On Block Limiting Norm and Structured Singular Value.

Abstract

The notion of limiting norm, introduced by Pokrovskii (Soviet Math. Dokl., vol. 20, pp. 1314-1317, 1979), is generalized to that of block limiting norm. A resemblance of inequalities shared by both the block limiting norm and the structured singular value, introduced by Doyle (Proc. IEE, vol. 129, pp. 245-250, 1982), motivates further investigation of their relationships. To that effect, the concept of generalized spectral radius of a set of linear operators is introduced. It is then shown that, for block-structure of size less than 4, the block limiting norm is equal to the structured aingular value and that, in the general case, the block limiting norm is always no less than the structured singular value. Finally, better bounds are obtained for both the block limiting norm and the structured singular value.