REDUCIBILITY IN SOME CATEGORIES OF PARTIAL RECURSIVE OPERATORS

We consider two categories with one object, namely the set of all partial functions of one variable from the set of natural numbers into itself; the morphisms are the partial recursive operators in one case, and certain continuous partial mappings in the other case. We show that these categories are recursion categories and we characterize the domains and the complete domains. Some observations are made on a notion of reducibility obtained by using the total morphisms of these categories, and, subsequently, the general recursive operators.     

Heckman-Opdam's Jacobi polynomials for the BCn root system and generalized spherical functions

We prove that the radial part of the Laplacian on the space of generalized spherical functions on the symmetric space GL(m+n)/GL(m)×GL(n) is the Sutherland differential operator for the root system BC_n and the radial parts of the differential operators corresponding to the higher Casimirs yield the integrals of the quantum Calogero-Moser system. It allows us to give a representation theoretical construction for the three parameter family of Heckman-Opdam's Jacobi polynomials for the BC_n root system.     

Canonical conjugate momentum of discrete label operators in quantum mechanics I: formalism

We discuss the inadequacy of the standard definition of canonical conjugation for a quantum operator having adiscrete spectrum. A different definition is proposed, based on the analogy betweencontinuous anddiscrete translations (or rotations). This definition can be applied to special operators which we calllabel operators. The general form of the conjugate momentum of a label operator is found and the resulting canonical commutation rules are discussed. It is shown that the canonical commutator acts like ac number in itsdomain _I , but the domain does not coincide with the whole Hilbert space. The properties of the subspace _I are also discussed.     

The spectrum of a one-dimensional hierarchical model

The spectrum of a discrete Schrödinger operator with a hierarchically distributed potential is studied both by a renormalization group technique and by numerical analysis. A suitable choice of the potential makes it possible to reduce the original problem to a two-dimensional map. Scaling laws for the band-edge energyE _be and for the integrated density of states are predicted together with the global properties of the spectrum. Different scaling regimes are obtained depending on a hierarchy positive parameterR: for R<1/2 the usual scaling laws for the periodic case are obtained, while forR>1/2 the scaling behavior depends explicitly onR.     

Ladder operators for Szego polynomials and related biorthogonal rational functions

We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to biorthogonal rational functions on the unit circle.

     

On perturbations of M-accretive operators in Banach spaces

In this paper, we consider the solvability of nonlinear equations of the form

where is an m-accretive operator on a Banach space , is a mapping on and .

     

Elementary reverse Hölder type inequalities with application to operator interpolation theory

We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces.

     

New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces

Let be a real Banach space and a bounded, open and convex subset of The solvability of the fixed point problem in is considered, where is a possibly discontinuous -dissipative operator and is completely continuous. It is assumed that is uniformly convex, and A result of Browder, concerning single-valued operators that are either uniformly continuous or continuous with uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let The effect of a weak boundary condition of the type on the range of operators is studied for -accretive and maximal monotone operators Here, with sufficiently large norm and Various new eigenvalue results are given involving the solvability of with respect to Several results do not require the continuity of the operator Four open problems are also given, the solution of which would improve upon certain results of the paper.

     

Analytical Functional Models and Local Spectral Theory

In 1959 E. Bishop used a Banach-space version of the analyticduality principle established by e Silva, Köthe, Grothendieckand others to study connections between spectral decompositionproperties of a Banach-space operator and its adjoint. Accordingto Bishop a continuous linear operator T L(X) on a Banach spaceX satisfies property (rß) if the multiplication operator is injective with closed range for each open set U in the complex plane. In the present articlethe analytic duality principle in its original locally convexform is used to develop a complete duality theory for property(rß). At the same time it is shown that, up to similarity,property (rß) characterizes those operators occurringas restrictions of operators decomposable in the sense of C.Foias, and that its dual property, formulated as a spectraldecomposition property for the spectral subspaces of the givenoperator, characterizes those operators occurring as quotientsof decomposable operators. It is proved that, unlike the situationfor commuting subnormal operators, each finite commuting systemof operators with property (rß) can be extended toa finite commuting system of decomposable operators. Meanwhilethe results of this paper have been used to prove the existenceof invariant subspaces for subdecomposable operators with sufficientlyrich spectrum. 1991 Mathematics Subject Classification: 47A11,47B40.     

Bismut superconnections and the Chern character for Dirac operators on foliated manifolds

In this paper, we show how to define a Bismut superconnection for generalized Dirac operators defined along the leaves of a compact foliated manifoldM. Using the heat operator of the curvature of the superconnection, we define a (nonnormalized) Chern character for the Dirac operator, which lies in the Haefliger cohomology of the foliation. Rescaling the metric onM by 1/a and lettinga 0, we obtain the analog of the classical cohomological formula for the index of a family of Dirac operators. In certain special cases, we can also compute the limit asa and show that it is the Chern character of the index bundle given by the kernel of the Dirac operator. Finally, we discuss the relation of our results with the Chern character in cyclic cohomology.