Regularized semigroups,iterated Cauchy problems and equipartition of energy

The iterated Cauchy problem under consideration is $$\Pi _{k = 1}^n (d/dt - A_k )u(t) = 0(t \geqslant 0).(*)$$ Here {A ₁,..., A_n} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When eachA _k is a (C ₀) semigroup generator, this semigroup is of class (C ₀) and was studied by J. T. Sandefur [26]. This result is extended to the case when eachA _k generates aC-regularized semigroup (withC independent ofk). This means one can solveu′=Au, u(0)=f∈C (Dom (A)) and getu(t)→0 wheneverC ^?1f→0; hereC is bounded and injective. When theA _k are commuting generator withA _k-A_j injective fork≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form \(u = \sum\nolimits_{i = 1}^n {u_i } \) whereu _i is a solution ofu′ _i=A_iu_i. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy.     

Functional calculus for generators of uniformly bounded holomorphic semigroups

We present a method for constructing a functional calculus for (possibly unbounded) operators that generate a uniformly bounded holomorphic semigroup, e^−zA. (A will be called a generator.) These are closed, densely defined operators whose spectrum and numerical range are contained in [0,∞), with respect to an equivalent norm.     

Finite rank,relatively bounded perturbations of semigroups generators

Summary This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator A_F=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b H. While Part I studied the question of generation of a s.c. semigroup on H by A_F and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_F, given A and a H, via a suitable vector b H; alternatively, given A, via a suitable pair of vectors a, b H; (ii) spectrality of A_F—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {_n} of A, the given vector a H and a given suitable sequence {_n} of nonzero complex numbers, which guarantee the existence of a suitable vector b H such that A_F possesses the following two desirable properties: (i) the eigenvalues of A_F are precisely equal to _n+_n; (ii) the corresponding eigenvectors of A_F form a Riesz basis (a fortiori, A_F is spectral). While finitely many _ns can be preassigned arbitrarily, it must be however that _n 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups

In this paper, we obtain optimal bounds for convergence rate for Yosida approximations of bounded holomorphic semigroups. We also provide asymptotic expansions for semigroups in terms of Yosida approximations and obtain optimal error bounds for these expansions.     

A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that -sectorial operators generally do not admit a bounded H^∞ calculus over the right half-plane. In contrast to this, we prove that the H^∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as as and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that -sectorial operators admit a bounded calculus over the Besov algebra of the right half-plane. We also discuss the link between -sectorial operators and bounded Tadmor-Ritt operators.     

Numerical semigroups bounded by the translation of a plane monoid

Aequationes mathematicae - Let $$\mathbb {N}$$ be the set of nonnegative integer numbers. A plane monoid is a submonoid of $$(\mathbb {N}^2,+)$$ . Let M be a plane monoid and $$p,q\in \mathbb {N}$$...     

Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators

We define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small -balls aroundα-n-names necessary to cover most of the system, for any generating partitionα. We show that this rate is essentially bounded if and only if the system is a translation of a compact group, and compute it for several classes of systems of entropy zero, thus getting examples of growth rates inO(n),O(n ^k ) fork ε ℕ, oro(f(n)) for any given unboundedf, and of various relationships with the usual notion of language complexity of the underlying topological system.     

A bounded property for gradients of diffusion semigroups on Euclidean spaces

We consider uniformly elliptic diffusion processes X(t,x) on Euclidean spaces , with some conditions in terms of the drift term (see assumptions A2 and A3). By using interpolation theory, we show a bounded property which gives an estimate of involving |x| and but not ||∇f||_∞, and a power of smaller than 1.     

Bases of identities for semigroups of bounded rank transformations of a set

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank ≤ k transformations of a set, namely based on these results we prove that the semigroup T _k (X) of rank ≤ k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and |X| ∈ «3, 4» or k ≥ 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank ≤ 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).     

A direct approach to the Weiss conjecture for bounded analytic semigroups

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded H ^∞-calculus and is based on elementary analysis.     

Linear differential operators with unbounded operator coefficients and semigroups of bounded operators

Associated with a family of evolution operators in a complex Banach space is a linear unbounded operator, which is studied with the aid of a semigroup of difference operators and a difference operator in a sequence space. Some formulas for the spectra of the linear operators in question (in particular, for abstract hyperbolic differential operators) and the spectrum mapping theorem for the semigroup of difference operators are obtained. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 811–820, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00032 and by the International Science Foundation under grant No. NZA000 and grant No. NZA300.     

On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices

Generalizing a recent result of E.B. Davies [4], we show that generators of bounded positive C₀-semigroups on atomic Banach lattices with order continuous norm have trivial peripheral point spectrum. Moreover, we give examples that the peripheral spectrum can be any closed cyclic subset of . Received: 20 September 2005; revised: 23 January 2006     

Applications of B‐bounded nonlinear semigroups to particle transport problems

In this paper we study an application of nonlinear B‐bounded semigroups introduced in a previous paper. The application is similar to the particle transport problem which led to B‐bounded linear semigroups. We deal with a nonlinear particle transport problem, which can be solved by using B‐bounded nonlinear semigroups. Copyright © 2010 John Wiley & Sons, Ltd.     

A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L ^p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups.     

Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L ¹(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L ^p (0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L ¹(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators if one of them generates a C ₀-semigroup. Günter Lumer in memoriam