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Some quadratic correct extensions of minimal operators in Banach spaces. (English) Zbl 1192.47005
Summary: Let \(A_0\) be a minimal operator from a complex Banach space \(X\) into \(X\) with finite defect, \(\text{def}A_0=m\), and \(\widehat A\) is a linear correct extension of \(A_0\). Let \(E_c(A_0,\widehat A)\) (resp., \(E_c(A^2_0,\widehat A^2))\) denote the set of all correct extensions \(B\) of \(A_0\) with domain \(D(B) = D(\widehat A)\) (resp. \(B_1\) of \(A^2_0\) with \(D(B_1) = D(\widehat A^2))\) and let \(E^m_c(A_0,\widehat A)\) (resp., \(E^{m+k}_c(A^2_0,\widehat A^2)\), \(k\leq m\), \(k,m\in\mathbb N\)) denote the subset of \(E_c(A_0,\widehat A)\) (resp., \(E_c(A^2_0,\widehat A^2)\)) consisting of all \(B\in E_c(A_0,\widehat A)\) (resp., \(E_c(A^2_0,\widehat A^2)\)) such that \(\dim R(B-\widehat A)=m\) (resp., \(\dim R(B_1-\widehat A^2)=m+k)\). In this paper, 6mm
(1)
we characterize the set of all operators \(B_1\in E^{m+k}_c(A^2_0,\widehat A^2)\) with the help of \(\widehat A\) and some vectors \(S\) and \(G\) and give the solution of the problem \(B_1x=f\),
(2)
we describe the subset \(E^{2m}_{2c}(A^2_0,\widehat A^2)\) of all operators \(B_2\in E^{2m}_c(A^2_0,\widehat A^2)\) such that \(B_2=B^2\), where \(B\) is an operator of \(E^m_c(A_0,\widehat A)\) corresponding to \(B_2\),
(3)
we give the solution of problems \(B_2x=f\).

MSC:
47A20 Dilations, extensions, compressions of linear operators
34B05 Linear boundary value problems for ordinary differential equations
45J05 Integro-ordinary differential equations
45K05 Integro-partial differential equations
46N20 Applications of functional analysis to differential and integral equations
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