A Besov algebra calculus for generators of operator semigroups and related norm-estimates (original) (raw)

The theory of Besov functional calculus: Developments and applications to semigroups

Journal of Functional Analysis, 2021

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra 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 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.

Lower estimates near the origin for functional calculus on operator semigroups

Journal of Functional Analysis, 2015

This paper provides sharp lower estimates near the origin for the functional calculus F (−uA) of a generator A of an operator semigroup defined on the (strictly) positive real line; here F is given as the Laplace transform of a measure or distribution. The results are linked to the existence of an identity element or an exhaustive sequence of idempotents in the Banach algebra generated by the semigroup. Both the quasinilpotent and non-quasinilpotent cases are considered, and sharp results are proved extending many in the literature.

Functional Calculus for the Series of Semigroup Generators Via Transference

Global Journal of Science Frontier Research , 2018

e apply an established transference principle in order to obtain the boundedness of certain functional calculi for sequence of semigroup generators. In particular, it is proved that by Markus Haase and Jan Rozendaal [20] if – be the sequence generates - semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on (sufficiently large) right half – plane that satisfies the sequence of . The bounded of this calculus grows at most logarithmically as . Asa consequence decay at . Then we show that each sequence of semigroup generator has a so-called (strong) m- bounded calculus for all andthat this property characterize the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so - called semigroups, the Hilbert space results actually hold in general Banach spaces.

Semigroup Kernels, Poisson Bounds, and Holomorphic Functional Calculus

Journal of Functional Analysis, 1996

Let L be the generator of a continuous holomorphic semigroup S whose action is determined by an integral kernel K on a scale of spaces L p (X ; \). Under mild geometric assumptions on (X, \), we prove that if L has a bounded H-functional calculus on L 2 (X; \) and K satisfies bounds typical for the Poisson kernel, then L has a bounded H-functional calculus on L p (X ; \) for each p # (1,). Moreover, if (X, \) is of polynomial type and K satisfies second-order Gaussian bounds, we establish criteria for L to have a bounded Ho rmander functional calculus or a bounded Davies Helffer Sjo strand functional calculus.

Calderón reproducing formulas and new Besov spaces associated with operators

Advances in Mathematics, 2012

Let L be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of Poisson type acting on L 2 (X) where X is a (possibly non-doubling) space of polynomial upper bound on volume growth. The aim of this paper is to introduce a new class of Besov spaces associated with the operator L so that when L is the Laplace operator − or its square root √ − acting on the Euclidean space R n , the new Besov spaces are equivalent to the classical Besov spaces. Depending on the choice of L, the new Besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers L α. Since our approach does not require the doubling volume property of the underlying space, it is applicable to any subset Ω of Euclidean spaces without any smoothness requirement on the boundary. We will also develop a number of Calderón reproducing formulas which play an important role in the theory of function spaces and are of independent interest. As an application, we study Besov spaces associated with Schrödinger operators with non-negative potentials satisfying reverse Hölder estimates.

Dichotomy Results for Norm Estimates in Operator Semigroups

Operator Theory: Advances and Applications, 2015

Dedicated to Charles Batty on the occasion of his sixtieth birthday. In line with an Oxford tradition, Charles Batty was sent on a mission: to instruct all the troops in C 0 semigroupsthis indeed was a brilliant decision!

Remarks on generators of analytic semigroups

Israel Journal of Mathematics, 1979

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex rn.mthers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups in L~, 1 < p < ~~, while the sufficient condition in the other characterization is meaningful in the case of non-j linear operators.