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1.
Let C be a closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, A be an α-inverse strongly-monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. We introduce an iteration scheme of finding a point of F (T)∩(A+B)−10, where F (T) is the set of fixed points of T and (A+B)−10 is the set of zero points of A+B. Then, we prove a strong convergence theorem, which is different from the results of Halpern’s type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.  相似文献   

2.
LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤AB, andB−A is of finite rank,A t ≤B t for somet>1 if and only if the null space ofB−A is invariant forA.  相似文献   

3.
In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT onY, and the spectrum ofT is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT onXX, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .  相似文献   

4.
For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s n (A) = O(exp(-anα)), where s n (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zIA)−1 of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).   相似文献   

5.
Let −A be a linear, injective operator, on a Banach spaceX. We show that ∃ anH functional calculus forA if and only if −A generates a bouned strongly continuous holomorphic semigroup of uniform weak bounded variation, if and only ifA(ζ+A) −1 is of uniform weak bounded variation. This provides a sufficient condition for the imaginary powers ofA, {A−is} sεR, to extend to a strongly continuous group of bounded operators; we also give similar necessary conditions.  相似文献   

6.
IfA=A *≥0 on the real Hilbert spaceH=L 2 (Ω, dμ) withKerA=A −1 ({0})∈0, (I+A)−1 compact andf(u)=c|u| p−1 u withc>0,p>1, the solutions ofu”+u’+Au+f(u)=0 tend to 0 in norm at least liket −1/(p−1) ast→∞. Here it is shown that the set of initial data of those solutions tending to 0 exponentially fast has near 0 the structure of a manifold with codimension dim(Ker A). If, in addition,A=−Δ with Neumann homogeneous boundary conditions, we show that the following alternative holds true: eitheru(t) tends to 0 exponentially fast, or ‖u(t)‖≥γt −1/(p−1) with γ>0 fort≥1.  相似文献   

7.
We propose a family of gradient algorithms for minimizing a quadratic function f(x)=(Ax,x)/2−(x,y) in ℝ d or a Hilbert space, with simple rules for choosing the step-size at each iteration. We show that when the step-sizes are generated by a dynamical system with ergodic distribution having the arcsine density on a subinterval of the spectrum of A, the asymptotic rate of convergence of the algorithm can approach the (tight) bound on the rate of convergence of a conjugate gradient algorithm stopped before d iterations, with d≤∞ the space dimension.  相似文献   

8.
The main purpose of this paper is the following Theorem: letE be a Dedekind complete Riesz space (vector lattice), letA be a vector subspace of it which majorizesE and letx 0 be an element ofE−A. IfT 0 is a positive projection onA, then there existsy 0E such thatT 0 can be extended to a positive projection onA+ℝx 0+ℝy 0.  相似文献   

9.
Martingales valued in certain subspaces ofL 1   总被引:1,自引:0,他引:1  
A Banach subspaceE ofL 1 is constructed such thatE fails the Radon-Nikodym property, yetE has no bounded dyadicδ-tree for anyδ>0. The unit ball ofE is also relatively compact in the topology of convergence in probability, which implies thatE has the strong Schur property. The construction uses martingales and probabilistic techniques. Supported in part by NSF-MCS-7827056.  相似文献   

10.
Let E 0 be a holomorphic vector bundle over P1(C) and0 be a meromorphic connection of E 0. We introduce the notion of an integrable connection that describes the movement of the poles of0 in the complex plane with integrability preserved. We show the that such a deformation exists under sufficiently weak conditions on the deformation space. We also show that if the vector bundle E0 is trivial, then the solutions of the corresponding nonlinear equations extend meromorphically to the deformation space.  相似文献   

11.
We explore connections between Krein's spectral shift function ζ(λ,H 0, H) associated with the pair of self-adjoint operators (H 0, H),H=H 0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K *(H 0−λ−i0)−1 K) associated with the operator-valued Herglotz functionJ+K *(H 0−z)−1 K, Im(z)>0 inH, whereV=KJK * andJ=sgn(V). Our principal results include a new representation for ζ(λ,H 0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E J((−∞, 0))), ℝ, whereA(λ)=Re(K *(H 0−λ−i0−1 K),B(λ)=Im(K *(H 0−λ-i0)−1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H 0, H) coincides with the trindex associated with the pair (Ξ(J+K *(H 0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.  相似文献   

12.
In this paper we show that the pluripolar hull ofE={(z, ω)∈C2:ω=e −1/z,z≠0} is equal toE. This implies thatE is plurithin at 0, which answers a question of Sadullaev. The result remains valid ife −1/z is replaced by certain other holomorphic functions with an essential singularity at 0.  相似文献   

13.
Let (X,A) be a measureable space andT:XX a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA 0A is a convergence class for ℳ such that, for everyAA 0, the sequence ((1/n) Σ i =0/n−1 1 A T i) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.  相似文献   

14.
In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence where Ad = (I + A) (IA)−1. We show that if A and A−1 generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then Ad is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of Ad is equivalent to the uniform boundedness of the semigroup generated by A.  相似文献   

15.
We prove that every locally compact non-discrete abelian groupG contains a compact subsetE such thatA(E) — the restriction algebra ofA(G) toE — admits spectral synthesis, although it contains a closed, regular, self-adjoint subalgebra which is isomorphic to an algebra of infinitely differentiable functions on [−1, 1]. We also give some general results concerning the failure of spectral synthesis in regular Banach algebras. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem, under the supervision of Professor Y. Katznelson, to whom the author wishes to express his gratitude for his valuable remarks, and the interest he showed in the preparation of this paper.  相似文献   

16.
We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C *-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E *(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E *) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].   相似文献   

17.
In this paper, generalizing the notion of a path we define ak-area to be the setD={g(t):tJ} on thek-skeleton of a convex compact setK in a Hilbert space, whereg is a continuous injection map from thek-dimensional convex compact setJ to thek-skeleton ofK. We also define anE k-area onK, whereE k is ak-dimensional subspace, to be ak-area with the propertyπ(g(t))=t,tπ(K), whereπ is the orthogonal projection onE k. This definition generalizes the notion of an increasing path on the 1-skeleton ofK. The existence of such sets is studied whenK is a subset of a Euclidean space or of a Hilbert space. Finally some conjectures are quoted for the number of such sets in some special cases.  相似文献   

18.
For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L 2 |r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz basis property are known. However, a sufficient and necessary condition is only known in the special case of an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions satisfying an additional assumption.   相似文献   

19.
Let A be the closed unbounded operator inL p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ 1≦argλ≦θ 2} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2θ<θ 23π/2 ; and there exists a constantc such that (I)││(λ-A)-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE andE + such thatA is completely reduced by the direct sum decompositionL p(G)=ELp (G) ⊕E+Lp (G) and each of the operatorsAE and—AE + is the infinitestimal generator of an analytic semigroup.  相似文献   

20.
We assume given a ringA with unit, and a subcomplex of the reduced bar complex ofA. We assume that this subcomplex is a deformation retract of the whole complex and thus has homology equal to the Hochschild homology ofA, but it will typically be smaller and easier to calculate with. We use these to construct (accordingly small) deformation retracts for the reduced bar complexes ofA[t] andA[t,t −1]. WhenA is a Banach algebra, we also do this construction forC (S1;A). Partially supported by N.S.F. Grant No. DMS 92-03398.  相似文献   

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