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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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