Criteria of Unitary Equivalence of Hermitian Operators with a Degenerate Spectrum
نویسنده
چکیده
Nonimprovable, in general, estimates of the number of necessary and sufficient conditions for two Hermitian operators to be unitarily equaivalent in a unitary space are obtained when the multiplicities of eigenvalues of operators can be more than 1. The explicit form of these conditions is given. In the Appendix the concept of conditionally functionally independent functions is given and the corresponding necessary and sufficient conditions are presented. Let P, Q be the operators from a unitary n-dimensional space Un in Un, and P , Q be the matrices of these operators in some orthonormal basis. Description of a system of invariants of these matrices which enables one to find out whether the given operators are unitarily equivalent is the classical problem of the theory of invariants (see, e.g., [1, §2.2], [2] and the references cited therein). In the author’s paper [3] it is shown that two matricess P,Q ∈ Mn(C) are unitarily equivalent iff the following conditions are fulfilled: tr { P l +P−P m + P 2 − } = tr { Q+Q−Q m +Q 2 − } , 0 ≤ l ≤ m ≤ n− 1, tr { P l + } = tr { Q+ } , 1 ≤ l ≤ n, (1) where A+ (A−) denotes the Hermitian (skew-Hermitian) part of the matrix A ∈ Mn(C): A± = (A±A∗)/2. Formulas (1) contain n(n + 3)/2 of complex (but only n2 + 1 of real) conditions, and all these conditions are independent if no additional restrictions are imposed on the entries of the matrices P±, Q±. However, if such restrictions are imposed, in particular, if some eigenvalues of the operator P+ have multiplicity ≥ 2, then not all of conditions (1) are independent [3]. 1991 Mathematics Subject Classification. 15A72, 13A50, 15A15, 15A54.
منابع مشابه
On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators
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