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SPECTRAL PROPERTIES OF CONTRACTIONS 3 3
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&inite algebra End(E arbitrary automorphism basis boundary spectrum coincidei coincides condition Consequently contains contraction converges convex COROLLARY critical exponent cross-norm defined denote diagonalizable dual End(E End(E,L equality equation Euclidean space eveiy example fact finite finite-dimensional follows formula giaph gioup Hence iepieiented iet o& iiometiy implies indecompoiable inequality infimum inner product invertible iome ipace ipectium isometry group iubalgebia iuch LEMMA linear functional linear operator linear space Markov chains matrix matrix norms maximal neceaaiy nilpotent noimed ipace nonnegative norm on End(E norm q normed space Obviously oithocomplemented opeiatoi noimi operator minorants operator norm orthogonal orthoprojector path of length piopeity projector PROOF rank-one operators reachable ring norm ring property scalar semigroup seminorm space End(E spec spectral radius strongly connected subalgebra subspace supp Suppose supremum theie exiiti Theorem unit ball unit-pieieiving unit-preserving unitary vectors vertex wheie whence