By Frazho A.E., Kaashoek M.A.

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**Extra resources for A band method approach to a positive expansion problem in a unitary dilation setting**

**Example text**

In particular, R-*R -1 = L=*L -1. PROOF. Notice that if 9 is unitary, then R ~ -- L, and thus, R - * R -~ = L-*~*~L -~ = L-*L -1. 36). To do this, let (~ and L be the respective (g_, E+)-symbols of 9 and L, both with respect to U. Since 9 - R-1L, we have ~(e i~) -- R(ei~)-lL(ei~). 34). We have L(e'~)v = (~§ '~) = (~:~§ i~) (v c E_). 25) and the previous formula to show that L is the function, bounded and analytic on the open unit disk, given by L(A)v = S ~ ( I - AT*)-IYYol/2v (v C E_). 20)) that R(A) = X~/2 + AE~_(I - A T * ) - I T * X X o 1/2.

S T E P 5. Now let us show that there exists a one to one correspondence between the set of all solutions to the generalized Carath6odory interpolation problem with data {Z, F, F} and the commuting expansion problem with data {A; T, U}. Let F be a solution to the generalized Carath4odory interpolation problem. Put B = LF. Then/C+ = g~_(/d) is an invariant subspace for B. Obviously, B commutes with U. 21) W B I ]C+ = W L F I t2+(U) = W T F = W = W H . Hence H = P n B I ~+. So A is the compression of B to 7-/.

57) along with ft~ + ft+ = L - * L -1 = R - * R -1, we have B* + B = (R + LG)-*(R*fF+ - G*L*f~+) + (ft+R - fF+LG)(R + LG) -1 = (R + LG)-*{(R*a; - C*L*a+)(R + L C ) + +(R* + G*L*)(ft+R - fF+LG) } ( R + LG) -1 = (R + LG)-*{R*(fF+ + ft+)R - G*L*(f~+ + fF+)LG}(R + LG) -1 = (R + L G ) - * ( I - G*G)(R + n a ) -1 . This shows that B + B* admits a factorization of the form B + B* = (R + L G ) - * ( I - G*G)(R + n c ) -~ . 58) Because G is strictly contractive, the previous identity shows that B + B* is also strictly positive.