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By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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L cp;e ... • , m) are known for the system ~. 44a). 44b) we obtain the vector cp+ =Sep-. e11. 44) leads us to the concept of an operator complex. •• , m). • , m) the square of which is the unit matrix. Definition The set of operators T, the system of vectors e .. • , m) in H, and the matrix J are termed an operator comple31 if the condition m 1 (T-T)4' • -. = I is satisfied. 48) Basic Concepts and Equations We shall denote the operator complex by lT; ecx; J]. 44). It has been shown above that to each open system associated with channels of class i there belongs a certain operator complex.

The first k components of an arbitrary vector in H k vanish under the transformation. The analogue of the Jordan canonical form for infinite dimensional spaces has been developed by Brodskii [6/1, 6/2, 8, 9, 23]. The operator is now called 'unicellular' if one of two arbitrary invariant subspaces is contained in the other. In order to perform the resolution into chains in the case where dimH = oo we have to transform T into triangular form in an infinite dimensional space. This will be discussed in Chapter 6.

5 four-pole device shown in Fig. 4 possesses an operator complex of the form [T i = O; e1 = 0, e2 = VG; J = (01 ~)]· (1. 12). It can be shown in the same way that the network of Fig. 5 has an operator complex of the form [ T = O; ei = i , VL e2 = O; J = (0I ~)]· (1. _11, Loo (1. 77) Let us now consider the network of Fig. 6. The corresponding equations are di L" at=-U1-U2-·. 2-· ··-fit+!. U" being the voltage drop across the capacitors. 1t we have 32 Basic: Concepts and Equations Fig. 6 where ~-1 = L,,, a2k =Ck (k = 1, 2, ...