A bilinear Orlicz-Pettis Theorem

Suppose E,F,G are locally convex spaces, is a bilinear operator and λ is a scalar sequence space. A series j_∑xj in E is λ b multiplier convergent if for every t={tj}∈λ there exists xt∈E such that for every y∈F. Under continuity assumptions on the linear operators b(x,⋅), we establish several versions of the Orlicz-Pettis Theorem for multiplier convergent series. Applications to spaces of continuous linear operators are given.     

关于Orlicz-Pettis定理的一个推广

研究了局部凸空间中级数无条件收敛和子级数收敛的各种等价形式,证明了Orlicz-Pettis定理在局部凸空间中一般拓扑意义下仍成立,并且利用此结果刻划了取值于局部凸空间的强可加矢值测度.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [5].     

一类乘子收敛级数空间

仅仅依靠序列空间λ的内蕴性质, 作者给出了λ -乘数收敛级数空间X(λ)上的一个局部凸拓扑T_Β, 并证明了(X(λ), T_Β)是AK -空间, 具有序列完备性和Banach-Steinhaus性质. 特别是作者给出了此空间上的一个改进的Orlicz-Pettis定理.     

The differentiation operator in the space of uniformly convergent Dirichlet series

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half-planes for each . The properties of the formal inverse of the differentiation are also investigated.     

Some New Applications of the Subspace Theorem

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(q ⁿ ) is an algebraic integer for infinitely many n, then h(f(q ⁿ )) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.     

Riesz points of upper triangular operator matrices

Two results are proved which concern Riesz points of upper triangular operator matrices. Applications are made to questions involving when Weyl's Theorem holds for an upper triangular operator matrix.

     

A simple bound for the remainder of the neumann series in the case of a self-adjoint compact operator

We consider the functional equation of the second kind \gf − λK\gf = f with K a compact self-adjoint linear operator on a Hilbert space: a Fredholm integral equation of the second kind, for example. We establish the simple bound where λ is any regular value of K; ø is the solution of the equation corresponding to λ; λ₁ is the characteristic value of K smallest in absolute value; and N = 0, 1, 2, .... For \s|λ\s| < \s|λ₁\s|, this is an estimate for the remainder of the partial sums of the Neumann series.     

A continous version of Orlicz-Pettis theorem via vector-valued Henstock-Kurzweil integrals

We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.     

The index of triangular operator matrices

For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.

     

Differential invariants of the transformation group of a homogeneous space

We demonstrate that the invariant operators on a homogeneous space generate differential invariants and invariant differentiation operators. The coordinate-free method of this article makes it possible to simply the computations essentially, namely to reduce them to operations of linear algebra. Some examples are exhibited.     

Transformations in the space of almost convergent sequences

Under consideration is the space of almost convergent sequences as well as the operators acting on it. We prove that the space is invariant under the transformations of some class including, in particular, the Hardy and averaging operators.     

Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator

We consider a multiplier transformation and some subclasses of the class of meromorphic functions which was defined by means of the Hadamard product by N.E. Cho, O.S. Kwon and H.M. Srivastava in [N.E. Cho, O.S. Kwon, H.M. Srivastava, Inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations, J. Math. Anal. Appl. 300 (2004) 505-520].     

Discrete spectrum of a model operator in Fock space

We consider a model describing a “truncated” operator (truncated with respect to the number of particles) acting in the direct sum of zero-, one-, and two-particle subspaces of a Fock space. Under some natural conditions on the parameters specifying the model, we prove that the discrete spectrum is finite. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 518–527, September, 2007.     

A New Proof for the Interpolating Theorem

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Norm equalities and inequalities for operator matrices

Several norm equalities and inequalities for operator matrices are proved in this paper. These results, which depend on the structure of circulant and skew circulant operator matrices, include pinching type inequalities for weakly unitarily invariant norms.     

Viscosity approximation method for accretive operator in Banach space

Let X$X$ be a uniformly smooth Banach space, C$C$ be a closed convex subset of X$X$, and A$A$ an m-accretive operator with a zero. Consider the iterative method that generates the sequence {_xn}$\left\{x_n\right\}$ by the algorithm

_xn+1=_αnf(_xn)+(1−_αn)_Jrnxn,$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)J_{r_n}{x}_n\text{,}$