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On the Matrix Spectral Function of a Generalized Second-Order Differential Operator in a Ramified Space

✍ Scribed by Matthias Weber

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 623 KB
Volume: 206
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19992060107

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✦ Synopsis

The existence of a unique 71 x n matrix spectral function is shown for a selfadjoint operator A in a Hilbert space Lg(m). This Hilbert space is a subspace of the product of spaces L2(rn;) with measures rn,, i = 1 , . . . , n , having support i n [O,m). The inner product in Li(m) is the weighted sum of the inner products in the L 2 ( r n , ) , i. e., ( f r g ) m . a = oii(fi,gi)m,, f = (fl,. . . ,fn), g = (91,. . . , g n ) E L:(m), with positive constants a,, i = 1,. . . ,TI. The operator A is given by (Af), = -Dn,i LIZ f, with generalized second or&r derivatives ,!In,, 0:. The elements of thc domain of A havc, continuous representa1.ives satisfying f ? ( O ) = f,(O), i, j = 1,. . . ,7i, and an additional gluing condition a t 0. li]) < 03. Here the entries ai are positive constants.

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