Integr. equ. oper. theory 38 (2000) 284-301 0378-620X/00/030284-18 $1.50+0.20/0 9 Birkh~user Verlag, Basel, 2000 I IntegralEquations and Operator Theory ON THE CONTINUITY OF THE SPECTRUM IN CERTAIN BANACH ALGEBRAS Israel Feldman and Naum Krupnik . In this paper we describe some classes of linear operators T E L(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spaces H such that the spectrum cr(T,L(H)) is continuous at the points T from these classes. We also describe some subalgebras Jt of the algebras .4 for which the spectrum or(a:,J[) becomes continuous at the points x when or(x, A) is restricted to the subalgebra .4. In particular, we show that the spectrum or(x, Jl) is continuous in Banach algebras g[ with polynomial identities. Examples of such algebras are given, i INTRODUCTION In this paper ~ always denote a Banach algebra with identity e over the field (P . All snbalgebras of ~ (closed or non-closed) will be assumed with identity e. For any subalgebra .4 C .~ we denote by GA the group of invertibIe elements x E ,4, and by T~(A) the radical of the algebra .4, i.e. re(.a) = {~ c .a: ~ - ,~= c a.a w ~ .a}. Denote by or(c, ~) the spectrum of the element x in algebra ~ and by 7- the set of all (nonempty) compact subsets of g' endowed with the Hausdorff metric d(F1, F2) = max(sup d(~, El), sup d(~, r~)). 9 EF~ zEF~ (0.,) It is well known [N] that the function f: ~ -+ 5 defined by equality f ( x ) = ~7(x,j~.) is upper semi-continuous, i.e. for each x0 E ~ and each neighborhood U((7(xo, A)) of the spectrum cr(x0,~) there exists 5 > 0 such that cr(x,~) C U(cr(xo,4)) whenever l]z - x01/< & In general the function f is not continuous on .4 ( see, for example, [A], Ch 1, w [H], problem 85 and many examples below in our paper). A modification of the example from [HI is considered below in Section 2. iThis research was partially supported by the Israel Science Fmmdation founded by tile Israel Academy of Sciences arid Humanit.ies.