An analogue to a result of Takahashi

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• 作者：Dragana S. Cvetković-Ilić ; ^{dragana@pmf.ni.ac.rs" class="auth_mail" title="E-mail the corresponding author} ; Vladimir Pavlović ^{vlada@pmf.ni.ac.rs" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Operator matrix ; Injectivity ; Invertibility ; Operator range
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：264-275
• 全文大小：366 K

文摘

In this paper we completely answer the following question: for given A∈B(H,K)$A\in \mathcal{B}(\mathcal{H},\mathcal{K})$, ac6828424936c91d3f5e21f14" title="Click to view the MathML source">C∈B(L,K)$C\in \mathcal{B}(\mathcal{L},\mathcal{K})$ does there exist X∈B(H,L)$X\in \mathcal{B}(\mathcal{H},\mathcal{L})$ such that ac7307655f68c95a627" title="Click to view the MathML source">A+CX$A+CX$ is injective? As an application of the obtained results, for given operators A∈B(H)$A\in \mathcal{B}(\mathcal{H})$, B∈B(K)$B\in \mathcal{B}(\mathcal{K})$ and C∈B(K,H)$C\in \mathcal{B}(\mathcal{K},\mathcal{H})$, separate necessary conditions and sufficient conditions on the triple (A,B,C)$(A,B,C)$ for the existence of an operator X   such that the 2×2$2\times 2$ operator matrix acffef00ae">$M_X=\left[\begin{array}{cc}\hfill A\hfill & \hfill C\hfill \\ \hfill X\hfill & \hfill B\hfill \end{array}\right]$ is injective are presented.