The boundedness below of 2×2 upper triangular operator matrices

Abstract

Wen$A \in \mathcal{L}(\mathcal{H})$ and$B \in \mathcal{L}(\mathcal{K})$ are given we denote byM _C an operator acting on the Hilbert space$\mathcal{H} \oplus \mathcal{K}$ of the form

$$M_C : = \left( {\begin{array}{*{20}c} A & C \\ 0 & B \\ \end{array} } \right),$$

where$C \in \mathcal{L}(\mathcal{K},\mathcal{H})$. In this paper we characterize the boundedness below ofM _C . Our characterization is as follows:M _C is bounded below for some$C \in \mathcal{L}(\mathcal{K},\mathcal{H})$ if and only ifA is bounded below and α(B)≤β(A) ifR(B) is closed; β(A)=∞ ifR(B) is not closed, where α(·) and β(·) denote the nullity and the deficiency, respectively. In addition, we show that if ρ _ap (·) and ρ _d (·) denote the approximate point spectrum and the defect spectrum, respectively, then the passage from$\sigma _{ap} \left( {\begin{array}{*{20}c} A & 0 \\ 0 & B \\ \end{array} } \right)$ to ρ _ap (M _C ) can be described as follows:

$$\sigma _{ap} \left( {\begin{array}{*{20}c} A & 0 \\ 0 & B \\ \end{array} } \right) = \sigma _{ap} (M_C ) \cup Wfor everyC \in \mathcal{L}(\mathcal{K},\mathcal{H}),$$

whereW lies in certain holes in ρ _ap (A), which happen to be subsets of ρ _d (A)∩ρ _ap (B).