Some quadratic correct extensions of minimal operators in Banach spaces

Abstract

Let A0 be a minimal operator from a complex Banach space X into X with finite defect, def A0 = m , and  is a linear correct extension of A0 . Let Ec(A0, Â) ( resp. Ec(A0,  2) ) denote the set of all correct extensions B of A0 with domain D(B) = D(Â) ( resp. B1 of A0 with D(B1) = D(Â2) ) and let Em c (A0, Â) ( resp. Em+k c (A0,  2),k m, k,m ∈ N ) denote the subset of Ec(A0, Â) ( resp. Ec(A0,  2 ) consisting of all B ∈ Ec(A0, Â) ( resp. Ec(A0,  2) ) such that dimR(B− Â) = m ( resp. dimR(B1− Â2) = m+ k ) . In this paper: 1. we characterize the set of all operators B1 ∈ Em+k c (A0, Â2) with the help of  and some vectors S and G and give the solution of the problem B1x = f , 2. we describe the subset E2m 2c (A 2 0,  2) of all operators B2 ∈ E2m c (A0, Â2) such that B2 = B2 , where B is an operator of Em c (A0, Â) corresponding to B2 , 3. we give the solution of problems B2x = f . Mathematics subject classification (2010): 46N20, 47A20, 34B05, 45J05, 45K05.