Let R(D) be the algebra generated in Sobolev space W22(D) by the rational functions with poles outside the unit disk D. In this paper the multiplication operators Mg on R(D) is studied and it is proved that Mg ~ Mzn if and only if g is an n-Blaschke product. Furthermore, if g is an n-Blaschke product, then Mg has uncountably many Banach reducing subspaces if and only if n > 1. 相似文献
Let D(m, n; k) be the semi-direct product of two finite cyclic groups and , where the action is given by yxy−1 = xk. In particular, this includes the dihedral groups D2m. We calculate the automorphism group Aut (D(m, n; k)). 相似文献
This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?n, w(x)) and F (?n, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2(x) b(x, D) b1(x), where b1(x) and b2(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L2. 相似文献
Summary Let
a plane angle of opening α∈(π, 2π). LetPD andPN the Dirichlet and Neumann problems associated to the Poisson equation in
. ForPD andPN it is proved non existence of solution in Lp(
) whenp=2/(1±π/α). In other words, the ranges of elliptic operators naturally associated toPD andPN are not-closed in Lp(
) forp=2/(1±π/α).
Sunto Sia
} un angolo piano di apertura α∈(π, 2π). SianoPD ePN i problemi di Dirichlet e di Neumann associati all'equazione di Poisson in
. PerPD ePN si prova non esistenza di soluzioni in Lp(
) quandop=2/(1±π/α). Vale a dire i ranges degli operatori ellittici naturalmente associati aPD ePN sono non-chiusi in π--AgBrαKLp(
) perp=2/(1±π/α).
(as usual, R(D,b) denotes the conformal radius of a domain D with respect to a point b D) in the family of all quadruples of nonoverlapping simply connected domains {Dk},bk Dk,k=1,...,4, is obtained. Here, {b1,...,b4} are four arbitrary distinct points on
is an arbitrary positive number. The proof involves the solution of the problem on maximizing a certain conformal invariant, which is related to the problem under consideration. Bibliography: 5 titles. 相似文献
The self-affine measures μM,D corresponding to the case (i) M=pI3, D={0,e1,e2,e3} in the space and the case (ii) M=pI2, D={0,e1,e2,e1+e2} in the plane are non-spectral, where p>1 is odd, In is the n×n identity matrix, and e1,…,en are the standard basis of unit column vectors in . One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D) and to find them. In the present paper we show that, in both cases (i) and (ii), there are at most 4 mutually orthogonal exponentials in L2(μM,D) each, and the number 4 is the best. 相似文献
For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality
whenever x0, x1 ∈D and
. This paper presents some results on the boundedness and continuity of γ-convex functions. For instance, (a) if there is some x*∈D such that f is bounded below on D∩b̄(x*,γ), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball b̄(x*,γ/2)⊂ D and D′ is a closed bounded subset of D, then f is bounded on D′ iff it is bounded above on the boundary of D′; (c) if dim X>1 and the interior of D contains a closed ball of radius γ, then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x*,γ/2) in which f has at most countably many discontinuities, then the set of all points at which f is continuous is dense in D.The authors thank the referees for constructive remarks 相似文献
Let F(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound-soft cylinder of cross section D. Suppose F has the Fourier expansion where an = an(k, . Then if ?2 is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |bn| where the bn are independent of such that for all directions Domains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ?D or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions. 相似文献
Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C n, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff Sc(a, b; D) — the class of functions q such that q ? Sc(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z1,z2,…zn. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and Sc(a, b; D). 相似文献
Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝn. We study the representation of this function in the form of a series u(x) = u0(x) + |x|2u1(x) + |x|4u2(x) + …, where uk(x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula.
Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162. 相似文献
Let, where A={a1,..., an} and B={b1,...,bm} are systems of distinguished points, and let H be a family of homotopic classes Hi, i=1, ..., j + m, of closed Jordan curves in C, where the classes Hj+, =1, ..., m, consist of curves that are homotopic to a point curve in b. Let ={1,...,j+m} be a system of positive numbers. By P=P(,A,B) we denote the extremal-metric problem for the family H and the numbers : for the modulusU=U(,A,B) of this problem we have the equality
, whereD*={D1*
,...,Dj+m*
} is a system of domains realizinga maximum for the indicated sum in the family of all systemsD={D1,...,Dj+m} of domains, associated with the family H (byU(Di)) we denote the modulus of the domain Di, associated with the class Hi). In the present paper we investigate the manner in whichU=U(,A,B) and the moduliU=(D
1*
) depend on the parameters i, ak, b; moreover, we consider the conditions under which some of the doubly connected domains D
i*
,i=1,...,j, from the system D* turn out to be degenerate (Theorems 1–3). In particular, one obtains an expression for the gradient of the function M, as function of the parameter a=ak (Theorem 4). One gives some applications of the obtained results (Theorem 5).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 136–148, 1985. 相似文献
Let be a Banach algebra and let X be a Banach -bimodule. In studying (,X) it is often useful to extend a given derivation D: → X to a Banach algebra containing as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate
identities to obtain the extension as a limit of operators b ↦ D(ba) − b.D(a), a ε in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation.
In this paper we make clear which part of this approach is analytic and which algebraic by presenting an algebraic scheme
that gives derivations in all situations at the cost of enlarging the module. We use our construction to give improvements
and shorter proofs of some results from the literature and to give a necessary and sufficient condition that biprojectivity
and biflatness is inherited to ideals. 相似文献