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On the Matrix Spectral Function of a Generalized Second‐Order Differential Operator in a Ramified Space
Author(s) -
Weber Matthias
Publication year - 1999
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19992060107
Subject(s) - mathematics , hilbert space , differential operator , order (exchange) , product (mathematics) , operator (biology) , combinatorics , space (punctuation) , matrix (chemical analysis) , subspace topology , inner product space , pure mathematics , mathematical analysis , domain (mathematical analysis) , geometry , biochemistry , chemistry , linguistics , materials science , philosophy , finance , repressor , transcription factor , economics , composite material , gene
The existence of a unique n × n matrix spectral function is shown for a selfadjoint operator A in a Hilbert space L 2 α (m). This Hilbert space is a subspace of the product of spaces L 2 (m i ) with measures m 1 , i = 1, n , having support in [0,∞]. The inner product in L 2 α (m) is the weighted sum of the inner products in the L 2 ( m i ), i.e., ( f, g ) m ,α =∑α i ( f i ,g i ) m i , f =( f 1 ,… f n ), g =( g 1 ,… g n )∈ L α 2 (m), with positive constants α i = 1, …, n . The operator A is given by (A f ) i = − D m i D x + f , with generalized second order derivatives D mi D + x . The elements of the domain of A have continuous representatives satisfying f 1 (0) f j (0), i, j = 1, n , and an additional gluing condition at 0.