Abstract: Preface .- I. Banach Algebras .- II. Operators .- III. Essential Spectrum .- IV. Taylor Spectrum .- V. Orbits and Capacity .- Appendix .- Bibliography

Abstract: In close analogy to the fundamental role of random numbers in classical information theory, random operators are a basic component of quantum information theory. Unfortunately, the implementation of random unitary operators on a quantum processor is exponentially hard. Here we introduce a method for generating pseudo-random unitary operators that can reproduce those statistical properties of random unitary operators most relevant to quantum information tasks. This method requires exponentially fewer resources, and hence enables the practical application of random unitary operators in quantum communication and information processing protocols. Using a nuclear magnetic resonance quantum processor, we were able to realize pseudorandom unitary operators that reproduce the expected random distribution of matrix elements.

Abstract: A sequence of positive linear operators which approximate each continuous function on [0,1] while preserving the functione 2 (x) =x 2 is presented. Quantitative estimates are given and are compared with estimates of approximation by Bernstein polynomials. Connections with summability are discussed.

Abstract: We discuss boundedness and compactness of composition operators followed by multiplication as operators between Bloch-type spaces of analytic functions on the unit disk.

Abstract: Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators. The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.

Abstract: In this paper we provide various approximation results concerning the classical Korovkin theorem via A -statistical convergence. We also study the rates of A -statistical convergence of approximating positive linear operators and give some examples. Mathematics subject classification (2000): 41A25, 41A36; 40A05.

Abstract: The technique based on a *-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. Within the proposed formalism, there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress-energy tensor are assumed to be related to the interface between the quantum and classical formalism. The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of the Klein-Gordon equation. Considering the averaged stress-energy tensor with respect to Hadamard quantum states, the presented definition turns out to be equivalent to an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is determined by the local geometry and the parameters which appear in the Klein-Gordon operator. In particular, no extra added-by-hand term g αβQ and no arbitrary smooth part of the Hadamard parametrix (generated by some arbitrary smooth term ``ω 0 '') are involved. The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ-function approach whenever that technique can be implemented.

Abstract: Let K be a closed convex subset of a reflexive Banach space X. We consider self-mappings of K which are bounded on bounded subsets of K and satisfy a relaxed form of nonexpansivity with respect to a given convex function f. The family of these operators is endowed with the topology of uniform convergence on bounded subsets of K. We show that “almost all” such operators T share the property that they have a fixed point z T such that, for any x ∈ K, the orbit converges weakly to z T . Here the meaning of “almost all” is in the sense of Baire's categories: the collection of all those operators which do not have this property is contained in a countable union of nowhere dense sets.

Abstract: The seminal 1989 work of Douglas and Paulsen in the theory of analytic Hilbert modules precipitated a number of major research efforts. This in turn led to some intriguing and valuable results, particularly in the areas of operator theory and functional analysis. With the field now beginning to blossom, the time has come to collect those results under one cover. Written by two of the most active and often-cited researchers in the field, Analytic Hilbert Modules reports on the progress made by the authors and others, including the characteristic space theory, rigidity, the equivalence problem, the Arveson modules, extension theory, and reproducing Hilbert spaces on n-dimensional complex space.

Abstract: We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6 and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on [12], where the relationship of the normal standard tractor bundle to the ambient construction is described.

Abstract: We study hypoelliptic operators with polynomially bounded coefficients that are of the form K=∑ i=1 m X i T X i +X 0+f, where the X j denote first order differential operators, f is a function with at most polynomial growth, and X i T denotes the formal adjoint of X i in L 2. For any ɛ>0 we show that an inequality of the form ||u||δ,δ≤C(||u||0,ɛ+||(K+iy)u||0,0) holds for suitable δ and C which are independent of yR, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Herau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x≥|y|τ−c,τ(0,1],cR}.

Abstract: Preface. Introduction. 1: Boundary value problems with the transmission property. 1.1. Symbolic calculus and pseudo-differential operators. 1.2. Parameter-dependent boundary value problems. 1.3. General kernel cut-off constructions. 1.4. Notes and complementary remarks. 2: Operators on manifolds with conical singularities. 2.1. Mellin operators and cone asymptotics. 2.2. The cone algebra. 2.3. Analytic functionals and asymptotics. 2.4. Notes and complementary remarks. 3: Operators on manifolds with exits to infinity. 3.1. Scalar operators. 3.2. Calculus with operator-valued symbols. 3.3. Boundary value problems on manifolds with exits to infinity. 3.4. Notes and complementary remarks. 4: Boundary value problems on manifolds with edges. 4.1. Manifolds with edges and typical operators. 4.2. Weighted Sobolov spaces. 4.3. Operator conventions in the edge pseudo-differential calculus. 4.4. Operator-valued edge symbols. 4.5. The algebra of edge boundary value problems. 4.6. Further material on edge operators. 4.7. Notes and complementary remarks. 5: Crack theory. 5.1. Differential operators in crack configurations. 5.2. Parameter-dependent calculus in the model cone. 5.3. Local crack theory. 5.4. The global calculus. 5.5. Notes and complementary remarks. Bibliography. List of Symbols. Index.

Abstract: Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.

Abstract: In order to treat one-parameter semigroups of linear operators on Banach spaces which are not strongly continuous, we introduce the concept of bi-continuous semigroups defined on Banach spaces with an additional locally convex topology τ. On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous for τ and such that the orbit maps are τ-continuous. We then apply the result to semigroups induced by flows on a metric space as studied by J. R. Dorroh and J. W. Neuberger [21], [22], [5], [6], [7], [23].

Abstract: A systematic construction is presented of 1/4 BPS operators in N=4 superconformal Yang-Mills theory, using either analytic superspace methods or components. In the construction, the operators of the classical theory annihilated by 4 out of 16 supercharges are arranged into two types. The first type consists of those operators that contain 1/4 BPS operators in the full quantum theory. The second type consists of descendants of operators in long unprotected multiplets which develop anomalous dimensions in the quantum theory. The 1/4 BPS operators of the quantum theory are defined to be orthogonal to all the descendant operators with the same classical quantum numbers. It is shown, to order $g^2$, that these 1/4 BPS operators have protected dimensions.

Abstract: Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ t joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D t corresponding to t=1/3 is the so-called ``cubic'' Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator on spinors and an eigenvalue estimate for the first eigenvalue of D 1/3. This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T≠ 0 is a 3-form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and a detailed discussion of a 5-dimensional example.

Abstract: We study conditions equivalent to the invertibility of f m g when f and g are idempotents in a unital ring, and give applications to bounded linear operators in Banach and Hilbert spaces. In the setting of rings we are able to show that many conditions previously linked to finite dimensionality, rank equalities, norm topology of bounded linear operators or to properties of C *-algebras can be in fact proved by simple algebraic arguments.

Abstract: In the paper, binary 1-Lipschitz aggregation operators and specially quasi-copulas are studied. The characterization of 1-Lipschitz aggregation operators as solutions to a functional equation similar to the Frank functional equation is recalled, and moreover, the importance of quasi-copulas and dual quasi-copulas for describing the structure of 1-Lipschitz aggregation operators with neutral element or annihilator is shown. Also a characterization of quasi-copulas as solutions to a certain functional equation is proved. Finally, the composition of 1-Lipschitz aggregation operators, and specially quasi-copulas, is studied.

Abstract: This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,e) such that f(t,e)→f(t) when e goes to zero. This led to several new notions as representations: fractal functions and e-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schrodinger equation v...

Abstract: The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of pos- itive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition un- der which any positive finite-rank operator B can be expressed as a sum of rank-one operators with norms specified by a sequence of positive numbers {ci}. Equivalently, this result proves the existence of a frame with B as it's frame operator and with vector norms { p ci}. We further prove that, given a non-compact positive operator B on an infinite di- mensional separable real or complex Hilbert space, and given an infinite sequence {ci} of positive real numbers which has infinite sum and which has supremum strictly less than the essential norm of B, there is a se- quence of rank-one positive operators, with norms given by {ci}, which sum to B in the strong operator topology. These results generalize results by Casazza, Kovaycevic, Leon, and Tremain, in which the operator is a scalar multiple of the identity op- erator (or equivalently the frame is a tight frame), and also results by Dykema, Freeman, Kornelson, Larson, Ordower, and Weber in which {ci} is a constant sequence.

Abstract: We discuss a notion of wave front set which allows us tocontrol the behaviour `at infinity' of temperate distributions. Weobtain the microlocality and microellipticity properties with respect toa class of global pseudodifferential operators and a propagation theoremfor the corresponding class of Fourier Integral Operators. Through theseresults, we prove an adapted global version of the classical theoremconcerning the singularities of solutions of hyperbolic Cauchy problemsfor linear operators with multiple characteristics of constantmultiplicities. Finally, we make a comparison with the scattering wavefront set introduced by Melrose.

Abstract: Despite all the analogies with ‘‘usual random’’ models, tight binding operators for quasicrystals exhibit a feature that clearly distinguishes them from the former: the integrated density of states may be discontinuous. This phenomenon is identified as a local effect, due to the occurrence of eigenfunctions with bounded support.

Abstract: We consider a singular differential-difference operator Λ on the real line which includes, as particular case, the Dunkl operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of Λ into the first derivative operator d/dx. We exploit these transmutation operators to develop a new commutative harmonic analysis on the real line corresponding to the operator Λ. In particular, we establish a Paley–Wiener theorem and a Plancherel theorem for the Fourier transform associated to Λ.

Abstract: We consider a basic d-adic model for the scattering transform on the line. We prove L2 bounds for this scattering transform and a weak L2 bound for a Carleson type maximal operator (theorem 1.4). The latter implies boundedness of d-adic models of generalized eigenfunctions of Dirac type operators with potential in L2(). We show that this result cannot be obtained by estimating the terms in the natural multilinear expansion of the scattering transform (proposition 5.1).