A band method approach to a positive expansion problem in a by Frazho A.E., Kaashoek M.A. PDF

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

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Then we say that {Z, B , C , D} is a state space realization of F if F is the function defined by F(A) = D + A C ( I - A Z ) - I B . Notice that the values of F are linear operators from b/into y and F is an analytic function in some neighborhood of the origin. In fact, using (I - ,~Z) -1 = ~n~=0 ,~nZn, it follows that F(A) admits a power series expansion of the form: oo F()~) = D + E )~nvzn-lB' n=l in some neighborhood of the origin. Finally, it is noted that if Z is stable and [Z, B, C, D} is a realization of F, then F is a function in Hcc(/A, y).

Then f~ is well defined and has the following properties: (i) f~' is strictly positive, (ii) f~' commutes with U, (iii) Q -- Pnf~'17/. 342 Frazho, Kaashoek Furthermore, the operator L maps g_ into 74. 30) also hold for L. 34) L2M+($_) c_ JM+(E+). 8 coincide. This is an immediate consequence of the following result. 9 Let {T, U; Q} be the data for a commuting extension problem, and assume that Q is strictly positive. 33). 19). 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.

1) follows by directly verifying that G(A)F(A) = I and F(A)G(A) = I. 2). 1), we obtain F()~)-IC(I - ~ Z ) -1 = = D - 1 C ( I - AZ) -1 _ A D - t C ( I - A J ) - I B D - 1 C ( I - AZ) -1 = = D - 1 C ( I _ )~j)-I [(i _ A J) - )~BD-1C] ( I - ~Z) -1 = D - 1 C ( I - A J) -1 . This completes the proof. Let Z1 and Z2 be two stable operators on 7/, and let C~ and Cz be two operators from 7/into y . Then there is a unique solution P on 7 / t o the Stein equation P = Z2PZ1 + C~C1. 3) Frazho, Kaashoek 367 Moreover, this solution is given by oo P = E Z~C~C1Z'~.

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A band method approach to a positive expansion problem in a unitary dilation setting by Frazho A.E., Kaashoek M.A.


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