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On Local Invertible Operators in L 2 (ℝ 1 , H )
Author(s) -
Hasanov Mahir
Publication year - 2001
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 - 10.1002/1522-2616(200108)228:1<145::aid-mana145>3.0.co;2-v
Subject(s) - mathematics , invertible matrix , commutator , neighbourhood (mathematics) , operator (biology) , factorization , pure mathematics , differential operator , algebra over a field , discrete mathematics , mathematical analysis , algorithm , biochemistry , chemistry , lie conformal algebra , repressor , transcription factor , gene
We study operators of the form Lu = $\textstyle {d^2u \over dt^2}$ — G ( t ) u ( t ) in L 2 ([ t 0 — δ , t 0 + δ ], H ) with $\overline {D(L)}$ = L 2 ([ t 0 — δ , t 0 + δ ], H ) in the neighbourhood [ t 0 — δ , t 0 + δ ] of a point t 0 ∈ ℝ 1 . Such problems arise in questions on local solvability of partial differential equations (see [6] and [7]). For these operators,one of the major questions is if they are invertible in a neighbourhood of a point t ∈ ℝ 1 . To solve this problem we establish needed commutator estimates. Using the commutator estimates and factorization theorems for nonanalytic operator‐functions we give additional conditions for the nonanalytic operator ‐function G ( t ) and show that the operator L (or $\overline L$ ) with some boundary conditions is local invertible.