On Gauss-Weierstrass semigroups and inversion of potentials associated with the Bessel differential operator

In this paper, we define the generalized Gauss Weierstrass semigroups with Weierstrass kernel, and give some of their properties. Using them, we study the inversion formulas for the generalized Riesz and Bessel potentials, generated by the generalized shift operators and associated with the Laplace Bessel differential operator.     

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

We search for conditions on a countably compact (pseudocompact) topological semigroup under which: (i) each maximal subgroup H(e) in S is a (closed) topological subgroup in S; (ii) the Clifford part H(S) (i.e. the union of all maximal subgroups) of the semigroup S is a closed subset in S; (iii) the inversion inv: H(S) → H(S) is continuous; and (iv) the projection π: H(S) → E(S), π: x ↦ xx ⁻¹, onto the subset of idempotents E(S) of S, is continuous.      

The Dirac-Ramond operator on loops in flat space

In this paper, a rigorous construction of the S¹-equivariant Dirac operator (i.e., Dirac-Ramond operator) on the space of (mean zero) loops in is given and its equivariant L²-index computed. Essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra.     

Transference principles for semigroups and a theorem of Peller

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows one to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to Lp-spaces and—involving the concept of γ-boundedness—to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.     

Certain topological properties and duals of the domain of a triangle matrix in a sequence space

The matrix domain of the particular limitation methods Cesàro, Riesz, difference, summation and Euler were studied by several authors. In the present paper, certain topological properties and β- and γ-duals of the domain of a triangle matrix in a sequence space have been examined as an application of the characterization of the related matrix classes.     

Completely p-summing maps on the operator Hilbert space OH

We prove the completely p-summing ideals of OH are all equal as sets for 1?p<2. A phase transition then occurs at p=2 as we also show for p?2, the completely p-summing ideals of OH turn out as sets to be Schatten ideal classes with the limiting case being the Schatten 4-class ideal S₄ when p→∞.     

The index of the spin Dirac operator on the weighted projective space and the reciprocity law of the Fourier-Dedekind sum

The main purpose of this paper is to give a geometric interpretation of the reciprocity law of the Fourier-Dedekind sum given by M. Beck and S. Robins. In fact, the V-index of the spin^c Dirac operator on the weighted projective space is equal to the dimension of the space of all weighted homogeneous polynomials of given degree, and this equality gives precisely the Beck-Robins reciprocity law. In this equality, the Fourier-Dedekind sums appear as the localization terms of the V-index of the spin^c Dirac operators and have a relationship to the eta invariants of lens spaces.     

On <Emphasis Type="Italic">K</Emphasis>-groups of operator algebra on the 1-shift space

In this paper we discuss the K-groups of Wiener algebra ;W. For the 1-shift space XGM2,We obtain a characterization of Fredholm operators on X^nGM2 for all n ∈ N. We also calculate the K-groups of operator algebra on the 1-shift space XGM2.     

Estimates of weighted Hardy-Littlewood averages on the p-adic vector space

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‖.     

与高阶抛物型Schrodinger算子相关的Riesz变换的L^p估计——献给陆善镇教授75华诞

令δ/δt＋（-△）^2＋V^2为R^n＋1（n≥5）上的高阶抛物型Schrodinger算子,其中非负位势V与时间t无关且属于逆Holder类Bq1（Rn）（q1〉n/2）.本文得到与高阶抛物型Schrodinger算子相关的Riesz变换▽^2（δ/δt＋（-△）^2＋V^2）^-1/2的L^p（R^n＋1）估计.     

Critical exponent for semilinear damped wave equations in the N-dimensional half space

We generalize a previous result of Ikehata (Math. Methods Appl. Sci., in press), which studies the critical exponent problem of a semilinear damped wave equation in the one-dimensional half space, to the general N-dimensional half space case. That is to say, one can show the small data global existence of solutions of a mixed problem for the equation u_tt−Δu+u_t=|u|^p with the power p satisfying p∗(N)=1+2/(N+1)<p?N/[N−2]+ if we deal with the problem in the N-dimensional half space.     

The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator

In this paper, we study the existence of countable many positive solutions for a class of nonlinear singular boundary value systems with p-Laplacian operator:

     

The norm of a composition operator with linear symbol acting on the Dirichlet space

We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.     

Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C ₀-semigroup.     

Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

We determine the limit of the bottom of spectrum of Schrödinger operators with variable coefficients on Wiener spaces and path spaces over finite-dimensional compact Riemannian manifolds in the semi-classical limit. These are extensions of the results in [S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401-424]. The problem on path spaces over Riemannian manifolds is considered as a problem on Wiener spaces by using Ito's map. However the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the Riemannian curvature tensor on the underling manifold is not equal to 0. The difficulties are solved by using unitary transformations of the Schrödinger operators by approximate ground state functions and estimates in the rough path analysis.