On normal extensions of submatrices

✍ Scribed by Chung-Chou Jiang; Kung-Hwang Kuo

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 135 KB
• Volume: 370
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00414-2

✦ Synopsis

In this paper, we study the extension properties of a bounded linear transformation from a subspace of a Hilbert space into the whole space (e.g., which has a normal extension). Given an n × n normal matrix A and a k × n matrix B, k n, we obtain some sufficient conditions of subnormality for the submatrix (column matrix) A B by means of the geometric behavior of A and B. If, in particular, B is of rank one, we show that these sufficient conditions are also necessary for subnormality of A B . In order to prove these results, we establish the key lemma which says that XX * = B * B if and only if X * = V B for some k × k unitary matrix V .