The strong closure of the simultaneous similarity orbit of some pairs of finite rank operators
Let \(A\) and \(B\) be operators on a separable Hilbert space \(\mathcal H\). Denote by \(\mathcal S(A)\) the similarity orbit of \(A\), that is, the set of all operators \(W^{-1}AW\), where \(W\) is an invertible operator on \(\mathcal H\), and by \(\mathcal S(A,B)\) the simultaneous orbit of the pair \((A,B)\), that is, the set of all pairs \((W^{-1}AW, W^{-1}BW)\) where \(W\) is an invertible operator on \(\mathcal H\). Denote by \(\mathcal L(\mathcal H)\) the algebra of all operators on \(\mathcal H\). On \(\mathcal L(\mathcal H)\), consider the strong operator topology. Let \(\bar{\mathcal S}(A)\) and \(\bar{\mathcal S}(A,B)\) denote the strong closure of \(\mathcal S(A)\) and \(\mathcal S(A,B)\), respectively. Let \(d(A,B)\) denote the dimension of the span of \(({\text{ker A}})^\perp\) and \(({\text{ker B}})^\perp\). ``Let \(\mathcal C(A,B)\) be the set of pairs \((X,Y)\) in \(\bar{\mathcal S}(A)\times\bar{\mathcal S}(B)\) such that \(d(X,Y)\leq d(A,B)\). In [the author, Linear Algebra Appl. 432, No. 8, 1873--1877 (2010; Zbl 1182.47022)] it was shown that \(\bar{\mathcal S}(A,B)\subset \mathcal C(A,B)\) if \(A\) and \(B\) are finite rank operators. In the paper we study the equality \(\bar{\mathcal S}(A,B) = \mathcal C(A,B)\).'' The following theorem is the main result of the paper. Theorem. Let \(A_1\) and \(A_2\) be finite rank operators on \(\mathcal H\). Then \(\bar{\mathcal S}(A,B) = \mathcal C(A,B)\) if and only if \((\mathrm{ran} A)\cap(\mathrm{ran} B)=\{0\}\).
scientific article; zbMATH DE number 6138290
scientific article