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Additional info for Integral Equations and Operator Theory - Volume 64
64 (2009) On the Structure of L1 of a Vector Measure J. A. es Submitted: December 16, 2008. 33 Integr. equ. oper. 1007/s00020-009-1679-9 Integral Equations and Operator Theory State Space Formulas for a Solution of the Suboptimal Nehari Problem on the Unit Disc Ruth F. Curtain and Mark R. Opmeer Abstract. We give state space formulas for a (“central”) solution of the suboptimal Nehari problem for functions defined on the unit disc and taking values in the space of bounded operators in separable Hilbert spaces.
2 −2 ∗ Furthermore, we have B ∗ XBTX = TX B XB so that the above is equivalent to 2 ∗ 2 2 ∗ −TX B XLB LC B + B ∗ XBB ∗ XB + TX − TX B LC B = I. Using the definition of TX from (8) this is equivalent to − B ∗ XLB LC B − B ∗ XBB ∗ XLB LC B + B ∗ XBB ∗ XB + B ∗ XB − B ∗ LC B − B ∗ XBB ∗ LC B = 0. (29) Combining the first and fourth term on the left-hand side gives −B ∗ XLB LC B + B ∗ XB = B ∗ X(I − LB LC )B = B ∗ LC B, so that these terms cancel against the fifth term on the left-hand side of (29).
Noting that all the transfer functions G, V, and Z are holomorphic on D, we perform some elementary calculations on D. 7 Z(¯ z ) = V12 (z)∗ V11 (z)−∗ holds on the open unit disc. Using the just established identity we have ∗ (G(z) + Z(¯ z )) (G(z) + Z(¯ z )) − I = I G(z) + Z(¯ z) = V11 (z)−∗ 0 × I 0 G(z) I ∗ −I 0 ∗ V(z) 0 I I G(z) + Z(¯ z) ∗ I 0 G(z) I V11 (z)−∗ 0 V(z)∗ −I 0 × 0 I . 7 we obtain ∗ (G(z) + Z(¯ z )) (G(z) + Z(¯ z )) − I ≤ V11 (z)−∗ 0 ∗ −I 0 0 I V11 (z)−∗ 0 (50) = −V11 (z)−1 V11 (z)−∗ ≤ 0.