Spectral Decomposition of Symmetric Operator Matrices

✍ Scribed by Reinhard Mennicken; Andrey A. Shkalikov

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 649 KB
• Volume: 179
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19961790115

✦ Synopsis

The authors study symmetric operator matrices A B = ( B ' C ) in the product of Hilbert spaces H = Hi xH2, where the entries are not necessarily bounded operators. Under suitable assumptions the closure Lo exists and is a selfadjoint operator in H. With Lo, the closure of the transfer function M(X) = C -X -B * ( A -X)-'B is considered. Under the assumption that there exists a real number p < inf p(A) such that M ( P ) << 0, it follows that E p ( z). Applying a factorization result of A. I. VIROZUB and V. I. MATSAEV [VM] to the holomorphic operator function M( A), thespectral subspaces of corresponding to the intervals ] -00, p ] and [ p, 00[ and the restrictions of Lo to these subspaces are characterized. Similar results are proved for operator matrices which are symmetric in a Kr&n space.