On conditions for invertibility of difference and differential operators in weight spaces

NASA Astrophysics Data System (ADS)

Bichegkuev, Mairbek S.

2011-08-01

We obtain necessary and sufficient conditions for the invertibility of the difference operator D_E\\colon D(D_E)\\subset l^p_\\alpha \\to l^p_\\alpha, (D_E x)(n)=x(n+1)-Bx(n), n\\in {Z}_+, whose domain D(D_E) is given by the condition x(0)\\in E, where l^p_\\alpha=l^p_\\alpha({Z}_+,X), p\\in \\lbrack 1,\\infty \\rbrack , is the Banach space of sequences (of vectors in a Banach space X) summable with weight \\alpha\\colon{Z}_+\\to (0,\\infty) for p\\in \\lbrack 1,\\infty) and bounded with respect to \\alpha for p=\\infty, B\\colon X\\to X is a bounded linear operator, and E is a closed B-invariant subspace of X. We give applications to the invertibility of differential operators with an unbounded operator coefficient (the generator of a strongly continuous operator semigroup) in weight spaces of functions.