An index formula for the product of linear relations
A linear relation \(A\) from a linear space \(X\) to a linear space \(Y\) is a linear subspace of the space \(X\times Y\). Denote by \(L(X,Y)\) the class of all linear relations from \(X\) to \(Y\). The domain and the range of a linear relation \(A\) are defined by \(\text{dom}\,A=\{x:\{x,y\}\in A\}\) and \(\text{ran}\,A=\{y:\{x,y\}\in A\}\), respectively. The kernel and the multi-valued part of \(A\) are linear spaces given by \(\text{ker}\,A=\{x:\{x,0\}\in A\}\) and \(\text{mul}\,A=\{y:\{0,y\}\in A\}\), respectively. The nullity and co-nullity of \(A\) are defined by \(n(A)=\dim\,\text{ker}\,A\) and \(n_c(A)=\dim\,\text{mul}\,A\) and the defect and co-defect are given by \(d(A)=\dim Y/\text{ran}\,A\) and \(d_c(A)=\dim X/\text{dom}\, A\), respectively. The index and the co-index of a linear relation \(A\) are defined as \(r(A)=n(A)-d(A)\) and \(r_c(A)=n_c(A)-d_c(A)\). The main result of the paper says that if \(X,Y,Z\) are linear spaces and \(A\in L(X,Y)\) and \(B\in L(Y,Z)\) are linear relations with finite indices, then \(r(BA)-r(A)-r(B)=r_c(BA)-r_c(A)-r_c(B)\) and their common value coincides with \(\dim(Y/(\text{ran}\,A+\text{dom}\,B))-\dim(\text{mul}\,A\cap\ker B)\). Further, this result is used to prove that \(r(A^{p+q})=r(A^p)+r(A^q)\) and \(r_c(A^{p+q})=r_c(A^p)+r_c(A^q)\) for \(A\in L(X)\) and all \(p,q\in{\mathbb N}\) under the natural assumption of the non-emptiness of the so-called extended resolvent set of \(A\).
scientific article; zbMATH DE number 5614894