By Frazho A.E., Kaashoek M.A.
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Additional resources for A band method approach to a positive expansion problem in a unitary dilation setting
10. 9). 1), and thus Q - TQT* = K+E~_ + E + K ~ . Frazho, Kaashoek 347 Recall that E ~ _ X X o I = Ie+. 47). Next notice that M + XoIX*TQTXXo 1 = M + XolX*(TQT = - Q)XXo Xo 1 - M* = Xo 1 - K+XXo I + Xo 1 1. Here we used that Q X = E+, and hence X * Q X = X o . We conclude that with our choice of M we have Do = Do, M. 11 shows that 12+ is a well-defined bounded linear operator on ]C which commutes with U and leaves ~+ invariant. Moreover, f~+ + 12~ is strictly positive. It remains to show that Pu12+17/= A.
In this part we assume that B is a commuting expansion of A with respect to U and B + B* is strictly positive. 57). 61) G = -(I + L*AL)-IL*AR, where A = B - f~+. We claim that the operator I + L * A L is invertible, and thus, G is a well defined operator. To see this, recall that L - * L -1 = R - * R -1 = f~+ + f~_. Hence (I + L*AL) + (I + L*AL)* = I + L* (L-*L -I + A + A*) L = I+L-*(B+B*)L -1. 13 shows that I + L * A L is invertible. 61) is well defined. Obviously, G commutes with U. 61) uniquely determine each other.
57) may also be written in the form B = ~+ - L - * G ( I + ~ G ) - I R -1. 59) To see this, recall that L - * L -1 = f~+ + ~*+. Thus B = ( f ~ + R - f~*+LG)(R+ LG) -1 = [f~+(R + LG) - (f~+ + f~+)LG] (R + LG) -1 = f~+ - (~+ + f~*+)LG(R + LG) -1 = f~+ - L - * L - 1 L G ( R + LG) -1 = f~+ - L - * G ( I + ~ G ) - I R -1. Clearly, the operator L - * G ( I + @G)-IR -1 commutes with U. 60) L - * G ( I + ,~G)-IR-1]C+ C ~g+ 0 7-t. 30) shows that R -1 leaves ]C+ invariant. 15 the operator (I + @G) -1 has the same property.