the $\mathcal{B}(\mathbb{D})$-calculus which extends several previous results., Comment: To appear in Journal of Operator Theory

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalizing and sharpening estimates, and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways which are interesting and significant. As consequences of our calculi, we derive several well-known operator norm-estimates and provide generalizations of some of them., Comment: 86 pages. This is a version of the paper to appear in Journal of the Institute of Mathematics of Jussieu

Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces $B^{ s, A }_{ q, X }$ for non-negative operators $A$ on Banach spaces $X$ with a full range of indices $s \in \mathbb{R}$ and $0 < q \leq \infty$. The approach we use is the dyadic decomposition of resolvents for non-negative operators, an analogue of the Littlewood-Paley decomposition in the construction of the classical Besov spaces. In particular, by using the reproducing formulas for fractional powers of operators and explicit quasi-norms estimates for Besov spaces we discuss the connections between the smoothness of Besov spaces associated with operators and the boundedness of fractional powers of the underlying operators.

It is shown, with two sets of indicators that separately load on two distinct factors, independent of one another conditional on the past, that if it is the case that at least one of the factors causally affects the other, then, in many settings, the process will converge to a factor model in which a single factor will suffice to capture the covariance structure among the indicators. Factor analysis with one wave of data can then not distinguish between factor models with a single factor versus those with two factors that are causally related. Therefore, unless causal relations between factors can be ruled out a priori, alleged empirical evidence from one-wave factor analysis for a single factor still leaves open the possibilities of a single factor or of two factors that causally affect one another. The implications for interpreting the factor structure of psychological scales, such as self-report scales for anxiety and depression, or for happiness and purpose, are discussed. The results are further illustrated through simulations to gain insight into the practical implications of the results in more realistic settings prior to the convergence of the processes. Some further generalizations to an arbitrary number of underlying factors are noted.

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra $\mathcal B$ of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of $\mathcal B$ and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory., Comment: 51 pages. This is a version of the paper to appear in Journal of Functional Analysis

We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them., Comment: This is the authors' accepted version of a paper which will be published in Mathematische Annalen

We study the rate of energy decay for solutions of a coupled wave-heat system on a rectangular domain. Using techniques from the theory of $C_0$-semigroups, and in particular a well-known result due to Borichev and Tomilov, we prove that the energy of classical solutions decays like $t^{-2/3}$ as $t\to\infty$. This rate is moreover shown to be sharp. Our result implies in particular that a general estimate in the literature, which predicts at least logarithmic decay and is known to be best possible in general, is suboptimal in the special case under consideration here. Our strategy of proof involves direct estimates based on separation of variables and a refined version of the technique developed in our earlier paper for a one-dimensional wave-heat system.

Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some estimates for the resolvent of $B$, and also to extend bounded operators related to $A$, for example a semigroup generated by $A$. When $X$ is a Hilbert space, one may also want $Y$ to be Hilbert space. Results of this type for bounded operators have been given by Arens, Read, M\"uller and Badea, and we give some extensions of their results to unbounded operators and we raise some open questions. A related problem is to improve properties of a $C_0$-semigroup satisfying lower bounds by extending it to a $C_0$-group on a larger space or by finding left-inverses. Results of this type for Hilbert spaces have been obtained by Louis and Wexler, and by Zwart, and we give some additional results., Comment: This is the authors' accepted version of the paper. It will be published in due course in the Journal of Operator Theory

We obtain integral representations for the resolvent of $\psi(A)$, where $\psi$ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and $A$ is a sectorial operator on a Banach space. As a corollary, for a wide class of functions $\psi$, we show that the operator $-\psi(A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $\psi$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for $A$ can be described, at least on Hilbert spaces, in terms of the existence of a bounded $H^{\infty}$-calculus for $A$. As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators., Comment: The paper has been accepted for publication in Advances in Mathematics. This is the authors' accepted version

We study maximal regularity in interpolation spaces for the sum of three closed linear operators on a Banach space, and we apply the abstract results to obtain Besov and H\"older maximal regularity for complete second order Cauchy problems under natural parabolicity assumptions. We discuss applications to partial differential equations., Comment: 18 pages

We prove $L^p$-analogues of the classical tauberian theorem of Ingham and Karamata, and its variations giving rates of decay. These results are applied to derive $L^p$-decay of operator families arising in the study of the decay of energy for damped wave equations and local energy for wave equations in exterior domains. By constructing some examples of critical behaviour we show that the $L^p$-rates of decay obtained in this way are best possible under our assumptions., Comment: Minor corrections have been made

We study the product formula $(fg)(A) = f(A)g(A)$ in the framework of (unbounded) functional calculus of sectorial operators $A$. We give an abstract result, and, as corollaries, we obtain new product formulas for the holomorphic functional calculus, an extended Stieltjes functional calculus and an extended Hille-Phillips functional calculus. Our results generalise previous work of Hirsch, Martinez and Sanz, and Schilling., Comment: This is the authors accepted manuscript for a paper being published in Mathematische Zeitschrift. The final publication is available at Springer via http://dx.doi.org/10.1007/s00209-014-1378-3

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus, and complex, real and harmonic analysis. It also leads to several results of independent interest., Comment: Version 2 includes numerous minor corrections, and is the authors' final version. The pape will be published in the Journal of the European Mathematical Society in April 2016