Local spectral multiplicity of a linear operator with respect to a measure

Let T be a bounded linear operator in a separable Banach space X and let μ be a nonnegative measure in χ with compact support. A function m_T,μ is considered that is defined μ-a.e. and has nonnegative integers or +∞ as values. This function is called the local multiplicity of T with respect to the measure μ. This function has some natural properties, it is invariant under similarity and quasisimilarity; the local spectral multiplicity of a direct sum of operators equals the sum of local multiplicities, and so on. The definition is given in terms of the maximal diagonalization of the operator T. It is shown that this diagonalization is unique in the natural sense. A notion of a system of generalized eigenvectors, dual to the notion of diagonalization, is discussed. Some examples of evaluation of the local spectral multiplicity function are given. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 293–306.