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 C with compact support. A function mT,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. The spectral multiplicity of a linear operator, which is studied in many papers (let us point out [4, 6, 9]), is one of its invariants. The notion of the local spectral multiplicity has been known only for normal operators. Here we discuss a definition of the local spectral multiplicity of a linear operator in the general case. It is based on the notion of a “maximal diagonalization” and seems to be natural enough. The local multiplicity of a linear operator with respect to a measure μ on the plane is introduced as a measurable nonnegative function defined μ-a.e. It possesses a series of natural properties. In particular, the spectral multiplicity is greater or equal than the essential supremum of the local multiplicity, and the local spectral multiplicity of a direct sum of linear operators equals to the sum of local multiplicities. A connection between the notions of generalized eigenvectors, of the local spectral multiplicity, and of diagonalizations is also explained. Generalized eigenvectors in various forms are actively exploited in the operator theory, see, for example, [1]. The author expresses his gratitude to A. A. Borichev and E. M. Dyn’kin for valuable consultations. 1. Reducing subspaces. Let N be a normal operator on a separable Hilbert space H. A (closed) subspace K in H is called reducing if it is invariant with respect to N and N∗. Assume now that N in its spectral representation has the