Algebra Of Symbols Research Articles

Fredholm Toeplitz Operators and Slow Oscillation

The purpose of this paper is to show how Fred hoi m criteria for Toeplitz operators, whose symbols lie in an algebra,A, may often be generalized to cover a larger symbol algebra generated by A and SO, the slowly oscillating functions. Mere A and SO are algebras of continuous functions on the real line, so that we are concerned principally with the effect of a single discontinuity in the symbol function.We shall treat the cases when A is the almost periodic functions, the semi-almost periodic functions and the multiplicatively periodic functions. Sufficient criteria for Fredholmness are obtained in Section 5. The more difficult task of establishing necessary and sufficient criteria is only achieved here for the slowly oscillating almost periodic functions and this is done in Section 6.

A PERIODICITY THEOREM IN THE ALGEBRA OF SYMBOLS

We introduce the concept of an elliptic family on the manifold M in a trace algebra. We define the Chern character of an elliptic family. We also introduce the algebra of formal symbols on Rn with coefficients in a trace algebra. We establish a connection between the Chern characters of an elliptic family on M in the algebra of formal symbols on Rn and of the elliptic family on M × R2n formed by the leading terms of the symbols. Bibliography: 8 titles.

The classical limit for Weyl quantization

Here we formulate some theorems on convergence of solutions of non-relativistic quantum mechanics equations and probability distributions for quantum observables to the corresponding classical solutions and distributions. Some new algebra of Weyl symbols, which consists of temperated distributions, is the basic structure used here.

V. On the calculus of symbols, with applications to the theory of differential equations

The calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients. The laws of combination of ordinary algebraical symbols may be divided into the commutative and distributive laws; and the number of symbols in the higher branches of mathematics, which are commutative with respect to one another, is very small. It became then necessary to invent an algebra of non-commutative symbols. This important step was effected by Professor Boole, for certain classes of symbols, in his well-known and beautiful memoir published in the Transactions of this Society for the year 1844, and the object of the paper which I have now the honour to lay before the Society is to perfect and develope the methods there employed. For this purpose I have constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those assumed in Professor Boole’s memoir, and I thus arrive at equations of great utility in the integration of linear differential equations with variable coefficients.