1-.. Tromba, I'Degree theory on oriented infinite dimensional varieties and the Morsenumber of minimal surfaces spanning a curve in Rn,'1 Trans. Am. Math. Soc., 290, No. 1.385-413 (1985). -

J. Wolf. "The geometry and structure of isotropy irreducible homogeneous spaces.11 Acta

Ma"th.. 120. No. 1. 59-148 (1968).M. Wang-;-;;:d W. Ziller. 'IOn normal homogeneous Einstein manifolds." Ann. Sci. Ecole

Norm. Sup. ger. 4. ~. No. 4, 563-633 (1985).

OF OPERATORS IN THE BOSON FOCK SPACE

UDC 519.46Yu. A. Neretin

135@ 1990 Plenum Publishing Corporation0016-2663/90/2402-0135$12.50

Hilberts space with the scaLar proQuc~:

(/, g) = ~~ f (z) g (z) exp (-(z, z») dzdz.

are' interested in operators of the form

The main problem considered in this article is the problem of the boundedness or

operators.Unitary operators of the form (0.1) appeared in [1], in such a form Berezin hag writtenthe automorphisms of the ~anonical commutation relations. I~ numerous papers of the

70-80 (we mention only [2-5]) the fundamental role of the a~tomorphisms of canonicaland anticommutation relations in the representation theory for infinite dimen-

groups hag been clarified (this role is the same as for the operators of variablesand multiplication by a function in the representation theory of Lie groups). After

had been discovered that a representationof an infinite dimensional group is, in fact,visible part of a representation of an essentially bigger'and invisible with the un-

eye semigroup (see [6]), and, actually, even not a semigroup, hut a category, at firstproblemof semigroupwith the Weil representation hag arisen. Ol'shanskii indicated that

semigroup is semigroup BO of all operators of the form (0.1), and then a problem hagconcerning the algebraic nature of this semigroup, as well as the problem of the

of the operators. It turns out (Ol'shanskii), that for n < ~ the boundedness of

the ope1'ators (0.1) is equivalent to the pair of conditions: 1)~(~ ~)11<1; 2) IIKII<1, IINII<1

(here, as everywhere in this paper, under the norm of a matrix we understand the Euclideannorm). In the joint paper by Ol'shanskii, Nazarov, and the author [9] it hag been clarifiedthat the considered semigroup is isomorphic to same semigroup of linear relations.

In Sec. 1 of this paper we introduce an accurate definition of operators of the form(0.1), in Sec. 2 we discuss a realization of the semigroup BO as a semigroup of linear rela-tions, and a semigroup of generalized fraction-linear Krein transformations of an infinite

matrix ball. In Secs. 3 and 4 we formulate and prove theorems on the boundednessthe operators. In Sec. 5 we consider a somewhat more general c1ass of operators.

For applications of the semigroup BQ to the representation theory of the Virasoro alge-bra, and to the conformal quantum field theory (cf. [7; 10]), see the Fermion analog of this

paper (cf., [8]).The author is grateful to G. I. Ol'shanskii and M. L. Nazarov for cooperation, and also

to E. B. Tsekanovskii for useful remarks.

cember 15, 1989.

IIXX21

1. 0

!fJ .B~son Fock Spa~ Let V be a complex Hilbert space of dimension n = 0, 1, 2, ..., ~ith the scalar product (', .). We are interested mainly in the case n =~. We shall assurne

implya noninvariant point of view and suppose that V is the coordinate space Cn or ~2. Wehall consider the space F(V) of holomorphic functions on V with the scalar product

({, g) = 55f (z)~) exp (-{z, z» dzdz.

f n < ~ then this formula should be understood literally [1], tor n = ~ the simplest is to

onsider the inductive limit limF{Ch) tor n ~ ~ take its completion and obtain F(~2)'--It is known [1] that any bounded operator in F(V) is an integral operator in the direct

eaning of this term. Indeed, let A be a bounded operator. Let us consider the function

K (u, v) = <exp (z, v); A exp (z, u», (1.1)

ere e(z,v) and e(z,u) are considered as functions of z. Then

Af (u) = SS K (u, ii) f (v) exp (-{v, v» dvdv. (1.2)

The kernel K(u, v) is a function which is holomorphic in the variable u and antiholo-

rphic in the variable v. Taking into account that the norm of the function exp(z, v)

luals exp(-}(v, v»), and also (1.1), we obtain that the kerne 1 K(u, v) satisfies the estirna-

"i

tha~

:ton thetin

(1.3)

I

~. O~er!tors B[§!..:.th a kerne! of the formw~Operator B[S] in the boson Fock space F(V) is called an operator

K{z, ü) = expwhe1lie~

ing

bol

:J~

onlycond(b),

! T=Tt,IITII<1. (1.We hal:l define vector~ b [T] = bz [T] = exIl (1/2 (zTzt)) in the space F(V). the condition IITII < f

gu antees that b [T] E F (V).

1361

0°. s = St.;

1".11811,1; \2". IIKII < 1, 11M 11 < 1;

3". K and Mare Hilbert-Schmidt operators.

!~OP~~!~i~g_I.1 ~~._I. O!'shansk~ If operator A with a kernel of the form (1.4) is

ded, then the conditions 0°-3° are satisfied.

Proof~ 1° follows tram (1.3).

2°. A.1 = exp{-}zKzl}. This function lies in F(V) if and only if "KII < 1, where K is

ilbert-Schmidt operator. To obtain an analogical statement tor M A*.1 should be con-

ered.

Counterexample. Let A be a diagonal matrix with eigenvalues Al' A2'...' where rlAil2 <

d rlAil2 = 00. Let S=(1~A 1~A). Then the operator B[S] is unbounded.

It follows

be a block 2n x 2n-matrix, with

Let v = 11 XII + X]2T (1 -X22T)-1

~~I

+uTu' + au'}eXP(-(uJU»~UdÜ ==

= cb[K + LT(1-MT)-lLIIL~1- T .M)-la']swhere c is same constant. The integral is absolutely convergent anh the RHS of this relationlies in F(V).

The proposition can be proved by a direct computation.

The explicit form of the RHS is not needed further, we are interested only in the follow-ing.

COROLLARY: The set Fo(V) is invariant with respect ot B[S].

Hence all the operators B(S) have a joint dense invariant domain of definition.

~. Product of the Operators B(S]. THEOREM!.:1..:.. Let

SI = (~t ~), S2 = (~r ~).

I

Then

B [Si] B [SI] = dei «1 -MP)-l!l) B [Si. S2];

Iwhere

8 *8 = (K+LP(1-MP)-lL' L(t-PM)Q )1 2 Q'(1-MP}-lL' R+Q'(1-MP)-lMQ.

The formula can be verified by a direct computation. For the proof we needonly the fact that the matrix SI *S2 satisfies the conditions 0°-3°. The v~rification ofcondition 3° is obvious, condition 2° follows immediately from the Krein-Shmul'yan lemma(b), and to obtain 1° we have to apply the Krein-Shmul'yan lemma to the matrices

,'K IL I 0 1

LI I J'J t 0

(1.6)Proof.

T =(~t ~).(XII Xli ) = IX21 Xu

137

wnere ~ne integral converges and the RHS of the equality is contained in F(V).

Proof.. For the proof we need only the condition I1 K + LT (1 -MT)-l Lt 11 < 1

tram the next lenuna.

~ 1.1 (Krein and Shmul 'Ya~ Let X = (~:~ ~~). ~"'

11 XII< 1, 11 X2211 < 1. Let T be a matrix of dimension n x n. L,

X21U.

a) If IITil<1,thenv<1.

b) If ,lI T 11< 1, then v < 1.

c) IfIISII<1, IITII<1,then v < 1.

This is a well-known result (see (11-13]). and a little bit later we shall understandthat it hag a clear geometrical sense.

Further. we shall introduce the vectors b[Tla] = exP{+zTz'+ azt}, where T satisfies all

these conditions and a E: V. It is easy to see that b [T I a] E F (V). The set F 0 (V) of allfinite linear combinations of the vector b(Tla] is dense in F(V).

~roposition 1.3:- Let (:t;), T, a be the same as before. Then

~ Weil Representation2.1. Shale-Berezin Symplectic Group. By SPn we shall denote the group of all block

Zn x 2n -matrices (n = 0, 1,..., ~) of the form Y = (; ~) ,. satisfying the conditions:

01)-10 ;1) Y preserves the symplectic form

2) 'I' is a Hilbert-Schmidt operator.

ß~mark.o If n < 00 then SPn is isomorphie to the group Sp (2n, R). To see this, we shallnotice that the group SPn preserves the n-dimensional subspace consisting of vectors of theform (a ä).

THEOREM 2.1 (Berezin [1]~" see also [14]). The formula

p (Y) = det «1 -'1"1'*)1/2) B [W:-~l <1It-l-(1)-1 'l'

gj

thIt

(tSe

veca

def:ines a projective unitary representation of tbe group SPn in F(V) (tbe Weil representa-

tion).( ;;<1>-1 <1>'-1 )Remark. Matrix <1>-1 -<I>-lV satisfies tbe conditions 0°-3°. Moreover. as it bas been

noticed by G. I. Ol'sbanskii. tbis matrix is unitary.

2.2. Symplectic Subgroup. Let us consider two copies V+ and V- of tbe space V. wbere

Olle of tbem consists of sequences of tbe form (vt. v; ). and tbe otber of sequences of

tbe form (vi. vi ). We sball introduce in V+ .V- a skew-symmetric form {'. .} witb tbe

mat1:-ix -~~) and a Hermitian form 1\(.. .) witb tbe matrix (~-~). Let W+ .W- be a second

cop;, of tbe space V+ .V_.

Let matrix S =(:t;) satisfy tbe conditions 0°-2°. We sball construct for it a ~

l(S:~ in (V+ E:f) V_) E:f) (W+ ffi W_)~ Tbis subspace consists of all vectors of tbe form (v+. v-.

w+. w_) satisfying tbe condition

+

Eh.f<)'

(:~) = (~t ~) (:~).

The resulting subspace, generally speaking, is not a graph of an operator V+ ffi V- ~~+ EBWe shall consider t(S) as a linear relation. We recall that linear relation can be multi-,pliE!d, namely, if R1, R2, R3 are linear spaces, and L1C Rl ffi Rz, LzC Rs EB Ra are linear

1138

Proposition 1.4. Let 11, 11 E F 0 (V). Then <B [8] 11, I:a> = <11' B [8*112).

The proof needs only a direct verification. (Although it would not be cautiously enoughto claim that B[S*] = B[S*].)

1.6~ Co~verg~nce.- P~oposition 1.5. Let us consider a sequence of bounded operatorsB[Si]' In order that B[Si] converge weakly to B[S] it necessary and sufficient that the twoconditions hold

1) B[Si] is uniformly bounded.

2) Si ~ S weakly.

Moreover, if 1) and 2) are satisfied then operator B[S] is bounded.

Proof. We shall show, tor instance, the necessity. We shall consider the set R of allvectors tram F(V) of the form la (z) = exp (azl), a E V. If B[Si] weakly converges in B[S] then<tu, B [Si] Iv> = Ki (u, iJ) converges tor arbitrary u and v to <lu, B [Si] Iv> = K (u, iJ). Hence, the

sequence of kerneis converges pointwise, that means, also the sequence (uV) Si (;t) , should con-

verge pointwise, and this means the weak convergence of the operators Si'

We notice that usually condition 2) can be easily verified, and, at the contrary, 1) isvery difficult to verify.

:, then, by definition, their product LILl = L3 CR1 ffi R3' consists of all pairs"3) E Rl ffi R3' tor which there exists an rl E RI such that (rl' r2) E LI, (r2' rs) E L2" Linear

can be naturally considered as graphs of not everywhere defined and multivalued

operators.

Linear relation ~ = ~(s) satisfies the conditions: 0+. ~ preserves the form {', .},.e., if and, moreover, ~ is maximal among spaces satisfying this condition,

{PI' qI} = {P2' q2} (this is a consequence of condition 0°).

1+. ~ "stretches" the form A(', '), i.e., if (p, q)El,then A(p, p)<A(q, q) (a conse-quence of condition 2°).

2+ .If (p, 0) E l, then A (p, p) < Eil P 111, if (0, q) E l, then A (q, q) > eil q Ir, where E doesnot depend on p, q (thisstrengthening of condition 1+ follows tram 2°).

Remark. We shall introduce in V + Ef) V -EB W + ffi W -the forms

{(VI' wJ, (V2' W2)}' = {VI' V2} -{Wl' W2},

A' ((VI' wJ, (V2' W2» = -A (VI' v2) + A (W11 W2),

'ihere Vi E V + Ef) V_, Wi E W + ffi W -" Then condition 0+ means that ~ is a Lagrangian subspace'iith respect to the form {', .}I, and condition 1+ means that the form AI is positive defi-

nite on ~.

Conversely, one can show that an arbitrary linear relation ~ satisfying the conditions0+-2+ hag the form ~ = ~(s), where S satisfies the conditions 0°-2°.

Proposition 2.1. 1 (8J 1 (82) = 1 (81 * 82), where the multiplication * is defined by formula(1.6).

The proposition can be verified by a computation.

We have not used Jet condition 3°. It determines in the sem~group of all linear rela-tions satisfying conditions 0°-2° a subsemigroup which we shall ckll a symplectic semigrouprsp.

Then Theorem 1.1 can be reformulated in the following way.

THEOREM 2.2. Let l~ fsp. Let S = S(t) be the corresponding matrix. Then t ~ B[S(t)]is a projective representation of semigroup rsp.

This representation generates the Weil representation of the Shale-Berezin group,and thus it can be also called the Weil representation.

2.2. Infinite Dimensional Cartan Domain~. By ~ =~n we shall denote the set of com-0 ~.. --plete symmetric Hilbert -Schmidt matrices of dimension n X n (n< (X)) with the norm <1. Ifn < 00 then ~n is one of realizations ~n of Hermitian symmetric space Sp (2n, R) / U (n).

Let PF (V) be a projective Fock space. We shall consider the mapping x: ~n -'toPF (V),given by the formula x (T) = b [TI.

Semigroup rsp, acting in the Fock space, maps vectors of the form b[T] into vectors ofthe same form (Proposition 1.1). Thus, we obtain the action of the semigroup rsp on ~.It is given by the formula

(K L)J1 L' M T=K+LT(1-MT)-lLt.

(the Krein-Shmul'yan lemma tram 1.3 asserts that the mappings I1(S) map domain ~ into it-gelt).

Mapping I1(S) is invertible if and only if the matrix S is unitary. The group of all in-vertible mappings of the form I1(S) is the Shale -Berezin group, and in the finite dimensionalcase it is simply Sp (2n, R).

Further we shall mention that with each T~~ there is connected a subspace A(T) inv+ ~ V_. It consists of all vectors of the form (v+, Tv+). By the symmetricity of matrix Tthe subspace A(T) is Lagrangian with respect to the form {', .}. By the condition IITII < 1form A(., .) is positive definite on A(T).

Proposition 2.2. A (11 (S) T) ~'- l (S)]" (T).

Proof. This is based on a simple verification.

139

0+J1

°S'1 UO1~1sodo~d Aldde pue S(u/l-l)= Us a~uanbas aq~ ~ap1SUO~ o~ q~noua s1 ~1 ase~ s1q~ado~ °l °1 > OS. uaq~ ase~ aq~ o~ pa~npa~ aq ue~ ~U1q~A~aAa lle 10 ~s~14 °1oo~d

"lfill./l = I J1/ I a~aq~ ',/,-(1 If 1- 1) lap ~ II[S] H 11 Uaql ),'S = S ~a1 O(O( UO1~1sodO:Id

.(;f ,;) = so ~a'I.00 ) Aw1P = u uaq~ ase~ aq~ ~ap1suo~ 11eqs a~ a~aH 'ase~ 1~uo1suaw1a-a~1u14 "S"[

.(~ad~d s1q~ Jo Z"vpu~ I.Y .s~as o~ as01~ '[81] os1e aas) [91] U1 pa~1~ uaaq s~q ~~arqns s1q~ uo Aqd~~80J1q1q

aAJsua~xa uV .pasn ua1Jo uaaq seq 11 A11uanbasuo~ pue (a~edsqns 1U~1~~AU1 ue uo wa~oaq1Ju1a~]) [SI] u1a~] Aq peu011uaw uaaq s~q (s~no 01 ~e11w1s u011n~11s ~ U1) [ ~ Z u011~~11dw1aql "I ~ Z u011e~11dw1 aq1 asn 11eqs a~ 'as~no~ JO .A8o1o1n~~ e s1 UO1~1sodo~d s1~

.(l)n uo11ela~ ~eau11 aq1 O~ 1~adsa~ q11~ 1U~1~~AU1 s1 1 a~~dsqns aq~ ([

!(s)n 8u1ddew aq~ Jo 1U10d pax1J ~ S1 I (Z

![S]H ~o1~~ado aq1 Jo ~0~~aAuaS1a u~ S1 [l]q (1

:~ual~A1nba a~~ suO111puo~ 8U1~O11oJ aqI. .Z.( uo111S0äO~d .S1U1Od pax14 "y'[

.('f,-(.zW -J)) 'lap = IIISI EI 11I;;, uaqJ, .(: I;) = oS' pue

pa1J51~e5 aq wa~oaq~ aq~ JO 5U01~dwn55e aq~ ~a1 '(Z'1 pu~ 1'[ U01~150QO~d O~) XHY110HO~

'[O]q ~o~~aA wnn~~A aq~ uo pawn55~ 51[S]8 Jo ~ou aq~ 5nq~ pu~ 'I = II(S)811 a.I°Ja~aq~ .~ >11711 ~n8\\ "1 Jo .Ia50d :J1.I~aUDUA5-'){ aq~ s~

A~ß A.IaAa u1 8U1~:J~ 51 [S]8 .Io~~.Iado "'){ aa.I8ap JO 5U01~:JUnJ snoaua80moq 11~ Jo ~a5 aq~ 51A~ß a~aq~ 'A~ß 5a~~d5 ~~aq11H Jo wn5 ~~a~1P ~ 0~u1 5a50dwo:Jap (A)~ ~~q~ a:J1~ou a~ ~oN

'(Z1)8 = (Z)J[S]8 a8u~q~xa a1q~1.IeA JO

~o~~.Iado aq~ A1dm15 51 [S]9 .Io~~.Iado 'A1~Uanba5UO~ pU~ ,(~ I~) ~oJ aq~ 5~q S X1.I~em uaq~ ~n8

"I = [O]q = [ili]q 5nqili "0 = ili ~eq~ awn55~ u~~ a~ A~11~.Iaua8 JO 5501 ~noq~1~ a~oJe~aq~ pU~.,.~ U1~WOP aq~ uo A1aA1~15U~.I~ 8un:J~ 51 dS dno.I8 u1Za.Iag-a1~qs aq~ ~~q~ AJ1~aA o~ A5ea 51

~1 " [ili]q .IO~~aA Uo pawn55~ 51 ~o~~~ado aq~ Jo ~ou aq~ ~eq~ aAo.Id o~ a~~ aM -"JoO.Id

"Y = 11 [ili]911 uaqJ. "[J.]qy= [ili]q[S]8 P10q "%31 aW05 ~oJ pu~ ),{S = S ~a1 "1"[ U01~150QO~d "5~0~:JaAua(f1~ "["[

"~S = S a5~~ aq~ o~ 5aA1a5~no au1Juo~ ue~ a~

Z"[ pue 1"[ 5~wwa1 aq~ Aq 'a~uaH 'V),{V ~o~~.Iado ~U10rp~-J1a5 aq~ Jo 55aupapunoq aq~ AJ1~aAo~ ~ua1:J1JJn5 51 ~1 V .Io~~.Iado ue Jo 55aupapunoq aq~ aAo.Id o~ ~~q~ 11~~a.I 11~q5 a~ ~ON

"1'1 wa.IoaqJ, Jo Joo.Id aq~ Jo ~1.I1d5 aq~ U1 ewwa1 u~AI1nwqs-u1a~~ aq~ Jo d1aqaq~ q~1~ pa1J1.IaA aq u~~ q~1q~ '(ql"[ ~da:Jxa '5n01Aqo a.Ie 5~uawa~e~5 aq~ 11V -"JOO~d

"Z"[ ma~oaqJ. JO U01~-dwn55~ aq~ 5a1J51~~5 lS~IS uaq~ 'Z'( ma.Ioaqili Jo u01~dwn55e aq~ AJ51~~5 lS 'IS J1 (q

"Z" ( wa~aqJ, Jo u01~dwn55~aq~ 5a1J51~e5 ~S uaq~ 'Z"( wa.IoaqJ. Jo u01~dwn55e aq~ 5a1J51~e5 S J1 (~ "Z"( VWWR1

"1"[ lUe.IoaqJ. Jou01~dwn55~ aq~ 5a1J51~e5 lS~IS uaq~ '1"( wa.Ioaqili Jo u01~dwn55~ aq~ AJ51~~5 lS CIS J1 (q

'1"( ma.Ioaqili Jo u01~dwn55e aq~ 5a1J51~e5 ~S uaq~ '1"[ lUe.Ioaqili Jo5u01~dwn55~ aq~ 5arJ51~~5 S JI (e "1"[ VWWH1 'a5e~ ~uro~PV-J1as aq~ o~ u01~:JnpaH "Z"(

"papunoq 51 :,'S uaq~ 5.I0~e~ado ~ea1~nu a~~ W pue ~ J1

"papunoq 51 [S]g uaq~ 1 > IISII J1

"Z'1 WO~J 0(-00 5U01~1pUO:J aq~ AJ51~~5 S ~a1

°ZO[ WlIHO3HJ.

°1°[ WlIHO:!lHJ.

°1°[

s1Ue~oeqJ. sseupepunog

'sma~oaq~

aq~Jo uoT~eTnw~o~

"[

'sno1Aqo sawo~aq ewwoT ueAITnwqs-u1a~~ aql Jo (e luawale~s 's~al q~ns u1 'a~uaH

HPII< 1 Then

~oof~I det (1 -PX)I > det (1 -X).

The lemma can easily be derived from the von Neumann-Horn inequality and theWeyl inequality for singular and eigenvalues of a linear operator (see [17J).

In such a way we have obtained a new proof of the theorem of G. I. Ol'shanskii on theboundedness of the operators B[SJ in the finite-dimensional case (the original proof of

shanskii gives an exact formula tor the norm of the operator B[SJ, see [9J).

~:~~~~~ We shall denote by Pn the projection from F(V) onto theset of functions depending only on the variables Zl,.",Zn' It is easy to see that

0

0-1..

0P..=B der= B[nn].,

I~'0 0

.,.LJt1

MJJ

!MzJ

(K1J 00 0s

Lft L:lL' L'

12 22

Sn=then

~:)~ ( it 0 i110 0

A -1k

A;-

(1 -Z:ZJ-1(1 -Z:Z2)(1 -Z:Za)-l (1 -Z:ZJ.

The family of numbers Al' A2'... is also invariant with respect to Sp (2n,R),

We shall introduce also an auxiliary Riemann metric

(ds2)E = tr dZ*dZand the corresponding distance

141

~

2) 11 B [Snlll < det (1n- I Ku !)-1/2< det (1 -I K 1)-1/2.

i~~trY of Domain ~C». and Proof of ~~4.1. The Geometr of Finite Dimensional Domains ~n' Let n <~. We shall introduce in

the domain ~.. aRiemann metric invariant with respect to the group Sp (2n, R):

d$2 = tr (1 -Z*Z)-ldZ* (1 -ZZ*)-ldZ.

Then the distance between points Zl, ZI~~ can be calculated according to the formula

1 L 1 + rl/2p2(Zl,Z2) =T In2~1; rk =

..1-~

Proposition 4.!. P (fJ. (S)Zl' fJ. (8)Zz) -< P (Zl' Zz).

Proof. Thanks to the continuity, without lass of generality we can assume that IISII < 1and that ~(S) is injective. Any such S (see [20]) can be represented in the form S = UL,where U E Sp (2n, R), and 1 = 1*. Since U preserves the metric, without lass of generalitywe can assume that the fixed point of mapping ~(S) is the point 0, and thus, ~(S)Z = LZLt.Now it remains to verify that ~(S) does not increase the Riemann metric ds2.

~;~. Geo~etrv of- the Domain ~~~ We shall introduce on ~~ a compound distance, metricP [it is easy to see that series (4.1) is convergent if Zl,Z2E;?t~],and also a metric PE'Metric PE determines on ~~ the usual topology in the space of Hilbert-Schmidt operators.

Proposition 4.2. Domain ~~ is complete with respect to the metric p.

Proof. We shall consider a ball Bc consisting of points ZE~~, satisfying the condi-tion p (O,Z)-< C. ODe can show that on this ball the metrics P and PE are equivalent, i.e.,PE (Zl, Zz) -< P (Zl' Z2) -< kPE (ZI, Zz) tor same k depending only on C. The simplest thing to dois to verify the case of domains ~n tor Riemann metrics ds2 and (ds2)E, and then to take thelimit tor n ~~. But the space of all Hilbert-Schmidt operators is complete with respectto the metric PE, and the ball Bc is closed in the topology given by the metric PE [sincethe function P(Zl' Z2) is continuous in the metric PE]'

Proposition 4.3. P (fJ. (S)Zv fJ. (8)Z2) -< P (ZI' Z2) tor any Zl' Z. E ~~ .

Proof. By Proposition 4.1 holds

P (fJ. (nn * 8 * nn) Zl' fJ. (nn * 8 * 1tn)Zz) -< P (ZI, Zz)

[element 1Tn is introduced by equality (3.1)]. Further we take the limit tor n ~~.

4.3. Proof of Theorem 3.2. By Sec. 3.4 it is sufficient to show that the mapping ~(S)tor IISII < 1 has afixed p~i-nt-i~. ~~. For this end it is enough to verify that the mapping~(S) is contractive in metric p.

( 0 1-1: )We shall represent S in the form S = 81 * Ts, where 11 SI" -< f, and Ta = 1-E 0..'

LEMMA. P (fJ. (T s) ZI' fJ. (T s)Zz) -< (1 -e)p (Zl' Zz).

;:~~ This inequal~ty can be directly checked in the finite-dimensional case on thelevel of the Riemann metric ds2. Next we have to take the limit tor n ~~.

Now ~(S) = ~(Sl)~(Te:)' The mapping ~(Sl) does not increase distances and the mapping

~(Te:) is contractive. Theor~ 3.1 has been proved.

where 1.., JA E V.

To define precisely operators B[Slvt] we have to repeat all words said in Sec. 1. Weomit them and present only the formula tor the multiplications of operators

K L I ÄI ] B [ P Q l nl ] =cB [( K L ).(P Q)\ Ä+L(I-pM)-l(n+PJI)

L' M 111 Qt R Xl LI M QI R x+QI(I-MP)-l(Mn+/4)-

THEOREM 5.1.. If IISII < 1 then the operator B[Slvt] is bounded.

To prove this theorem we notice at first that in the case IISII < 1 the operator B[Slvcan be represented in the form

[ K L IÄ'] [ ° l-e IÄI ] [ 1 (K L )I O ] [ ° 1-8 1 0B L' M 11' =B 1-6 0 o .B (1-6r LI J.f 0 .B 1-6 0 111

By Theorem 3.1 the middle factor in the RHS is bounded tor a sufficiently small E >In such a way. the problem reduces to the boundedness of operators of the form

B

142:

~ Generalizations and Remarks

5.1. Nonhomogeneous Operators. The Shale-Berezin groups have an important and fre-quently used extension. It consists of all affine transformations of the Hilbert space,that their linear part is contained in the Shale-Berezin group. We shall consider ancal extension for semigroup rsp.

For this end we shall introduce the operators

B

Af(z) = B [

0 1-e/A:J-e 0 /J.

As before, without loss of generality, these operators can be considered self-adjoint,

i.e., we can assume that A =~.

Given by them mappings in F(V) can be rewritten in the form

0 1-e/ 0

11-e 0 bJ!(Z)=!((1-e)z+b)e(Z,b>.Now we notice that the space F(V) can be decomposed into a tensor product F(Cl)@n (where

n = dimV) with distinguished vectors fi(Z) = 1 (cf., [21]), and operator (5.1) decomposes

into a tensor product of the operators

AJ (zJ = f ((1 -e)zi + bJezibi.To verify the boundedness of the operator A it is enough to show that

n

IIAII=n"Aill<cx>, n=O,1,...,cx>.;=1

(5.1)

(5.2)

gn. (z;) = (- ez. + bi)m exp( + b".Zi) ,

The corresponding eigenvalues are

Gm = (1- e)mexp(+b"izJ.

where m = 0, 1, Z~

\ v IHence, IIAi" = (Jo, that is, the product (5.2) converges, and thus, the boundedness of the oper-

ator A hag been shown. In this way also Theorem 5.1 hag been proved.

~~~~~~~!!~~!&~ We have already seen that the operators B[S] corre-spond to linear symplectic relations. Exactly the same, also operators B[Slvt] form a repre-

sentation of affine symplectic relations (for details see [7]).

~_~~~.~egtic Categ~ Operators of the type B[S] acting from Olle Fock space F(V1)into another F(V2) are not less important than operators acting from space F(V) into itself.

They determine a representation of certain category of linear relations (see [7, 8]).

5.4. For the connection of operators B[S] with the canonical commutation relations see[7] .

5.5.

Paper [9J intersects with [22J

1.2.

3.

4.

5.

6.

7.

8.

9.

143

LITERATURE CITEDF. A. Berezin. The Second Quantization Method [in Russian]. Nauka. Moscow (1965).A. M. Vershik and S. V. Kerov. I'Characters and factor-representations of an infinite

dimensional unitary group." Dokl. Akad. Nauk SSSR. 267. No. 2. 272-276 (1982).Gr. Segal. "Unitary representations of some infinitedimensional groups.'1 Commum. Math.Phys., 80. No. 3. 301-342 (1981).V. I. Ol'shanskii, "Unitary representations of infinite dimensional pairs (G. K) andthe How formalism." Dokl. Akad. Nauk SSSR. 269. No. 1. 33-36 (1983).Yu. A. Neretin. I'Representations of the Vir~ro algebra and affine algebras," in: Con-

temporary Problems in Mathematics. Fundamental Directions [in Russian], 22, 163-224(1988). -

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"~ ",',-"

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21.22.

Yu. A. Neretin, "A complex semigroup containing the group of diffeomorphism of a circle,"Funkts. Anal. Prilozhen., 21, No. 2,82-83 (1987).M. G. Krein and Yu. L. Shm~'yan, "Fraction-linear transformations with operator coef-ficients," in: Mathematical Studies [in Russian], Val. 2, No. 3, Kishinev (1967), pp.64-96.Yu. L. Shmul'yan, "Theory of linear relations and spaces with an indefinite metric,"Funkts. Anal. Prilozhen., 10, No. I, 67-72 (1976).Yu. L. Shmul'yan, "Generalfraction-linear transformations of operator balls," gib.Mat. Zh., 19, No. 2, 419-425 (1978).D. Shale, "Linear symmetries of free boson fields," Trans. Am. Math. Soc., 103, 149-167(1962). -

M. G. Krein, "An application of the fixed point principle to the theory of linear mapswith indefiite metric," Usp. Mat. Nauk, 5, No. 2, 180-190 (1950).T. Ya. Azizov and I. S. Iokhvidov, Foundations of the Theory of Linear Operators inSpaces with Indefinite Metric [in Russian] , Nauka, Moscow (1986).V. B. Lidskii, "Inequalities for eigen and singular values," in: F. R. Gantmakher,Theory of Matrices [in Russian], 4th edn., Nauka, Moscow (1988), pp. 502-525.V. A. Khatskevich, "Generalized Poincare metric on an operator ball," Funkts. Anal.Prilozhen., ll, No. 4, 92-93 (1983).G. I. Pyatetskii-Shapiro, Geometry of Classical Domains and Theory of Automorphic Func-tions [in Russian] , Fizmatgiz, Moscow (1961).G. I. Ol'shanskii, "Invariant cones in Lie algebras, Lie semigroups and holomorphicdiscrete series," Funkts. Anal. Prilozhen., 15, No. 4, 58-66 (1984).

A. Guichardet, Symmetric Hilbert Spaces and R;lated Topics, Lecture Notes Math., Val.

261, Springer-Verlag (1972).R. How, "Oscillator semigroup," Proc. Symp. Pure Math., ~, 61-132 (1988).

COMPUTATIONAL COMPLEXITY OF IMMANENTS AND REPRESENTATIONS OF

THE FULL LINEAR GROUP

A.

I. Barvinok UDC 512.5+519.6

1. Statement of the Pr'ob1em. Let A = 11 aij ~ E Mat (C, 11) be a matrix of dimension n x n overthe field of comp1ex numbers Cn and X be a character of the complex irreducib1eof the symmetric group Sn' The expression

ndX(A) = ~ x(a) II 4i.o(i)

oeB ;-1n -

is called the immanent of the matrix A corresponding to X. The quest ion of the -complexity of immanents was put by Strassen (see [1]). In this paper, an upper bound on thecomplexity is given of computing an immanent, which in the case if X is an a1ternating char-acter, x(a) = sgn 0, or a unit character, Va X «1) = 1. coincides according to logarithmic orderwith the known bounds, respectively, for the determinant and permanent and is better in com-parison with the Olles known for same representations corresponding to intermediate Young dia-grams. The author doesnot know any such nontrivial upper bounds on complexity for Younggrams of general form. In the paper, use of the theory of representations of the fullgroup GL (see [2, pp. 319-347]) is essential.

2. Notation. Let V be an n-dimensional vector space over C with scalar product <, >and an orthonormal basis e1,...,en' In the space y@n let us, in the standard manner,duce the scalar product (denote also<, », the actions of Sn' permuting factors, and offull linear group GL(V) [the n-th tensor power n@n of the natural representation ~ ofin V]. Let us denote by ~X the symmetrization of ~ by a representation of Sn with a char-acter X. For the dimension dim~X' an explicit effective formula [2, p. 326] is known.

I. M. Sechenov Leningrad Institute of Evolutionary.Physiology and Biochemistry.lated tram Funktsional'nyi Analiz i Ego Prilozheniya. Val. 24. No. 2. pp. 74-75. April-1990. Original article submitted November 2, 1989.

144 0016-2663/90/2402-0144$12.50 e 1990 Plenum Publishing Corporation