Abstract: We characterize those positive linear operators with positive in- verse for which the dominated ergodic estimate holds. We also prove that for such operators one has mean and a.e. convergence. 1. Introduction. Akcoglu proved (1) a Dominated Ergodic Theorem for pos- itive contractions in Lp, 1 o ||(2n + 1)_1 Y17=-nTl\\p 0(2n + 1)_1 J27=-n I^VI- In this note we see that the same holds for the averages (n+ 1)_1 ^"=0^ an(^tne corresponding maximal operator. Therefore our result generalizes Akcoglu's theorem to mean bounded operators when T and T_1 are positive. As a consequence, we get a.e. convergence of the averages. Since there are examples of Feder (6) and Assani (2) which show that this is false if the positivity is dropped, our result closes the case of invertible operators, and seems to indicate that the corresponding theorem for power bounded operators (invertible or not) might be true. Throughout this paper T will denote a positive linear operator with positive inverse on Lp, 1 LP(X,S~,p), with 1 < p < oo, and suppose T~1 is also positive. Then, as is well known (7), T and T_1 are Lamperti operators and they have the following properties: (a) For each integer i, there exists a positive function gi such that

Abstract: Let For any n × n matrix A, define the c-numerical range and the c-numerical radius by and rc (A) = max{|z|: z∈ W c (A)} respectively. We give a characterization for those linear operators Twhich satisfy for all matrix A. Moreover, we give a full description for the extreme points of the unit ball in the real linear space of n × n hermitian matrices with respect to the norm rc (·).

Abstract: Publisher Summary This chapter discusses modified Szasz operators. Papanicolau obtained some results on Bernstein-type operators. In an analogous manner, Singh defined a sequence of Szasz-type operators that maps the space of bounded continuous functions into itself as ( S n , x f ) ( t ) = ∑ p n , k ( t ) f ( x + k / n ) where p n , k ( t ) = ( e − nt )(( nt ) k )/ k ! and x ∈ [0, ∞) is fixed and proved some approximation properties. This chapter, motivated by the recent work of Derriennic on modified Bernstein polynomials introduced by Durrmeyer for functions integrable on [0, 1], presents a sequence of modified Szafsz operators defined on the space of integrable functions on [0, ∞). It presents some results in the form of lemmas which are required to prove the main results of the chapter.

Abstract: We are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self-adjoint operators in Hilbert spaces. The case is included where the approximations of the eigenfunctions don't belong to the domain of definition of the operator. For the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don't satisfy the boundary conditions of the problem. The error bounds suggest a certain defectminization method for solving the eigenvalue problems. The method is applied to the membrane problem.

Abstract: We prove that a bounded analytic function / on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by H°° and / does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if / and g are bounded analytic functions on the unit disk such that the commutator TfT* —TgTf (here Tf denotes the operator of multiplication by / on the Bergman space of the disk) is compact, then (1 — |z|2) min{|/'(z)|, |g'(z)|} —* 0 as \\z\\ t 1.

Abstract: Various energy and convergence factor parametrizations of the Mo/ller operator are discussed. These can usually be applied to eigenstates of the free Hamiltonian, in contrast to the Mo/ller operator itself, which is only correctly defined in Hilbert space. Formal differences of the parametrized operators are exemplified in one dimension using a particular separable potential for which all computations can be carried out analytically. The behavior of the energy parametrized operators are contrasted when the convergence parameter is taken to zero and it is shown how the parametrized operators may be consistently used if a proper interpretation of the formulas is maintained. The second virial coefficient is also examined for the particular potential and it is shown how the Mo/ller operator can be used in its evaluation.

Abstract: CONTENTS § 0. Introduction § 1. Auxiliary propositions § 2. The general scheme of investigation of degenerate operators and generalized Weyl formulae § 3. The Schrodinger operator with homogeneous degenerate potential § 4. Differential operators in a bounded domain § 5. Differential operators in an unbounded domain § 6. Hypoelliptic pseudodifferential operators with multiple characteristics and discrete spectrum § 7. Pseudodifferential operators with multiple characteristics and spectrum accumulated to –0References

Abstract: This work is principally concerned with the operator approach to the orbifold compactification of the bosonic string. Of particular importance to operator formalism is the con formal structure and the operator product expansion. These are introduced and discussed in detail. The Frenkel-Kac-Segal mechanism is then examined and is shown to be a consequence of the duality of dimension one operators of an analytic bosonic string compactified on a certain torus. Possible generalizations to higher dimension operators are discussed, this includes the cross-bracket algebra which plays a central role in the vertex operator representation of Griess's algebra, and hence the Fischer-Griess Monster Group. The mechanism of compactification is then extended to orbifolds. The exposition includes a detailed account of the twisted sectors, especially of the zero-modes and the twisted operator cocycles. The conformal structure, vertex operators and correlation functions for twisted strings are then introduced. This leads to a discussion of the vertex operators which represent the emission of untwisted states. It is shown how these operators generate Kac-Moody algebras in the twisted sectors. The vertex operators which insert twisted states are then constructed, and their role as intertwining operators is explained. Of particular importance in this discussion is the role of the operator cocycles, which are seen to be crucial for the correct working of the twisted string emission vertices. The previously established formalism is then applied in detail to the reflection twist. This includes an explicit representation of the twisted operator cocycles by elements of an appropriate Clifford algebra and the elucidation of the operator algebra of the twisted emission vertices, for the ground and first excited states in the twisted sector. This motivates the 'enhancement mechanism', a generalization of the Frenkel-Kac-Segal mechanism, involving twisted string emission vertices, in dimensions 8, 16 and 24. associated with rank 8 Lie algebras, rank 16 Lie algebras and the cross-bracket algebra for the Leech lattice, respectively. Some of the relevant characters of the 'enhanced" modules are determined, and the connection of the cross-bracket algebra to the phenomenon of 'Monstrous Moonshine' and the Monster Group is explained. Algebra enhancement is then discussed from the greatly simplified shifted picture and extensions to higher order twists are considered. Finally, a comparison of this work with other recent research is given. In particular, the connection with the path integral formalism and the extension to general asymmetric orbifolds is discussed. The possibility of reformulating the moonshine module in a 'covaxiant' twenty-six dimensional setting is also considered.

Abstract: A spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The operators themselves are elements in the algebra of linear bounded operators. The effect of these operators when operating on elements in the domain is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of spatially recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for developing corresponding control and trajectory design algorithms. Expressions interpreted within the operator algorithm framework led to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the high-level operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithms has been greatly simplified. The analytical formulation of the operator algebra and its implementation in Ada are discussed. >