Abstract multiparameter theory,II

In this paper we continue the study of multiparameter spectral theory commenced in our earlier paper of the same title. Here we relax the condition that the weight operator Δ₀ be strictly bounded away from zero. The case in which Δ₀ has 0 as a point of its continuous spectrum is discussed and a generalized eigenvector expansion and Parseval equality are obtained.     

Abstract multiparameter theory,III

We continue our study of the spectral theory of the multiparameter system [T_r, V_rs], where T_r, V_rs are self-adjoint operators in a Hilbert space H_r, r, s = 1,…, k. In contrast to our earlier works we assume the T_r to be bounded and the V_rs unbounded (although we require the V_rs to be pairwise commutative). We use the strong definiteness condition on Δ₀ = det{V_rs} in demanding that it be strictly positive definite in H, the tensor product of the spaces H₁,…, H_k. A generalized eigenvector expansion and Parseval equality are obtained.     

On multiparameter theory

In this paper we point out a mistake in the paper “Abstract Multiparameter Theory I” of P. J. Browne. We give new proofs for some of the results of P. J. Browne.     

Abstract oscillation theorems for multiparameter eigenvalue problems

We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem $\text{T}{}_{\mathrm{m}}\text{x}{}_{\mathrm{m}}\text{=}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}\text{x}{}_{\mathrm{m}}$, 0 ≠ x_m?H_m, m = 1 … k. (1) Here T_m and V_mn are self-adjoint operators on Hilbert spaces H_m. The index is based on the number of negative eigenvalues of T_m ? ∑_{n = 1}^kλ_nV_mn and on the sign of the determinant δ₀ with (m, n)th entry (x_m, V_mnx_m). We assume that certain cofactors of δ₀ are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (1) in the case where δ₀ is positive.     

Potential theory related to some multiparameter processes

In a first part, we present a potential theory constructed form a continuous kernel on a locally compact space. The notions of capacity, quasi-continuity, equilibrium measures and potentials are specially studied. In a second part, we particularize the framework, and, in the third part, we give probabilistic interpretations in this particular case. The process then involved is a sum of independent symmetric Levy processes in ^d , viewed as a multiparameter process. For instance, hitting probabilities for the process are estimated in terms of capacity.     

Clifford theory and Galois theory,I

We combine Galois theory with crossed products by letting grading groups act on ground fields. We show that Clifford theory for suitable normal subgroups preserves such actions.     

Abstract stationary theory of multichannel scattering

The abstract stationary theory of Kato and Kuroda is applied to multichannel scattering problems. It is shown that stationary wave operators exist and are asymptotically complete whenever the resolvent can be completely described in terms of primary singularities for a commuting family of channel Hamiltonians. The abstract theory is applied to prove asymptotic completeness in the three-body problem with pair potentials in L₂$\text{R}$³ for $\text{p >}\text{3}\text{2}$.     

Abstract theory of universal series and applications

We give an abstract framework for the theory of universal series,from which we deduce easily and in a unified way most of theexisting results as well as new and stronger statements.     

Stochastic integrals for nonprevisible,multiparameter processes

We develop a general theory for stochastic integrals of generalized stochastic processesX(t), depending on multidimensional time, within the framework of the space of Wiener distributions (D ^*).     

C-Cosine Functions and the Abstract Cauchy Problem, I

IfAis the generator of an exponentially boundedC-cosine function on a Banach spaceX, then the abstract Cauchy problem (ACP) forAhas a unique solution for every pair (x, y) of initial values from (λ − A)⁻¹C(X). The main result is a characterization of the generator of aC-cosine function, which may not be exponentially bounded and may have a nondensely defined generator, in terms of the associated ACP.     

On some applications of graph theory,I

In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results.     

Quasi-implication algebras,Part I: Elementary theory

Quasi-implication algebras (QIA's) are intended to generalize orthomodular lattices (OML's) in the same way that implication algebras (J. C. Abbott) generalize Boolean lattices. A QIA is defined to be a setQ together with a binary operation → satisfying the following conditions (a→b is denotedab). (Q1) $$\left( {ab} \right)a = a$$ (Q2) $$\left( {ab} \right)\left( {ac} \right) = \left( {ba} \right)\left( {bc} \right)$$ (Q3) $$\left( {\left( {ab} \right)\left( {ba} \right)} \right)a = \left( {\left( {ba} \right)\left( {ab} \right)} \right)b$$ Every OML induces a QIA, wherea → b=a ^⊥?(a?b). On the other hand, every QIA induces a join semi-lattice with a greatest element 1, where 1=aa,a≤b iffab=1, anda?b=((ab)(ba))a. A bounded QIA is defined to be a QIA with a least element 0 (w.r.t.≤). The QIA associated with any OML is bounded, the zero elements being the same. Conversely, every bounded QIA induces an OML, wherea ^⊥=a0, anda?b=((ab)(a0))0. The relationC of compatibility is defined so thataCb iffa≤ba, and it is shown that every compatible sub-QIA of a QIA is an implication algebra.     

On some applications of graph theory,I

In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results.     

Trace operators,Feynman distributions,and multiparameter white noise

Within the framework of white noise analysis on the probability space = ^* R ^d R ^M , the recent work by Johnson and Kallianpur on the Hu-Meyer formula, traces, and natural extensions is generalized to the multiparameter case:d>1. Besides providing a more general setting for these topics, the paper gives an alternative definition for the traces, a distributional version of the natural extension, and a generalized Kallianpur-Feynman distribution. The development illustrates how traces and natural extensions are intimately related to Wick products and the change of covariance formula from quantum field theory, as well as to the projective tensor product of Hilbert spaces from functional analysis.