ON PERMUTABILITY AND SUBMULTIPLICATIVITY OF SPECTRAL-RADIUS

Let r(T) denote the spectral radius of the operator T acting on a comp lex Hilbert space H. Let S be a multiplicative semigroup of operators on H. We say that r is permutable on S if r(ABC) = r(BAC), for every A , B, C is an element of S. We say that r is submultiplicative on S if r(AB) less than or equal to r(A)r(B), for every A,B is an element of S . It is known that, if r is permutable on S, then it is submultiplicat ive. We show that the converse holds in each of the following cases: ( i) S consists of compact operators (ii) S consists of normal operators (iii) S is generated by orthogonal projections.