In this paper the sequence Banach space ψ (Z) is defined for a class of convex functions ψ , and properties of the Kand Jinterpolation spaces (E0,E1)θ ,ψ,K and (E0,E1)θ ,ψ,J for a Banach couple E = (E0,E1) and θ ∈ (0,1) are studied. Mathematics subject classification (2000): 46B70.

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On ψ- interpolation spaces

Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkin-type Krylov methods for linear equations, namely, the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum, or the pseudospectrum. The convergence to exp$(\tau A)v$ is faster than that of corresponding Krylov methods for the solution of linear equations $(I-\tau A)x=v$, which usually arise in the numerical solution of stiff ordinary differential equations (ODEs). We therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.

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On Krylov Subspace Approximations to the Matrix Exponential Operator

General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

On the Stability of Functional Equations and a Problem of Ulam

A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces

Complete Normed Algebras

1. Numerical range in unital normed algebras 2. Hermitian elements of a complex unital Banach algebra 3. Operators 4. Some recent developments.

Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras

Numerical Ranges II is a sequel to Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras written by the same authors and published in this series in 1971. The present volume reflects the progress made in the subject, expanding and discussing topics under the general headings of spatial and algebra numerical ranges and further ranges.

Numerical Ranges II

If G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.

J. Duncan

Papers

Complete Normed Algebras

Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras

Numerical Ranges II

Amenability of inverse semigroups and their semigroup algebras

The Numerical Index of a Normed Space