The boundedness below of 2 x 2 upper triangular operator matrices

When A is an element of L(H) and B is an element of L(K) are given we denote by M-C an operator acting on the Hilbert space H circle plus K of the form M-C := (---), where C is an element of L(K, H). In this paper we characterize the boundedness below of M-C. Our characterization is as follows: M-C is bounded below for some C is an element of L(K, H) if and only if A is bounded below and alpha (B) less than or equal to beta (A) if R(B) is closed; beta (A) = infinity if R(B) is not closed, where alpha(.) and beta(.) denote the nullity and the deficiency, respectively. In addition, we show that if sigma (alphap)(.) and sigma (d)(.) denote the approximate point spectrum and the defect spectrum, respectively, then the passage from sigma (alphap) (---- ) to sigma (alphap)(M-C) can be described as follows: sigma (alphap) (----) = sigma (alphap)(M-C) boolean OR W for every C is an element of L(K, H), where W lies in certain holes in sigma (alphap)(A), which happen to be subsets of sigma (d)(A) boolean AND sigma (alphap)(B).