Stable norms – From theory to applications and back

AbstractThe purpose of this survey paper is to give a brief review of certain aspects of stability of norms and subnorms acting on algebras over a field F, either R or C. A norm N on an associative algebra A over F shall be called stable if for some positive constant σ,N(am)⩽σN(a)mforalla∈A,m=1,2,3…A norm shall be called strongly stable if the above inequality holds with σ=1.We begin the paper by discussing several results regarding norm stability, including conditions under which norms on certain algebras are stable. The second part of the paper is devoted to applications, where we employ the notion of norm stability to obtain criteria for the convergence of a well-known family of finite-difference schemes for the initial-value problem associated with the parabolic system∂u(x,t)∂t=∑1⩽j⩽k⩽sAjk∂2u(x,t)∂xj∂xk+∑1⩽j⩽sBj∂u(x,t)∂xj+Cu(x,t),where Ajk, Bj and C are constant matrices, Ajk being Hermitian. The third and last part of the paper deals with the question of stability for subnorms acting on subsets of power-associative algebras that are closed under scalar multiplication and under raising to powers. A subnorm f on such a set S is a real-valued function satisfying f(a)>0 for all 0≠a∈S, and f(αa)=∣α∣f(a) for all a∈S and α∈F