Continuity of linear operators commuting with continuous linear operators. II
B. E. Johnson, Allan M. Sinclair · Transactions of the American Mathematical Society · 1969
Introduction.We continue to study the question of when a continuous operator F in a Banach space 36 (or a pair of continuous operators R, T in spaces SQ, 3£) has a discontinuous operator S with ST=TS (or a discontinuous S; 36 -*■ s2) with ST=RS).The operator S is said to commute with F (or the pair {T, R}).The authors are indebted to Dr. S. Swierczkowski who showed them how the fact that (F-A/)/0 = /0, where l0 is the subspace of £ = l2(Q, oo) consisting of sequences with only finitely many nonzero terms, F is the left shift (T{ai})j = aj + 1 and A e C, could be used to construct a discontinuous S; 3E -> 3£ commuting with F. A study of this situation yields Lemma 2.4, a general method of constructing discontinuous commuting operators.When this method is added to the method used in [3], reproduced here in Lemma 2.1, we have complete knowledge of when a compact or a quasi-nilpotent operator F has discontinuous operators commuting with it, viz.if neither of these two methods is applicable then there are no discontinuous operators commuting with F. The methods carry over to the case of operators F for which 3£ splits in some rough way corresponding to decompositions of the spectrum o(T), in particular if F is a spectral operator or an operator with totally disconnected spectrum.This work appears in §4.In §3 we consider the possibility of discontinuous operators S commuting with a pair {F, R} where a(R) is countable (and hence totally disconnected).The methods of §4 can be used to prove results about operators commuting with pairs, the extension being fairly obvious and rather messy.In these extensions one needs to assume more or less that both R and F satisfy the conditions of Theorem 4.3.The point in assuming that a(R) is countable is that we can proceed with fewer and more natural conditions on T.2. Construction of discontinuous commutants.Throughout this section X, ty will be complex Banach spaces; R and F will be continuous linear operators in "3), 36 respectively.A complex number A is a critical eigenvalue of {F, R} if (F-AF)36 is of infinite codimension in 36 and A e ap(R), the point spectrum of R.2.1.Lemma.If {T, R} has a critical eigenvalue A then there is a discontinuous linear operator S; 3£ -^ ?) with ST=RS.