Holomorphic Semigroups Research Articles

Quantitative estimates for bounded holomorphic semigroups

We revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood–Paley–Stein theory for symmetric diffusion semigroups.

Introducing and solving generalized Black–Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in (L^1-L^infty )-interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved.

Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA) is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers, Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (\varphi_{t}) that can replace the semigroup of rotations in Sarason’s theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (\varphi_{t}) . In addition, we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch spaces, and surprisingly enough, we find that the semigroups for which the Bloch version of Sarason’s theorem holds are exactly the same as in the BMOA case.

Holomorphic semigroups of finite shift in the unit disc

We give three necessary and sufficient conditions so that a parabolic holomorphic semigroup (ϕt) in the unit disc is of finite shift. One is in terms of the asymptotic behavior of speeds of convergence, the second one is related to the hyperbolic metric of its Koenigs domain Ω and the latter one deals with Euclidean properties of Ω.

Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincare recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.

ЗАДАЧИ ШОУОЛТЕРА–СИДОРОВА И КОШИ ДЛЯ ЛИНЕЙНОГО УРАВНЕНИЯ ДЗЕКЦЕРА С КРАЕВЫМИ УСЛОВИЯМИ ВЕНТЦЕЛЯ И РОБЕНА В ОГРАНИЧЕННОЙ ОБЛАСТИ

Deterministic and stochastic initial boundary value problems for the Dzekzer equation describing the evolution of the free surface of a filtering fluid in a bounded region with a smooth boundary are considered. Wentzell and Robin conditions are set on the boundary of the domain, and either the Showalter-Sidorov condition or the Cauchy condition is taken as the initial condition. Note that for the filtration model under study, the Wentzell condition is considered, which is not a classical condition. In recent years, the boundary condition has been considered in the mathematical literature from two points of view (classical and neoclassical). Since Cauchy and Showalter–Sidorov initial conditions have been studied earlier in various situations, in this work, in the particular case of classical Wentzell and Robin conditions, by methods of the theory of degenerate holomorphic semigroups, exact solutions have been constructed, which allow to determine quantitative predictions of changes in geochemical regime of groundwater under unpressurized filtration. Nelson–Glicklich derivative theory was used in stochastic case. In particular, the investigation of the set problems in the context of Wentzell boundary conditions allowed to determine the processes occurring at the boundary of two media (in the region and at its boundary).

On sectoriality of degenerate elliptic operators

AbstractLet $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$, where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

Angular Extents and Trajectory Slopes in the Theory of Holomorphic Semigroups in the Unit Disk

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy–Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.

Numerical Computations for One Class of Dynamical Mathematical Models in Quasi-Sobolev Space

The article studies some mathematical models that represent one class of dynamical equations in quasi-Sobolev space. The analytical investigation of solvability of the Cauchy problem in the quasi-Sobolev space and theoretical results used to enhance and develop an algorithm structure of the numerical procedures to find approximate solutions for models, the steps of algorithm based on the theoretical investigation of models, new algorithm of numerical method allowing to find approximate solutions of mathematical models under study in quasi-Sobolev space. Construction a program implements an algorithm of numerical method that allow finding approximate solutions for models. To construct the theory of degenerate holomorphic semigroups of operators in quasi-Banach spaces of sequences, we used the classical methods of functional analysis, theory of linear bounded operators, spectral theory. To construct the operators of resolving semigroups we used the Laplace transform of operator-valued functions in quasi-Banach spaces of sequences. The numerical investigation for models generate some approximate solutions which are normally based on the modified projection method. The convergence of the approximate solution to the exact one theoretically is justified by the convergence of the corresponding series, the agreement of approximate computations with the theoretical solution is established.

A Case Study of Holomorphic Semigroups

In this paper, we investigate some characteristic features of holomorphic semigroups. In particular, we investigate nice examples of holomorphic semigroups whose every left or right ideal includes minimal ideal. These examples are compact topological holomorphic semigroups.

Fatou, Julia, and escaping sets of conjugate holomorphic semigroups

We define commutator of a holomorphic semigroup, and on the basis of this concept, we define conjugate semigroups of a holomorphic semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given holomorphic semigroup is nearly abelian. We also prove that image of each of Fatou, Julia, and escaping sets of a holomorphic semigroup under commutator (affine complex conjugating map) is equal respectively, to the Fatou, Julia, and escaping sets of the conjugate semigroup. Finally, we prove that every element of a nearly abelian holomorphic semigroup S can be written as the composition of an element from the set generated by the set of commutators !(S) and the composition of the certain powers of its generators..

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

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.

Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc

Let Δ ⊊ C \Delta \subsetneq \mathbb {C} be a simply connected domain, let f : D → Δ f:\mathbb {D} \to \Delta be a Riemann map, and let { z k } ⊂ Δ \{z_k\}\subset \Delta be a compactly divergent sequence. Using Gromov’s hyperbolicity theory, we show that { f − 1 ( z k ) } \{f^{-1}(z_k)\} converges non-tangentially to a point of ∂ D \partial \mathbb {D} if and only if there exists a simply connected domain U ⊊ C U\subsetneq \mathbb {C} such that Δ ⊂ U \Delta \subset U and Δ \Delta contains a tubular hyperbolic neighborhood of a geodesic of U U and { z k } \{z_k\} is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if ( ϕ t ) (\phi _t) is a non-elliptic semigroup of holomorphic self-maps of D \mathbb {D} with Koenigs function h h and h ( D ) h(\mathbb {D}) contains a vertical Euclidean sector, then ϕ t ( z ) \phi _t(z) converges to the Denjoy-Wolff point non-tangentially for every z ∈ D z\in \mathbb {D} as t → + ∞ t\to +\infty . Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the Denjoy-Wolff point but is oscillating, in the sense that the slope of the trajectories is not a single point.

On p-elliptic divergence form operators and holomorphic semigroups

Second order divergence form operators are studied on an open set with various boundary conditions. It is shown that the p-ellipticity condition of Carbonaro-Dragicevic and Dindos-Pipher implies extrapolation to a holomorphic semigroup on Lebesgue spaces in a p-dependent range of exponents that extends the maximal range for general strictly elliptic coefficients. Results have immediate consequences for the harmonic analysis of such operators, including Hoo-calculi and Riesz transforms.

Asymptotic behavior of orbits of holomorphic semigroups

Let (ϕt) be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let Ω be the starlike at infinity domain image of the Koenigs function of (ϕt). In this paper we characterize the type of convergence of the orbits of (ϕt) to the Denjoy-Wolff point in terms of the shape of Ω. In particular we prove that the convergence is non-tangential if and only if the domain Ω is “quasi-symmetric with respect to vertical axis”. We also prove that such conditions are equivalent to the curve [0,∞)∋t↦ϕt(z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Ω.

Logarithmic well-posed approximation of the backward heat equation in Banach space

In this paper, we apply a logarithmic approximation introduced by Boussetila and Rebbani in order to study an ill-posed Cauchy problem in Banach space. The proposed method induces a well-posed problem whose approximation parameter yields continuous dependence on modeling. Throughout, properties of holomorphic semigroups are utilized culminating in the motivating application which is the backward heat equation in Lp(R), 1<p<∞.

Characterization of Log-convex decay in non-selfadjoint dynamics

The short-time and global behaviour are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.

Resolvent representations for functions of sectorial operators

We obtain integral representations for the resolvent of ψ(A), where ψ 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 ψ, we show that the operator −ψ(A) generates a sectorially bounded holomorphic C0-semigroup on a Banach space whenever −A does, and the sectorial angle of A is preserved. When ψ 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∞-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.

Formula omitted]-semigroups and resolvent operators approximated by Laguerre expansions

In this paper we introduce Laguerre expansions to approximate vector-valued functions. We apply this result to approximate C0-semigroups and resolvent operators in abstract Banach spaces. We study certain Laguerre functions in order to estimate the rate of convergence of these expansions. Finally, we illustrate the main results of this paper with some examples: shift, convolution and holomorphic semigroups, where the rate of convergence is improved.

On Kernels and Images of Resolving Analytic Degenerate Semigroups in Quasi-Sobolev Spaces

The theory of holomorphic degenerate semigroups of operators was constructed earlier in Banach spaces and Frechet spaces. However, examples of study of mathematical objects show the necessity of their consideration in more general cases. In this article the theory of holomorphic degenerate semigroups of operators is transferred to quasi-Sobolev spaces of sequences which are quasi-normed and even quasi-Banach spaces. The article besides the introduction and references contains two paragraphs. In the first, resolving analytic degenerate semigroups are constructed. The second paragraph includes the study of kernels and images of analytic semigroups.