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1.
Li  Yuan-Chuan  Shaw  Sen-Yen 《Positivity》1998,2(3):281-299
Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on L p () (1L 1(), C0 , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.  相似文献   

2.
Global exponentially bounded convoluted semigroups in Banach spaces are systematically treated with the help of Laplace transform. A perturbation theorem in this context is proved and some characterizations of the introduced class of analytic convoluted semigroups are obtained. Illustrative examples of generators of convoluted semigroups, including differential operators, are presented. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
Buchwalter and Schmets reconciled Cc(X) and Cp(X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if -barrelled ⇔ ?-barrelled for Cc(X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the Cc(X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) and van Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable Cp(X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for Cc(X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when Cc(X) is a df-space (Jarchow's 1981 question).  相似文献   

4.
In this article, we prove that in UMD Banach spaces the complex inversion formula of the Laplace transform is valid, in the strong sense, for wide classes of families of bounded linear operators. Our approach allows us to recover (in a unified way) known results about C 0-semigroups, cosine functions and resolvent families as well as to prove new results for k-convoluted semigroups and integrated semigroups, among others.  相似文献   

5.
We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the unification of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23:413–429, 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded ‘commuting’ perturbations.  相似文献   

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Let A be the 2mth-order elliptic operator of divergence form with bounded measurable coefficients defined in a domain Ω of . For 1<p<∞ we regard A as a bounded linear operator from the Lp Sobolev space to H−m,p(Ω). It is known that when , we can construct the resolvent (Aλ)−1 and estimate its operator norm for some λ if the leading coefficients are uniformly continuous. In this paper, we try to extend this result to a general domain. It is successful when m=1 if Ω is the half-space or a domain with C2 bounded boundary. For m>1 it is shown that the problem is reduced to the case where Ω is the half-space and A is a homogeneous operator with constant coefficients. We also give a perturbation theorem.  相似文献   

9.
Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever gf=0 for any fCb(X,E), gE.  相似文献   

10.
The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space . The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces such that ?n?1Hn is dense in and πn(X)=XHn for each n?1. Here, is the orthogonal projection. It is also shown that when X is compact convex with and satisfies the above condition, then C1(X) is complete if and only if the C1-Whitney extension theorem holds for X. Finally, for compact subsets of , an extension of the C1-Weierstrass approximation theorem is proved for C1 maps with compact derivatives.  相似文献   

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It is shown that every almost linear bijection of a unital C-algebra A onto a unital C-algebra B is a C-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all unitaries uA, all yA, and n=0,1,2,…, and that almost linear continuous bijection of a unital C-algebra A of real rank zero onto a unital C-algebra B is a C-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all , all yA, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all xX and all unitaries uA, is an A-linear isomorphism. This is applied to investigate C-algebra isomorphisms between unital C-algebras.  相似文献   

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The following two theorems are proved: (1) the generator of an exponentially equicontinuousn-times integratedC-cosine function also generates an exponentially equicontinuous [(n+1)/2]-times integratedC-semigroup; (2) IfA and −A are generators of exponentially equicontinuousn-times integratedC-semigroups, thenA 2 generates an exponentially equicontinuousn-times integratedC-cosine function.  相似文献   

15.
We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in . If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t(nθ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order <m.  相似文献   

16.
Let ΩRn be a bounded domain of class C2+α, 0<α<1. We show that if n?3 and uΩ is the maximal solution of equation Δu=n(n-2)u(n+2)/(n-2) in Ω, then the hyperbolic radius is of class C2+α up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.  相似文献   

17.
In this paper we study smooth classification of hyperbolic vector fields based on their linear approximations only and obtain the following. On Rn, n?5, with only two kinds of exceptions, any two hyperbolic vector fields with generic nonlinear parts and where Ai are n×n matrices, are C1 conjugate to each other if and only if A1 and A2 are strictly similar, and they are C1 orbitally equivalent if and only if A1 and A2 are similar.  相似文献   

18.
Transform methods are used to establish algebra homomorphisms related to convoluted semigroups and convoluted cosine functions. Such families are now basic in the study of the abstract Cauchy problem. The framework they provide is flexible enough to encompass most of the concepts used up to now to treat Cauchy problems of the first- and second-order in general Banach spaces. Starting with the study of the classical Laplace convolution and a cosine convolution, along with associated dual transforms, natural algebra homomorphisms are introduced which capture the convoluted semigroup and cosine function properties. These correspond to extensions of the Cauchy functional equation for semigroups and the abstract d'Alembert equation for the case of cosine operator functions. The algebra homomorphisms obtained provide a way to prove Hille-Yosida type generation theorems for the operator families under consideration.  相似文献   

19.
In the p-adic vector space , we characterize those non-negative functions ψ defined on for which the weighted Hardy-Littlewood average is bounded on (1?r?∞), and on . Also, in each case, we find the corresponding operator norm ‖Uψ‖.  相似文献   

20.
In this paper, we mainly study properties of nullsolutions of the operator Dk (kN=N?{0}), so-called k-regular functions. Firstly, we study the set of all homogeneous polynomials of degree p in x1,…,xn which are k-regular in the whole Rn, clearly is a right module over C(Vn,n), we construct a basis for the right module . Secondly, we study the k-regular and analytic functions, and we give the Taylor expansions for these functions. At last, the corresponding Taylor expansions for k-regular functions are given since each k-regular function is a real analytic function.  相似文献   

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