A study is made of a symmetric functional calculus for a system of bounded linear operators acting on a Banach space. Finite real linear combinations of the operators have real spectra, but the operators do not necessarily commute with each other. Analytic functions of the operators are formed by using functions taking their values in a Clifford algebra.
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CONTENTS 1. Introduction...................................................................................................5 2. N-tuples of linear transformations in finite-dimensional space......................8 3. Toeplitz operators on the polydisc and the unit ball....................................18 4. Subspaces of weighted shifts.....................................................................23 5. Joint spectra for N-tuples of operators........................................................27 6. Algebras of operator weighted shifts...........................................................30 7. Functional calculus for N-tuples of contractions..........................................37 8. Dual algebras, invariant subspace problem and reflexivity..........................44 9. Reflexivity of jointly quasinormal operators and spherical isometries..........45 10. Reflexivity and existence of invariant subspaces......................................47 11. Questions and open problems..................................................................56 References.....................................................................................................58
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