Stability properties of finite meromorphic operator functions. III

✍ Scribed by H Bart; M.A Kaashoek; D.C Lay

• Publisher: Elsevier Science
• Year: 1974
• Weight: 834 KB
• Volume: 77
• Category: Article
• ISSN: 1385-7258
• DOI: 10.1016/1385-7258(74)90041-9

✦ Synopsis

The conjugate operator function

In this section we study some important relations between the H, and K,,, spaces for A and those for the conjugate operator function A*. We denote the conjugate spaces of X and Y by X* and Y*, respectively, and we define A* to be the function with values in 9'( Y*, X*) given by A*(l) = A@)*, for 3, in the domain of A. Clearly A* E &?(k, 9( Y*, X*)) and v(A*; 39) = =v(A; ;b). If A, denotes the 72th coefficient of the Laurent expansion of A at &, then (An)* is the corresponding coefficient of the Laurent expansion of A* at &. 6.1 REMARK.

Suppose that A is finite meromorphic at 39. Then it follows that A* is also finite meromorphic at ho. In Section 3 we have constructed in a canonical way operator functions S and T which are holomorphic at 20 and resemble in certain aspects the function A. It is not dificult to show that S* (resp., T*) bears the same relationship to A* that T (reap., 8) bears to A. Indeed, let B and 6' be operator functions given by (3-3) and (3-4). Then and '*(I) = A*(A) B*(1), ;z E A, Ao* Qo*, A=&. Observe that &I* is a continuous projection of Y* and dim R(&*) ~00. Furthermore, &o* =IY* -&I* and, from (32), A+_,Qo*=O, i=l, . . . . p. If one compares these results with formulas (3-ll), (3-12) and (3-13), one sees that S* has the desired properties. Similar formulas hold for T*.