Sublinearity and other spectral conditions on a semigroup

Subadditivity, sublinearity, submultiplicativity, and other conditions are considered for spectra of pairs of operators on a Hilbert space. Sublineari ty, for example, is a weakening of the well-known property L and means sigm a(A + lambda B) subset of or equal to sigma(A) + lambda sigma(B) for all sc alars lambda. The effect of these conditions is examined on commutativity, reducibility, and triangularizability of multiplicative semigroups of opera tors. A sample result is that sublinearity of spectra implies simultaneous triangularizability for a semigroup of compact operators.