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,The Ruelle-Araki Transfer Operator in Classical Statistical Mechani(
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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. Define the composi-
= tion operator T Aoe(D) -l> A...J.D) by
(z) ~ T f = tt'(z) f 0'f{z) Then we have: 1) 1r has exactly one fixed point z* in D. 2) T is nuclear of order zero. -1 .... 3) trace T det{ , ) !{(z). = -'VIo
:0 inDl~ ~r- Proof: The proof of this theorem is similar to the one given
in the more general case. We will use here the fact that 1, has a mo
notone Schauder basis for which we take the following vectors e., i~ W: ~
(e ) i k = dik for all ke w • (B1)
Then any z e I, can be written as i-,,;
l;jL
Z = (B2 ) L: Zi e i i=l-
00 and Vie have IIzl1 = ~ Iz.\ i=l ~ Assertion 1) of the above theorem is a special case of the Earle-Hamil-
ton fixed point Theorem [133J for holomorphic mappings in an arbitrary
complex Banach space.
I We next prove assertion 2). For this Vie remark that there exists a I! 1 1 ball K:; (z*) of radius t around the fixed point z* such that 1p{K/,(Z"'»)
I'f, C Kcf(z*) • This follmls immediately from the assumption l'Y'o~ ~ 15J I I· !: 00 r g(z) = 11k! Dkg{Z") (z - z..)k. (B4 ) r = I· k=O i" I: Because y{z") = z* and )i'(D) C D one gets using relation (B3) 00 ~ k .. goy (z) = ~ 11k! D g(z ) (yo{z - z*»k • (B5) k=O Since is nuclear of order zero it has the representation 'ro ,1 Ii (B6) yo=LA . e* @e.1 i ~ i ~ /I~l, 'lith e: 17, lie: and {eik the Schauder basis of 11 as defined in (Bl). The sequence{:lJ belongs to the space lp for any p> 0, that i U means I ~ ,--li_. ~ ~. ' ".~_,","..;'••:.i- ...... , ,_. ~.~ • .... ".\.~:-~ 133 P £::"IA.I < 00. i=l ~ Inserting representation (B6) into relation (B5) and taking into account s~~etry .1- the linearity and properties of the operators Dkg(z*) we get .y 00 0( .... 0( k "'here we used the follo",ing notations: 0< is a rnultiindex <:(= (0<.). II.' with o(.€ fi-lU[O] for all i and = - 11.E'I 1. 10<1:- 00 L: 0( • <00. i=l ~ '"1 oii .. ,,0< A~: = /\ . r e (z-z ) = e* (z-z*) 0( i and (B8) n ~ n i iE IN ie. n-l 0< 0< . 0( .~~ 11~1 e- ) -e: = (e" ... e" .. ,ei,··,ei ,·· 1 ~ This allows us to write the operator T f (z) = f o¥'(z) as = ~ T '" (B9 ) = L > A "'*e k 0< ® e k,~ k=O O(,I~I =k k,~ '- with A.c(iC ;!k 0< : = n l '- i£. IN ~*. 0< ek 0( (g) : = 1/c = (z-z*)~ A~(l- '\ 0< (z) eX UfO Aoo(D). '- The number E is chosen in such a way that """" 1-E..... JI/ r II ~li ei(z-z) ei~ ~ < 6 for all z ~ D (B 11 ) ~ rr>h.; .; .:1.-.1,.. \-,""' ..... l • 1';;-, 1/ L".:l. e"'(z-z*) e." ~ ef' < J for all z € D {il ~ ~ ~ Relation (B11) is obviously equivalent to ~I~.P-£ le~(z-z*)1 "" t". (B12) ~ ~ ~ Next we prove a lemma. I !; r.- ~ o(~.' ~.s ! Lemma B.1 Let 0< be a multiindex with I I = k and let 1';:: i I I ~ i be the non-vanishing entries of ~. Then there exist numbers t.>O, ~ ! s b 1 ~ i ~ s, with ~ t. < J such that i=1 ~ r olD. I' I. ~ek,~II L s >l 1) II - {:;; t i I~ s -eX I 2) IjE~~,£! f n t. fi II i=1 ~ " Proof: Denote by K1qCRS the following compact set ,. ! s ~ lr X xi~'O i .and x. 1 Kc!'¥= { = (x1 , .• ,xs)C IRs V L ~ f i=1 J' 1 1 Let "l: K{II-7 IR be the continuous function I I, I s o(~i i(X) = IT xi (B13) II i=1 Ii j there exists ~ ~ Since ! is continuous a xo '" KJ''' with t (xo ) (x) for (t , ..,t )· Because I 1-£ Ie'*p. * If",,;,q all x EO K C'" Let x = 1 s f--+- A..I (z-z) " 0 . ~=1 f~ >~ j for all zE D we get 11 s s 0(" . But this proves just our assertion 1) of Lemma B.1. I -X< s k * f'I Let e (g) = 1/ oX" ! D g (z ) (e" ,. .,ec: , .. ,e , .. ,e p ). k ,ex n. 1 ) . p - J.= ~ \1 1 's >s The right hand side of this expression is nothing else but · ~ , ~ .. .,~'"" 135 d k *" s 1/(2"71'i)s g(z + L z. e ) f'\ oCr 1 () O(t; s i=l ~ \i Iz i =0 z C' l .... czs We can therefore apply Cauchy's inequalities to the function '"'g in the polycylinder p. = {Z€ its : jZi.l< t.i} Hhich because of the inequality 2) L:s t. ~ J " belongs completely to the domain of holomorphy of this' i=l ~ function. The function '"g is thereby defined as s IV : = g(z* + e ). g(zl""zs) zi s =i=l ~ i Doing this He get v(~ 'Jk "-- sup Ig(z) I /t l .. t>sat" 1/(21T'~.) s.""('1 .f"\zx\,s 9(i) / ... 1 I z=o z E: D s r;:l z l ... 0 s But this completes the proof of assertion 2) of Lemma B.l. Coming back to the proof of our Theor~n B.l we introduce the quan- tities s O(~. 1 (nt.~)- e 0< : = e Oi k ,- i=l ~ k '- (B15) s IXf. -* ~ ek,~ : = ( n t. ) e"* v( i=l ~ k '- ... In terms of them the operator T can then be written as 1" 00 T = L > A. k IX <01. ® ek,o< (B16) k=O c(--,liX/=k '- '- - Hith 00 > IAk Jq < 00 for all q> 0 , ==k=O -'-0( Jo But this shows that the operator T is a nuclear operator of order ze- roo Since the operator T on the other hand is the composition of this operator with a bounded linear multiplication operator also T is nuc- lear of order zero in the space A~(D) . Assertion 3) of Theorem B.1 finally is proved exactly in the same way as the analogous trace formula in reference01~ so that we can omit the details here. l; II I, j i i I H i I ~ ~ , ._-----..._------_.... ~ :~~. APPENDIX C. POSITIVE OPERATORS IN BANACH SPACES _s In this Appendix we collect some important results of the theory of positive operators in general Banach spaces relevant for our work here. A detailed discussion of this theory can be found in D10] and [134J . Let B be a real Banach space. A subset K C B is called a proper cone if (Kl) with xE:K also ~ x E K for all ~ ~ 0 I (C 1 ) (K2) if x E. K and -x e. K then x = O. Let K be a proper cone. K is called reproducing if every z€B can be written as z = x - Y I with x, Y E K I that means I if B = K - K 0 Every proper cone induces a partial order ~ in B: let x lYE B. '£hen we say x s y ¢=) y-xE-K. (C2) A linear operator T: B---7 B is called a positive operator if T leaves the cone K invariant: T Kc K• Let u E- K , u rJ.. 0 0 o 0 Definition Col A positive operator T is called u -positive if there o exist for" every x € K , X F 0 a number pE IN and positive real numbers o(,(b > 0 such that £, TPx s 0<. U f-> U o o This class of positive operators has been extensively studied by Kras noselskii and he shol-led Q09J I [i 10] I [134J that these operators al low for a generalization of the Theorems of perron-Frobeniusr6-77] respectively of Jentzsch [84J 0 , ,- I.>ti I! 1 Theorem C.) (Krasnoselskii) Let K be a reproducing cone in a real Ba- nach space B. Assume T : B ---7 B to be an uo-positive compact linear operator in B. Chose pE n~ and O<,~>o such that ~uo ~ TPu ~ o(.u 0 o' Then one has: 1) There exists an eigenvector xl unique up to scalar multiplication in the cone K with T x, ~1 xl' The eigenvalue is strictly po = A1 sitive and can be estimated by I: (31/p ~ ~ ~ ~1 /p , - ~ . 2) The eigenvalue~, is simple and all other eigenvalues of T (consi dered as a complex linear operator in the complexified Banach space B~ ) are in absolute value strictly smaller than A 1 I I It is obvious that this theorem reproduces for finite dimensional Ba- I; nach spaces just the results of Perron and Frobenius and for integral 1 operators on ('(M) where M is some compact manifold, the result of Jentzsch. ii How can one see if a positive operator is in fact u -positive? o An answer to this question i's given by [11 oJ Lemma C., Let T be a positive operator and let u E K, u J, O. If o 0 T , there exist natural numbers q and p and real numbers !X,~ >0 such that I' -I TPx ~ P.> u respectively Tqx ~ :;( u ' , o o I then T is already uo-positive. 1 A simple application of this theory for certain composition operators 1 in complex Banach spaces of holomorphic functions can be found in 035] Here we recall only the most, important. result of this·-work. r: To formulate it we need some definitions. Let DC~n be an open boun j ded domain in ~n. Let A~(D) be the Banach space of all holomorphic i1 functions on D with the sup-norm. \Ve denote by H. (D) the set of all ~n ~ holomorphic mappings ~: D,~ D where D, is some small open neighbour hood of D., I.t ,is then knm-/n, that 'If! has- exactly' one fixed' 'point ·z* 'in' ~ ,~a __Il~ " ~-:::5~ a- D [133J. Define a set Dm(z* ) as ar n D (z*) (C 3) IR D n {z* + IR 1 n Consider then mappings '1)JE H, (D) Clith I ~n po- ''f (DIR(Z*) C DIR(Z*) (C4 ) i- This is just a certain reality condition on the mapping 'If. The set lce of all mappings '\VIS H (D)' Hhich fulfil (C4) we denote by rf. (D)" I in In Let 1b k € rf. (D), , ~ k ,. m, such that there exists a k ,, ~ k ~ m, T In 0 0 with zk"* £ DIR(zk-3f- ) for a£l k. o ,- Let 0< = (e>:.). n.' be a multiindex withol € wu[o} andlo ~1"" ~m C!11 :Xml ;;(12 ,xm2 .xl i C . 0(1'··'0{. __ IR Obviously the mapping,¥,- -m is again in 11. (D). ~n Let f€A",,(D) and let be a domain with6cD. Let G , .. ,G be 6, 1 r f the (2n-2) -dimensional analytic null sets of f in 6. [137]. at Definition C. 2 The mappings .y" .. , Y'm E J;.ri (D) are called separating, if 1) there exists a k , 1 6 k 6 m Hith Z~ C DIR(Z~ ) for all 1 6 k ~ m, o 0 o 2) for all regions 6. with~CD there exists a number No<~such that for 5 all N'7 No and all z"tln D for Hhich there exists a multiindex Condition 2) in the above definition just says that the set of points lur- 0(, ,.. ,oil f - m (z) ~ is so dense in D for every Z that there does not exist f IR -/ -. 140 " any function fGnoo(D) whose null sets contain all these points without being identically zero. After this bulk of definitions we can finally formulate the main result of our investigations in ['35J : Theorem C.2 Let DC ~n be an open bounded simply connected domain. Assume the mappings 'If" . ., tm E ~~n (D) separating. Let 'fke A.,.,(D,) for , ;;; k ,; m, where D, is some open neighbourhood of D, with '(kiD> o. 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