Abstract: Kohn's Proof of the Hypoellipticity of the Hormander Operators.- Compactness Criteria for the Resolvent of Schrodinger Operators.- Global Pseudo-differential Calculus.- Analysis of some Fokker-Planck Operator.- Return to Equillibrium for the Fokker-Planck Operator.- Hypoellipticity and Nilpotent Groups.- Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts.- On Fokker-Planck Operators and Nilpotent Techniques.- Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians.- Spectral Properties of the Witten-Laplacians in Connection with Poincare Inequalities for Laplace Integrals.- Semi-classical Analysis for the Schrodinger Operator: Harmonic Approximation.- Decay of Eigenfunctions and Application to the Splitting.- Semi-classical Analysis and Witten Laplacians: Morse Inequalities.- Semi-classical Analysis and Witten Laplacians: Tunneling Effects.- Accurate Asymptotics for the Exponentially Small Eigenvalues of the Witten Laplacian.- Application to the Fokker-Planck Equation.- Epilogue.- Index.

Abstract: We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Fina...

Abstract: This paper represents a broadened version of the plenary lecture presented by the author at the conference Analytic Methods of Analysis and Differential Equations (AMADE-2003), September 4–9, 2003,...

Abstract: The aim of these notes is to describe some recent re- sults concerning dispersive estimates for principally normal pseu- dodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove L q Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough poten- tials. Dispersive estimates are L q estimates for nonelliptic partial differ- ential operators which are a consequence of the decay properties of their fundamental solutions. These decay properties follow from spa- tial spreading of the singularities of the solutions. Since solutions prop- agate in directions conormal to the characteristic set of the operator, this spreading can be related to nonzero curvatures of the characteristic set. Dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis. Various types of dispersive estimates are known to be true for op- erators such as the wave operator, the Schrodinger operator and the linear KdV, see Ginibre-Velo (4), Keel-Tao (11). They have proved to be useful in the study of nonlinear problems, as well as of problems with unbounded potentials. More recently, similar estimates have been obtained for wave op- erators with variable coefficients, beginning with the smooth case in Kapitanskii (10), Mockenhaupt, Seeger and Sogge (14), up to operators with C 2 coefficients in Smith (15) and Tataru (21), (23). Similar results were obtained for the Schrodinger equation in Staffilani-Tataru (19) (C 2 coefficients) and in Burq-Gerard-Tzvetkov (1) (smooth coeffic ients). In the variable coefficient elliptic case one should also mention Sogge's L q

Abstract: Sturm's 1836 Oscillation Results Evolution of the Theory.- Sturm Oscillation and Comparison Theorems.- Charles Sturm and the Development of Sturm-Liouville Theory in the Years 1900 to 1950.- Spectral Theory of Sturm-Liouville Operators Approximation by Regular Problems.- Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of Recent Developments.- Asymptotic Methods in the Spectral Analysis of Sturm-Liouville Operators.- The Titchmarsh-Weyl Eigenfunction Expansion Theorem for Sturm-Liouville Differential Equations.- Sturm's Theorems on Zero Sets in Nonlinear Parabolic Equations.- A Survey of Nonlinear Sturm-Liouville Equations.- Boundary Conditions and Spectra of Sturm-Liouville Operators.- Uniqueness of the Matrix Sturm-Liouville Equation given a Part of the Monodromy Matrix, and Borg Type Results.- A Catalogue of Sturm-Liouville Differential Equations.

Abstract: Basic definitions Operators on Hilbert modules Hilbert modules over $W^*$-algebras Reflexive Hilbert $C^*$-modules Multipliers of $A$-compact operators. Structure results Diagonalization of operators over $C^*$-algebras Homotopy triviality of groups of invertible operators Hilbert modules and $KK$-theory Bibliography Notation index Index.

Abstract: In this paper we show that any order continuous operator between two Riesz spaces is automatically order bounded. We also investigate different types of order convergence.

Abstract: The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing sequence $\lambda=(\lambda_1,...,\lambda_n)$, with repetitions according to multiplicity, and the diagonal of A is a point of $R^n$ that bears some relation to $\lambda$. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We give a new proof of the latter result for positive trace-class operators on infinite dimensional Hilbert spaces, generalizing results of one of us on the diagonals of projections. We also establish an appropriate counterpart of the Schur inequalities that relate spectral properties of self-adjoint operators in $II_1$ factors to their images under a conditional expectation onto a maximal abelian subalgebra.

Abstract: We study various spectral theoretic aspects of non-self-adjoint operators Specifically, we consider a class of factorable non-self-adjoint perturbations of a given unperturbed non-self-adjoint operator and provide an in-depth study of a variant of the Birman-Schwinger principle as well as local and global Weinstein-Aronszajn formulas Our applications include a study of suitably symmetrized (modified) perturbation determinants of Schr\"odinger operators in dimensions n=1,2,3 and their connection with Krein's spectral shift function in two- and three-dimensional scattering theory Moreover, we study an appropriate multi-dimensional analog of the celebrated formula by Jost and Pais that identifies Jost functions with suitable Fredholm (perturbation) determinants and hence reduces the latter to simple Wronski determinants

Abstract: We compare the classical definitions of p-summing multilinear operators with the class of multiple p-summing operators, by showing which linear properties are properly generalized for each one of these classes.

Abstract: I Singular Manifolds and Differential Operators GEOMETRY OF SINGULARITIES Preliminaries Manifolds with conical singularities Manifolds with edges ELLIPTIC OPERATORS ON SINGULAR MANIFOLDS Operators on manifolds with conical singularities Operators on manifolds with edges Examples of elliptic edge operators II Analytical Tools PSEUDODIFFERENTIAL OPERATORS Preliminary remarks Classical theory Operators in sections of Hilbert bundles Operators on singular manifolds Ellipticity and finiteness theorems Index theorems on smooth closed manifolds LOCALIZATION (SURGERY) IN ELLIPTIC THEORY The index locality principle Localization in index theory on smooth manifolds Surgery for the index of elliptic operators on singular manifolds Relative index formulas on manifolds with isolated singularities III Topological Problems INDEX THEORY Statement of the problem Invariants of interior symbol and symmetries Invariants of the edge symbol Index theorems Index on manifolds with isolated singularities Supplement. Classification of elliptic symbols with symmetry and K-theory Supplement. Proof of Proposition 5.16 ELLIPTIC EDGE PROBLEMS Morphisms The obstruction to ellipticity A formula for the obstruction in topological terms Examples. Obstructions for geometric operators IV Applications and Related Topics FOURIER INTEGRAL OPERATORS ON SINGULAR MANIFOLDS Homogeneous canonical (contact) transformations Definition of Fourier integral operators Properties of Fourier integral operators The index of elliptic Fourier integral operators Application to quantized contact transformations Example RELATIVE ELLIPTIC THEORY Analytic aspects of relative elliptic theory Topological aspects of relative elliptic theory INDEX OF GEOMETRIC OPERATORS ON MANIFOLDS WITH CYLINDRICAL ENDS Operators on manifolds with cylindrical ends Index formulas HOMOTOPY CLASSIFICATION OF ELLIPTIC OPERATORS The homotopy classification problem Classification on smooth manifolds Atiyah-de Rham duality Abstract elliptic operators and analytic K-homology Classification on singular manifolds Some applications LEFSCHETZ FORMULAS Main result Proof of the theorem Contributions of conical points as sums of residues Supplement. The Lefschetz number Supplement. The Sternin-Shatalov method APPENDICES Spectral Flow Eta Invariants Index of Parameter-Dependent Elliptic Families Bibliographic Remarks Bibliography Index

Abstract: We characterize those holomorphic symbols ϕ: D → D for which the induced composition operator Cϕ: Bω → Bμ (respectively, Bω,0 → Bμ,0) is bounded or compact, where D is the unit disc in the complex plane C, ω is a normal function on [0, 1) and μ is a non-negative function on [0, 1) with μ(tn) > 0 for some sequence satisfying limn→∞ tn= 1.

Abstract: We study pseudodierential operators with amplitudes a"(x, ) depending on a singular parameter " ! 0 with asymptotic properties measured by dierent scales. We prove, taking into account the asymptotic behavior for " ! 0, refined versions of estimates for classical pseudodierential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudo- dierential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sucient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodierential equations.

Abstract: Large sets of baryon interpolating field operators are developed for use in lattice QCD studies of baryons with zero momentum. Operators are classified according to the double-valued irreducible representations of the octahedral group. At first, three-quark smeared, local operators are constructed for each isospin and strangeness and they are classified according to their symmetry with respect to exchange of Dirac indices. Nonlocal baryon operators are formulated in a second step as direct products of the spinor structures of smeared, local operators together with gauge-covariant lattice displacements of one or more of the smeared quark fields. Linear combinations of direct products of spinorial and spatial irreducible representations are then formed with appropriate Clebsch-Gordan coefficients of the octahedral group. The construction attempts to maintain maximal overlap with the continuum $SU(2)$ group in order to provide a physically interpretable basis. Nonlocal operators provide direct couplings to states that have nonzero orbital angular momentum.

Abstract: A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states Non-separable states are called entangled These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1] In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2] In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1 The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation In what follows, H, K , are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H Equipped with the trace-norm topology, S(H) is a complete separable metric space If π is a Borel probability measure on S(H), then the Bochner integral

Abstract: We study properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. Moreover, we characterize the compactness of differences of two weighted composition operators.

Abstract: TOPOLOGICAL LIE ALGEBRAS Fundamentals Universal enveloping algebras The Baker-Campbell-Hausdor series Convergence of the Baker-Campbell-Hausdor series Notes LIE GROUPS AND THEIR LIE ALGEBRAS Definition of Lie groups The Lie algebra of a Lie group Logarithmic derivatives The exponential map Special features of Banach-Lie groups Notes ENLARGIBILITY Integrating Lie algebra homomorphisms Topological properties of certain Lie groups Enlargible Lie algebras Notes Smooth Homogeneous Spaces Basic facts on smooth homogeneous spaces Symplectic homogeneous spaces Some homogeneous spaces related to operator algebras Notes QUASIMULTIPLICATIVE MAPS Supports, convolution, and quasimultiplicativity Separate parts of supports Hermitian maps Notes COMPLEX STRUCTURES ON HOMOGENEOUS SPACES General results Pseudo-Kahler manifolds Flag manifolds in Banach algebras Notes EQUIVARIANT MONOTONE OPERATORS Definition of equivariant monotone operators H*-algebras and L*-algebras Equivariant monotone operators as reproducing kernels H*-ideals of H*-algebras Elementary properties of H*-ideals Notes L*-IDEALS AND EQUIVARIANT MONOTONE OPERATORS From ideals to operators From operators to ideals Parameterizing L*-ideals Representations of automorphism groups Applications to enlargibility Notes HOMOGENEOUS SPACES OF PSEUDO-RESTRICTED GROUPS Pseudo-restricted algebras and groups Complex polarizations Kahler polarizations Admissible pairs of operator ideals Some Kahler homogeneous spaces Notes APPENDICES Differential Calculus and Smooth Manifolds Basic Differential Equations of Lie Theory Topological Groups References Index

Abstract: Integrable systems are usually given in terms of functions of continuous variables (on R), in terms of functions of discrete variables (on Z), and recently in terms of functions of q-variables (on Kq). We formulate the Gel’fand-Dikii (GD) formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations. In particular they yield integrable equations over integers (difference equations) and over q-numbers (q-difference equations). We formulate the GD formalism also in terms of shift operators for all regular-discrete time scales. We give a method allowing to construct the recursion operators for integrable systems on time scales. Finally, we give a trace formula on time scales and then construct infinitely many conserved quantities (Casimirs) of the integrable systems on time scales.

Abstract: We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of Cheeger and Harnack and of the maximum principle (in both elliptic and parabolic versions), and apply them to study spectral theory, the ground state and the heat semigroup associated to these operators.

Abstract: We prove several spectral radius inequalities for sums, products, and commutators of Hilbert space operators. Pinching inequalities for the spectral radius are also obtained.

Abstract: We introduce and study a system of variational inclusions involving H-accretive operators in Banach spaces. By using the resolvent operator technique associated with an H-accretive operator, we prove the existence and uniqueness of solution for the system of variational inclusions involving H-accretive operators and construct a new iterative algorithm to approximate the unique solution.

Abstract: In the present paper, we introduce the Bezier variant of a general sequence of linear positive operators introduced by Srivastava and Gupta (8) and estimate the rate of convergence of these operators for functions of bounded variation.

Abstract: It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work. We also give explicit formulae for the nonlocal parts of higher recursion, Hamiltonian and symplectic operators of integrable systems in (1+1) dimensions. Using these two results we prove, under some natural assumptions, the Maltsev–Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.

Abstract: Let H be a complex Hilbert space, B(H) the algebra of all bounded linear operators on H and S α (H) the real linear space of all self-adjoint operators on H. We characterize the surjective maps on B(H) or S α (H) that preserve the numerical ranges of products or Jordan triple-products of operators.