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Abstract

We discuss some results and problems connected with estimation of spectra of operators (or elements of general Banach algebras) which are expressed as polynomials in several operators, noncommuting but satisfying weaker conditions of commutativity type (for example, generating a nilpotent Lie algebra). These results have applications in the theory of invariant subspaces; in fact, such applications were the motivation for consideration of spectral problems. More or less detailed proofs are given for results unpublished before or published in short communications; in some other cases we give a scheme of proof. The author is obliged to J. A. Erdos, V. S. Guba and especially to Yu. V. Turovskiĭ for useful discussions.