Distribution semigroups and abstract Cauchy problems

We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator in a Banach space the following assertions are equivalent: (a) generates a distribution semigroup; (b) the convolution operator has a fundamental solution in where denotes the domain of supplied with the graph norm and denotes the inclusion ; (c) generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

     

Regularized semigroups of bounded semivariation

We use regularized semigroups to consider local linear and semilinear inhomogeneous abstract Cauchy problems on a Banach space in a unified way. We show that the inhomogeneous abstract Cauchy problem {fx43-1} has a unique classical solution, for allf εC([0,T], [Im(C)]),x inC(D(A)), if and only ifA generates aC-regularized semigroup of bounded semivariation, and has a strong solution for allf εL ¹ ([0,T], [Im(C)]),x εC(D(A)) if and only if theC-regularized semigroup is what we call of bounded super semivariation. This includes locally Lipschitz continuousC-regularized semigroups. We give similar simple sufficient conditions for the semilinear abstract Cauchy problem {fx43-2} to have a unique solution. Well-known results for generators of strongly continuous semigroups, as well as more recent results for Hille-Yosida operators, originally due to Da Prato and Sinestrari, regarding (0.1), are immediate corollaries of our results. Results due to Desch, Schappacher and Zhang, on (0.2), for generators of strongly continuous semigroups, are similarly generalized to Hille-Yosida operators with our approach. This article appeared in the last issue of the Forum. However, due to an error by the Journal Secetary, the Abstract was omitted, and with it the equations which are the focus of the article. We therefore are reprinting the article in its entirety. The Journal Secretary regrets the error.     

Regularization for Ill-posed Cauchy problems associated with generators of analytic semigroups

This paper is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. Our main result is that if −A is the generator of an analytic semigroup, then there exists a family of regularizing operators for such an ill-posed Cauchy problem by using the quasi-reversibility method, fractional powers and semigroups of linear operators. The applications to ill-posed partial differential equations are also given.     

Regularized Trace of the Cauchy Transform

We give an exact value of the regularized trace of Cauchy transform C _?? i.e. the sum of the series ${\sum\nolimits_{n=1}^{\infty}( s_{n}^{2}( C_{\Omega}) -\frac{\vert \Omega \vert}{\pi n})}$ .     

Integrated semigroups,C-semigroups and the abstract Cauchy problem

LetA be a linear operator on a Banach space. We consider when the following holds. (*)u′(t,x)=A(u(t,x)) (t≥0),u(0,x)=x, has a unique solution, for allx in the domain ofA ⁿ⁺¹ . We discuss the relationship between (*), integrated semigroups, and C-semigroups. We use this to obtain new results about integrated semigroups and the abstract Cauchy problem. We give several examples where (*) may be easily shown using C-semigroups. Many of these examples may not be done directly using integrated semigroups.     

Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

The Norm and Regularized Trace of the Cauchy Transform

In this paper, the norm of the Cauchy transform C is obtained on the space L ²(D, dμ), where dμ = ω(|z|) dA(z). Also, (for the case ω ≡ 1), the first regularized trace of the operator C* C on L ²(Ω) is obtained. The results are illustrated by examples, with different specific choices of the function ω and the domain Ω.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 844–853.Original Russian Text Copyright ©2005 by M. R. Dostanic.     

On stability of the Cauchy equation on semigroups

Summary We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S such that the set {f(x + y) – f(x) – f(y): x, y S} is bounded, there exists an additive functiona:S for which the functionf – a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S X is a function for which the mapping (x, y) F(x + y) – F(x) – F(y) is bounded, then there exists an additive function A : S X such that F — A is bounded.     

Asymptotic equipartition of energy for equations of elasticity

In this paper we give a review of the equipartition of energy results of Goldstein and Sandefur [3], [4], [5] as well as proving a new result in the case of a particular fourth order equation. These results are then applied to the equations of elasticity to give a weak asymptotic orthogonality for the shear and pressure waves. In the case of boundary value problems in the interior of a bounded domain we get weak asymptotic orthogonality in the average.     

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     

Hille-Yosida operators and Cauchy problems

A new proof of a temporal regularity result for an abstract Cauchy problem with a Hille-Yosida operator is given having consequences on the existence of other types of solutions. New types of weak solutions are introduced.     

Unsolvable algorithmic problems for semigroups,groups, and rings

A survey of the subject outlined in the heading (with many proof s sketched) is given. A special focus is on the original proofs of the unsolvability theorems of Markov, Post, and Novikov for word problems in semigroups and groups. A method of Shirshov is described, which has led to proof of the main unsolvability theorems for Lie algebras.Translated from Itogi Nauki I Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 3–66, 1987.The Russian names in this survey are given according to standard transliteration. Some Russian authors, however, are also known in the Western literature in a different spelling. The corresponding equivalents are given below, in alphabetical order:     

The kernel of the second order Cauchy difference on semigroups

Let S be a semigroup, H a 2-torsion free, abelian group and \(C^2f\) the second order Cauchy difference of a function \(f:S \rightarrow H\). Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of \(C^2f = 0\) are the functions of the form \(f(x) = j(x) + B(x,x)\), where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of \(C^2f = 0\) to Fréchet’s functional equation and to polynomials of degree less than or equal to 2.     

Trace theorems for holomorphic semigroups and the second order Cauchy problem

We use the theory of boundary values (also called traces) of holomorphic semigroups as developed by Boyadzhiev-deLaubenfels (1993) and El-Mennaoui (1992) to study the second order Cauchy problem for certain generators of holomorphic semigroups. Our results contain in particular the result of Hieber (Math. Ann. 291 (1991), 1--16) for the Laplace operator on .