p. VOPD> ph .r3 o· l~ 2, MA!1~ ~IS·l.f L £tC 3'6 o_~.~----~ 0- 0. __• _ _____ Lecture Notes in Physics Edited by J. Ehlers, Munchen, K. Hepp, Zurich R. Kippenhahn, Munchen, H. A. Weidenmuller, Heidelberg and J. Zittartz, Koln Managing Editor: W. Beiglbock, Heidelberg 123- i. ,$ dB ;gIS! !!\lP' d! -tp '* +9 A?++& ei Dieter H."Mayer" ,The Ruelle-Araki Transfer Operator in Classical Statistical Mechani( 311 + F -6-9 q..- , ...... -%"...cl· d }1M 'ws:* •• •• • APPENDIX A. GROTHENDI.ECK'S THEORY OF NUCLEAR OPERATORS IN BANACH SPACES In this appendix we recall the fundamentals of Grothendieck's the- ory about nuclear operators in Banach spaces as far as they are neces- sary for the considerations in this work. A much more extensive pre­ sentation of this theory can be found in ~ 1] and fi 28] • A.l. The projective topological tensor product of Banach spaces Let E, F be two complex Banach spaces with norms II liE and II I~. Let E ® F be the tensor product of these. two spaces with the norm II II'll defined by = inf z:::=.1Ie)..11 E II f .11 F' (A1) IIx "71" {i} ). ;!. where the infimum has to be taken over all possible finite representa- ,- !~ tions of x € E ® F in the form I, ~ ,i :1 x =Le. ®f. , (A2) {i}). ). ,- ij with e. E and f. e F• ). e ). The completion of the space E.® F under this norm is denoted by ~ E~ ., F and is called the projective topological tensor product of the \ -71" ~ two spaces E and F. The norm defined in (Al) and introduced first by i A f: R. Schatten [129] is called the 'If-norm. The elements of the space E @'!T F j;; Iji'i are the Fredholm kernels. i! II "fJ A.2. The tensor product of two linear mappings ii [1 The following important Theorem was proved by Grothendieck E30J: l' Theorem A. 1 Let- E,' F, G be three Banacli spaces and let T" ExF~G· ~ ~ dl ====--"=~=---~=--=--------------- _ r i.' - .be a bilinear continuous mapping of the.direct product E x F into G. Then there exists a~ uniquely determined linear continuous mapping A T:'" E 0'1r F ---7 G with T'" u = T (e,f) ifu=e@f. Furthermore one :he- has 11TH = IITII :es- Consider then two pairs (E , F ) and (E , F ) of Banach spaces and 1 1 2 2 'e- two linear continuous mappings T. Ei~Fi Define a map- ~ ' i = " 2. " as follows: ping T 1 X T2 : E, x E2----7 F, ®17'F2 = (A3) According to Theorem A.l there exists an uniquely determined linear mapping A 1 ) (M) ta- which is called the tensor product of the two linear mappings T, and 1\2 ) A.3. Nuclear operators in Banach spaces Let E, F be'complex Banach spaces. Let E* be the dual Banach space of E that means the space of all bounded linear functionals f on E with = sup If (e)I < 00 • (A5) eeE IIell"'" .x. 1\ *1'\ Consider then the space E ®'Jr F Every element L E. E ®'/rF defines in a canonical way a bounded linear operator ~: E-7F. In fact, every L has a representation of the form L=LX with Lil/ < 00 (A6 ) {i} ~ {if ~ ., : where e* € E* , f. € F all have norm one. Therefore.;t can be defined ~ ~ as Le = LA.. e"*(el f. for e e. E. (A7) {i} ~ ~ ~ The correspondence L ~:t:- defines a mapping :f: E* ®'11'" F -7 B (E, F) where B(E, F} denotes the space of all linear bounded mappings of E into F. Unfortunately however it is not known if this mapping is one- to-one. Definition A.' Let E, F be Banach, spaces. Let L' (E, F) : = Y(E*®r,rFlC B(E,.F}. The elements of L (E, F) are called nuclear operators or some- I· ~ I times also Fredholm operators. The norm induced by in the space I 1 L' (E, F) is the trace norm or.the nuclear norm. i The space L' (E,F) is in general a quotient space of the space E*cib F. ~ A.4. The trace functional Let E be a Banach space and E* its dual space. Let LEE*-"® E have 'll: the representation e~® (A8) I' L = L 1. e l {i~ ~ ~ i e~ ~1. with {.:tJ E 1" E E*, e i E E, lIe711 L', "eill I: Then consider the expression \i ,1 II H tr.ace L. = LA. ei*.(e ) (A9) {is ~ i .,jl ,.:,.1 This is well defined and in fact a linear continuous functional on the space E* ®~E.A Since it is not knO\-In in general if the mapping ~ in (A7l is one-to­ II one it is not possible·· t!o;' say·· trhat· a'· nuclearoperatnr 'has' a ·tr=e. Re-' il1: • lB.. _ .--- -' .- , . -- .== = ~ --~i':yi¥)?-:- 129 ed member "that- this ',is' d±fferent in the case of a Hilbert space where a nuclear operator has always a trace. To get trace class operators in a general Banach space Grothendieck introduced another class of nuclear 7) operators which he called p-summable operators. Let 0 < p ~ 1 and let E, F be two Banach spaces. A Fredholm kernel L E E* ®-;r p' is called p-'summable if L has the follo>ling representation le- L =-L;t, e f (A10) *i ® i til J. >lith e~ E E* , f, E- F, Ile.... 1I "'" 1 , IIf ,II '"1 and rA, 'I ~'-l which means F)C J. J. J. ]. lJ.5 p P Lll./ < 00. :ome- {q 1: A linear nuclear operator:l: E~ F is called p-summable if there ex- ists a p-summable Fredkolm kernel LE. E*'"®'l(F such that ':fILl = "'"..p Denote the space of these p-summable operators by L {p} (E, F). Grothen­ F. dieck showed in 03~ that this space is a complete metrizable topolo­ gical space if one introduces on it the metric which is induced by the metric S originally defined on the space of p-summable Fredholm ker­ p ve nels: S (L) = inf L IIt .1 p (A11 ) P i . 1 where the infimum is taken over all representations of L in the form (A10) • A.S. The order of a nuclear operator and its Fredholm determinant A Let L E. E ~F be a Fredholm kernel. Consider the lO>ler bound q of all real numbers p, 0 < p ~ 1 , such that L is p-summable. The number q e is called the order of the Fredholm kernel. The set of all Fredholm [q] kernels of order ~ q is denoted by E ® F. 0- In analogy an operator;t: E ~ F is called nuclear of order q if there e- +< .- 130 ~ Ij exists a Fredholm kernel L of order q .,ith ~ = !f(L). The set of all :..., < nuclear operators of order ~ q ",ill be denoted by L [q1 (E, F). For nuc- I lear operators of order zero in particular one has the following cha- racterisation: r A nuclear operator.t: E~F is of order zero if, and only if it has a representation as given in (A10) where the sequence{~il belongs to the space lp for all p >o. I:., Other properties of nuclear operators are the following ones: :i Let t:: E -:> F be nuclear of order q and let T1: F ~ G respectively i· T : G~ E be linear bounded mappings. Then the mapping T Q,t.\) T : G~ G i· Z 1 Z , is also nuclear of order q. I Furthermore the tensor product of t.,o nuclear operators of order q in I. the projective topological tensor product of the corresponding spaces i i' is again nuclear of order q. Finally, Grothendieck.proved the following interesting Theorem~~: , ., j1 l. Theorem A. Z Let La L [I3 (E, E) with 0 ~ P ~ Z13. Then the Fredholm de- I' I! Ii terminant det(1 z ,t.). is an· ent-ire· function of -order ~ r, ",here 1/r = II!I 1/p - 1/Z, and of genus O. The operatorJe is of trace class and one has , I det(l - z~) = lr[(1 - z 1.) fi} ~ respectively i d trace ;f. = LX, I' h\ ~ . I' wherefAi~ are the non-vanishing eigenvalues of ~ counted according to ~ their algebraic multiplicities. For these eigenvalues one has further- more ~ ~ I' ~I:tl p < 0iC> • i {ij 1: ru M !~ F, S I-1 . 1. .-c'"-. ::;:;;;., APPEi,m IX B. lUC- COMPOSITION OPERATORS ON BANACH SPACES OF HOLOMORPHIC FUNCTIONS ~e are going to prove in this Appendix a version of a theorem about composition operators on the Banach space of holomorphic func- tions over open domains in the space 1, which is stronger than the one ~ ~. we gave in 1 By. restricting the discussion to the Banach space 1 1 we can weaken quite a lot the technical ass~ptionswe had to make there ly for a general Banach space. This is related to the existence of a mo­ ~G notonic Schauder basis i~ the space 1 [13~ • 1 :1 Theorem B.l Let D be an open bounded region in 1, and zo a point in '; D. Let'l'o: 1 1 --7> 1 1 be a nuclear mapping of order zero with II 'If'oll< 1 such that the mapping 'If: 1,--71, defined as ~(z) = zo +lfo(z) maps D strictly inside itself, that means1'(D) C D. LetV~AocJD) be a hoC. ~- lornorphic function on D which is continuous on D.