Factorizations of Operator Matrices

This note gives explicit factorizations of a 2 \Theta 2 operator matrix as a product of an upper triangular operator matrix and an involutory, unitary or J - unitary operator matrix. A pattern is given for construction of factorizations of this kind. Let H and K be Hilbert spaces and let L(H;K) denote the space of all bounded linear operators from H to K. Put L(H) = L(H;H ). Throughout, M = " A B C D # denotes any given operator in L(K \Theta H). To describe factorizations of M , define T (Z) = (AZ +B)(CZ +D) \Gamma1 ; (1) S(W ) = (A \Gamma WC) \Gamma1 (WD \Gamma B) for Z; W 2 L(H;K ). By [4, Prop. 4], if M \Gamma1 exists then [A \Gamma T (Z)C] \Gamma1 exists whenever Z 2 L(H;K) and (CZ + D) \Gamma1 exists, and [CS(W ) + D] \Gamma1 exists whenever W 2 L(H;K) and (A \Gamma WC) \Gamma1 exists; moreover, S = T \Gamma1 . Theorem 1 Put R = " \Gamma(A \Gamma W 0 C) AZ 0 +W 0 D 0 CZ 0 +D # ; U = " \Gamma(I \Gamma Y X) 2Y \Gamma Y XY X I \Gamma XY # : a) Suppose ..