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Composition Preserves Rigidity
Author(s) -
Lotto B. A.,
McCarthy J. E.
Publication year - 1993
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/25.6.573
Subject(s) - mathematics , modulo , rigidity (electromagnetism) , composition operator , composition (language) , lebesgue measure , generalization , unit circle , pure mathematics , unit (ring theory) , operator (biology) , combinatorics , mathematical analysis , lebesgue integration , multiplication operator , hilbert space , linguistics , philosophy , biochemistry , mathematics education , chemistry , structural engineering , repressor , transcription factor , engineering , gene
Say that f in H p (1 ⩽ p ⩽ ∞) of the unit disk is rigid if it is determined in H p by its argument modulo 2π on the unit circle. We show that if f is rigid and u is an inner function, then the composition f o u is rigid. The proof uses the disintegration of Lebesgue measure with respect to u to compute the adjoint of the operator of composition with u . This result is a generalization of the work of Younis, who proved the special case p = 1 using operator theoretic methods.