0. Notation and Preliminaries In thi s chapte r a number o f fact s whic h wil l prove usefu l i n the seque l are assembled togethe r with eithe r som e indication o f proof o r an appropri - ate referenc e t o th e literature . Thes e ar e presente d i n a sequenc e o f shor t sections, eac h o f whic h bear s a descriptiv e titl e o f it s contents . W e begin, however, wit h a sectio n o n notation . I n particular , i t shoul d b e note d tha t the latter contains one nonstandard Conventio n which is imposed in order to make Statement s an d proof s wor k mor e o r les s simultaneously fo r bot h th e disc and the halfplane . 0.1. Notatio n The symbol s € , C jxk an d I R wil l denot e th e comple x numbers , th e space of complex j x k matrice s and the real numbers, respectively, wherea s C i s shor t fo r € x l C + [resp . €_ ] Stand s fo r th e ope n uppe r [resp . lower] half plane , T fo r th e uni t circle , I D for it s interior : B D = {k € C : \k\ 1 } and I E for it s exterio r wit h respec t t o the extende d comple x plan e € = € U {oo}: I E = {X e € : 1 \k\ oo} . If A is a set, then Ä denotes its closure. I f A is a matrix or Operator, then A* Stand s fo r it s adjoin t wit h respec t t o th e Standar d inne r produc t unles s indicated otherwise . Thu s if A i s a matrix, then A* designates its conjugat e transpose and A T it s ordinary transpose, whereas, if A is a complex number , then A* Stands fo r it s comple x conjugate . I f A i s a j x k matrix , the n th e symbol \A\ designates it s maximum s-number , alia s its norm a s an Operato r from (£ k int o C 7 whe n bot h o f thes e Space s ar e take n wit h th e Standar d inner product . The symbo l A + wil l b e use d t o designat e eithe r D o r C + whil e th e symbols which appear in the following table are interpreted accordin g to the choice of A+, as indicated. i http://dx.doi.org/10.1090/cbms/071/01

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