Some extensions for Ramanujan’s circular summation formulas and applications

The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we...     

Extreme points and isometries of φ-nuclear operators in Hilbert spaces

A modulus function φ is a continuous strictly increasing subadditive real valued function φ: [0, ∞) → [0, ∞) for which φ(0) = 0. The object of this paper is to define φ-nuclear operators in Banach spaces. The basic properties of these operators are studied. In particular it is proved that φ-nuclear operators are stable under injective tensor product. In case of Hilbert spaces, the extreme points of the unit ball and the isometrics of such class of operators are characterized.     

Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution, for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.     

On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Pólya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Pólya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

     

Čebyšev's type inequalities for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.     

Metric betweenness,Ptolemaic spaces,and isometric embeddings of pretangent spaces in ℝ

We study some properties of pretangent spaces describing the infinitesimal geometry of general metric spaces. The necessary and sufficient conditions characterizing Ptolemaic pretangent spaces and pretangent spaces, where every three points are situated “on a single straight line,” are found. As a corollary, we obtain a criterion of embeddability of pretangent spaces into the set ℝ of real numbers endowed by the natural metric.     

Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii–Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii–Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas–Rachford splitting method, the Fermat–Weber location problem as well as the alternating projection method by John von Neumann.     

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.     

Analogues of Miyachi, cowling-price and Morgan theorems for compact extensions of ℝ n

Let K be a compact subgroup of automorphisms of ? ⁿ . We formulate and prove an analogue of Miyachi’s theorem for the semi-direct product K ? ? ⁿ . This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the L ^p ? L ^q analogue of Morgan theorem for any compact extension of ? ⁿ . These upshots are proved using the representations theory and the Plancherel formula for the group Fourier transform.     

Strong convergence of Browder’s and Halpern’s type iterations in Hilbert spaces

We use results from theory of nonexpansive mappings to unify and deduce the recent results of Lin and Takahashi (Positivity 16:429–453, 2012) and of Takahashi (J Optim Theory Appl 157:781–802, 2013).     

Denjoy-Wolff theorems for Hilbert’s and Thompson’s metric spaces

We study the dynamics of fixed point free mappings on the interior of a normal, closed cone in a Banach space that are nonexpansive with respect to Hilbert’s metric or Thompson’s metric. We establish several Denjoy-Wolff type theorems which confirm conjectures by Karlsson and Nussbaum for an important class of nonexpansive mappings. We also extend and put into a broader perspective results by Gaubert and Vigeral concerning the linear escape rate of such nonexpansive mappings.     

Bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes

We construct explicit bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes with good parameters. This correspondence is a generalization of a work of Maharaj, Matthews, and Pirsic.     

Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces

The purpose of this paper is to suggest and analyze a number of iterative algorithms for solving the generalized set-valued variational inequalities in the sense of Noor in Hilbert spaces. Moreover, we show some relationships between the generalized set-valued variational inequality problem in the sense of Noor and the generalized set-valued Wiener-Hopf equations involving continuous operator. Consequently, by using the equivalence, we also establish some methods for finding the solutions of generalized set-valued Wiener-Hopf equations involving continuous operator. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory.     

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

The isometrical extensions of 1-Lipschitz mappings on Gteaux differentiability spaces

Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.     

The angular derivative of Fréchet-holomorphic maps in J*-algebras and complex Hilbert spaces

We study the problem of the existence of the angular derivative of Fréchet-holomorphic maps of the generalized right half-planes defined by non-zero partial isometries in infinite dimensional complex Banach spaces called J^*-algebras. These generalized right half-planes and open unit balls (which are bounded symmetric homogeneous domains) are holomorphically equivalent. The results obtained here also hold in C ^*-algebras, JC^*-algebras, B ^*-algebras and ternary algebras, containing non-zero partial isometries, and in complex Hilbert spaces. Some examples are given. The principal tool we use are general results of the Pick-Julia type in J^*-algebras.