By Frazho A.E., Kaashoek M.A.
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Extra info for A band method approach to a positive expansion problem in a unitary dilation setting
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. -/. 35) we see that the space UAA+(C_) is an invariant subspace for both L* and L-*. 60) holds. 60) implies that P n ( B - f~+)[7-/= 0.
In this case, F maps AJ one to one and onto At. PROOF. We claim that F is bounded below, that is, there exists a scalar 5 > 0 such that ][Ff]] > 5f]] for all f in/(2. To see this, let f be a unit vector in ]C. By assumption F + F* >_ eI for some ~ > 0. Hence 0 < e < ((F + F*)f, f ) = (F f, f ) + (f, F f ) <_ 2][Ff]]. This implies that F is bounded below. Replacing F by F* shows that F* is also bounded below. Therefore F is invertible. Now assume that A4 is an invariant subspace for F. Put F:~ = F]A4.
The values of ~+,M are bounded linear operators on E+. Since J is stable, the function ~+,M is a bounded analytic function on the open unit disk. 49). 49). 3 in the Appendix to show that 5(e = = fi+, (e + (0 < < Let ~+,M be the operator on t: commuting with U such that the g+-symbol of ~+, M is equal to ~+,M. The fact ~+,M is a bounded analytic function on the open unit disk implies that the space K:+ is invariant under ~+, M. Notice that gt = ~t+, M + ~*+, M" PART 2. 50) Q = T Q J * + E+Co. 21) gives Q - TQJ* = Q - (T - E + X o I X * T ) Q J * - E + X o ~ X * T Q J * = E+XolE~_ - E + X o l X * T Q J * = E+Co.
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