Spectral representation of local symmetric semigroups of operators

A spectral integral representation is established for locally defined symmetric semigroups of operators, with indices which are not restricted to a neighborhood of zero. This extends the well-known results of Fröhlich (Adv. Appl. Math. 1:237–256, 1980) and Klein and Landau (J. Funct. Anal. 44:121–137, 1981).     

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f _n } ∪L _p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ _n a _n T ⁿ g/ n whenT is anL ₂ -contraction,g∃L ₂ , and {a _n } is an appropriate sequence. Given a sequence {f _n }∪L _p (Ω, μ), 1<p≤2, of independent centered random variables, we study conditions for the existence of a set ofx of μ-probability 1, such that for every contractionT on andg∈L ₂ (π), the random power series Σ _n f _n (x)T ⁿ g converges π-a.e. The conditions are used to show that for {f _n } centered i.i.d. withf ₁∃L log⁺ L, there exists a set ofx of full measure such that for every contractionT on andg∃L ₂ (π), the random series Σ _n f _n (x)T ⁿ g/n converges π-a.e. We use Menchoff's own spelling of his name in the papers he wrote in French. Dedicated to Hillel Furstenberg upon his retirement     

Extensions of ordered semigroups

The present note summarizes the author's dissertation [2]. Let S and T be ordered semigroups, the latter with a zero element O. Let Σ be an ideal extension of S by T determined by a partial homomorphism ϕ of T^*=T/O into S. A partial solution is given to the problem of determining all ways of extending the given orders on S and T^* to Σ=S U T^*. Every such “extending order” (≤) on Σ carries with it a certain “null decomposition” T^*=X∪Y (X∩Y=ℴ) of T^*, and the existence and behavior of extending orders is discussed in terms of properties of null decompositions of T^*.     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

Extensions of a theorem of Balas

We extend Balas' theorem on facial constraints in two directions. First, we provide a generalization in the facial case, and characterize the set obtained. Second, we establish a ‘countable convergence’ in certain cases where faciality is dropped as an hypothesis. We relate the latter results to finite convergence of cutting-plane algorithms.     

On the generalizations of Denjoy-Wolff theorem

We consider generalizations of Denjoy-Wolff theorem on strongly pseudo-convex domains. Beardon [3] gave a general Denjoy-Wolff theorem in hyperboli domain in the complex plane. Our main results are generalizations of Beardon's result in a higher dimensional setting.     

Extensions of a theorem of Cauchy-Liouville

We deal with the equations Δpu+f(u)=0 and Δpu+(p−1)g(u)p|∇u|+f(u)=0 in RN, where g(t) is a continuous function in (0,∞), p>1 and f(t) is a smooth function for t>0. Under appropriate conditions on g and f we show that the corresponding equation cannot have nontrivial non-negative entire solutions.     

Some generalizations of the Hardy-Littlewood-Paley theorem

Translated from Matematicheskie Zametki, Vol. 51, No. 3, pp. 24–34, March, 1992.     

Further generalizations of the Gale-Nikaido-Debreu theorem

A fixed point theorem on compact compositions of acyclic maps on admissible (in the sense of Klee) convex subsets of a t.v.s. is applied to generalize Gwinner’s extensions of the Walras excess demand theorem and of the Gale-Nikaido-Debreu theorem.     

Extensions of the Hausdorff-Young theorem

In this paper the clasical Hausdorff-Young theorem, which states that iff ∈L ^p, 1≦p≦2, on the line and is its Fourier transform, then whereq ⁻¹+p ⁻¹=1, is extended in two ways for certain Orlicz spacesL ^Φ. IfL ^Φ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks are also included. Dedicated to the memory of my nephew, K. Ramakrishna, who appeared to be so brilliant. This research was supported by the NSF Grants GP-5921 and GP-7678.     

Sort-Jacobi and generalizations of the symmetric eigenvalue problem

In this paper, it is sketched how the Sort-Jacobi method for the symmetric eigenvalue problem extends to the (–1)-eigenspace of the Cartan involution on an arbitrary semisimple Lie algebra. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as the symmetric, Hermitian, skewsymmetric, symmetric and skew-symmetric ℝ-Hamiltonian eigenvalue problem and the singular value decomposition. It allows a unified treatment of the above eigenvalue methods, including local convergence analysis. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some generalizations of Rolle's theorem

In this paper, we consider particular tiling problems suitable for able high-school students. Three alternative student-generated proofs of the solution to the initial problem are given in detail, and various points in connection with these proofs are briefly discussed.     

Extensions of symmetric relations

Maximal accretive and dissipative extensions of symmetric relations defined by abstract boundary conditions are considered in Hubert space. Under the assumption that the resolvent of some extension has properties such as belonging to the Neumann-Shatten ideal or having given asymptotics of the s-numbers, either sufficient or necessary and sufficient conditions are found for a perturbation of the boundary conditions guaranteeing that the above properties are preserved for the resolvent of the perturbed extension.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 825–834, December, 1977.     

Unitary extensions of two-parameter local semigroups of isometric operators and the Krein extension theorem

A notion of two-parameter local semigroups of isometric operators in Hilbert space is discussed. It is shown that under certain conditions such a semigroup can be extended to a strongly continuous two-parameter group of unitary operators in a larger Hilbert space. As an application a simple proof of the Eskin bidimensional version of the Krein extension theorem is given.