Abstract. A bounded linear operator $ T, $ respectively an $n$-tuple $ T $ of commuting bounded operators, on a complex Banach space $ {\cal X} $ is strongly harmonic if it is contained in a unital commutative strongly harmonic closed subalgebra $ {\cal A} \subset B({\cal X}). $ Every strongly harmonic operator is decomposable in the sense of Foiaş and every strongly harmonic $n$-tuple is decomposable in the sense of Frunză. On the other hand, it is proven that the class of strongly harmonic operators is quite large and that operators in this class have very nice properties. If an elementary operator is determined by two strongly harmonic $ n$-tuples, then it is strongly harmonic, and its local spectra are in a simple connection with the analytic local spectra of $2n$-tuple of the coefficients.

AMS Subject Classification (1991): 47B40, 47B47, 47B48

Received February 27, 2001, and in revised form April 23, 2001. (Registered under 2869/2009.)