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

Let $T$ be a bounded linear operator in a separable Banach space $\mathcal X$ and let $\mu$ be a nonnegative measure in $\mathbb C$ with compact support. A function $m_{T,\mu}$ is considered that is defined $\mu$-a.e. and has nonnegative integers or $+\infty$ as values. This function is called the local multiplicity of $T$ with respect to the measure $\mu$. 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 ezamples of evaluation of the local spectral multiplicity function are given. Bibliography: 10 titles.