Matrix norms and their applications

1 - Operators in Finite-Dimensional Normed Spaces.- 1. Norms of vectors, linear functionals, and linear operators.- 2. Survey of spectral theory.- 3. Spectral radius.- 4. One-parameter groups and semigroups of operators.- Appendix. Conditioning in general computational problems.- 2 - Spectral Properties of Contractions.- 1. Contractive operators and isometries.- 2. Stability theorems.- 3. One-parameter semigroups of contractions and groups of isometries.- 4. The boundary spectrum of extremal contractions.- 5. Extreme points of the unit ball in the space of operators.- 6. Critical exponents.- 7. The apparatus of functions on graphs.- 8. Combinatorial and spectral properties of l?-contractions.- 9. Combinatorial and spectral properties of nonnegative matrices.- 10. Finite Markov chains.- 11. Nonnegative projectors.- 3 - Operator Norms.- 1. Ring norms on the algebra of operators in E.- 2. Characterization of operator norms.- 3. Operator minorants.- 4. Suprema of families of operator norms.- 5. Ring cross-norms.- 6. Orthogonally-invariant norms.- 4 - Study of the Order Structure on the Set of Ring Norms.- 1. Maximal chains of ring norms.- 2. Generalized ring norms.- 3. The lattice of subalgebras of the algebra End(E).- 4. Characterization of automorphisms.- Brief Comments on the Literature.- References.