Linear Und Complex Analysis Problem Book: 199 Research Problems

HP(B)

o

PROBLEM. Construct an unconditional basis in the space

H~)

p%
p ={

. There is also an auxilia-

Does there exist an orthonormal unconditional basis in

H~(B)

?

The case p < ~ is less clear. In HPc~) we have unconditional bases, cf. [3] ,[5]. However in ~everal variables the very existence of an unconditional basis in Hr~ B), p< ~ is still open. The proof of isomorphism between

H{(8)

and ~4(~)) given in [6]

25 can be extended to

~ < ~

(after some technical modifications)

provided the following question has positive answer. QUESTION.

I~s~P~),

~ < ~

isomorphic to a complemented sub-

HP(B) ?

REFERENCES !

I. P e E c z y n s k i

A.

Banach spaces of analytic functions and

absolutely summing operators.

CBMS regional conference series

N°30. 2. ~ 0 q K a p e B C.B. C y m e c T B o B a ~ e 6a3~ca B npocTpaHcTBe ~yH~-I ~ , 8 H S ~ T ~ q e c E N X B EpyDe, H HeEOTOpNe CBO~OTBa C ~ C T e ~ p a H E - /n~Ha. - ~ T e ~ . c 6 O p H ~ E , I9V4, 95 (I87), B~n.I, 3-18~ 3. S j 8 1 i n P., S t r o m b e r g J.-O° Basis properties of Hardy spaces. Stockholms Universitet preprint No 19, 1981. 4. W o j t a s z c z y k nal basis in

H~

P.

. - A r k i v f~r Mat.,1982,

5. W o j t a s z c z y k tems. - Studia Math. 6. W o j t a s z c z y k

P.

H P

-spaces,

20, No 2, 293-300. ~

and spline sys-

(to appear). P.

isomorphic to Hardy spaces Math.

The Franklin system is an unconditio-

Hardy spaces on the complex ball are on the disc,

~

~
. - Annals of

(to appear)

P.WOJTASZCZYK

~ath.lnst.Polish Acad.Sci° 00-950 Warszawa, /

Sniadeckich, POLAND

8

28 1.8.

SPACES WITH THE APPROXIMATION PROPERTY?

Recall that a Banach space ~ has the approximation property (a.p.) if for all compact E c ~ and for all 8 > 0 there is a bounded

linear o p e r a t o r T : × • × ~ o h that IJ~-T~H < 6 when ~ E . and such that T has finite rank. Not every Banach space has the a~p.

~] Does ~

have the a~p.?

Some mild evidence that this might be true comes from the recent result that L ~ / ~ (i.e.B~O) has the a°p, K2~ • Another interesting space £or which the a.p. is unknown is ~ ' ~ ( ~ ) ~ ~ ~. (When ~ I the answer is easy and positives) Here

REPERENCES I. E n f I o P. A counter-example to the approximation problem in Banach spaces. - Acta Hath., 1973, 130, 309-317. 2. J o n e s P.W. BM0 and the Banach space approximation problem. Institut Mittag-Leffler report No,2, 1983. PETER W.JONES

Institut Mittag-Leffler Aurav~gen 17 S-182 62 Djursholm Sweden Usual Address: Dept~of Mathematics University of Chicago Chicago, Illinois 60637 USA

27 OPERATOR BLOCKS IN BANACH LATTICES

1.9.

The operator Q8 of multiplication by the characteristic function of a measurable subset 6 c ~0,I~ has the unit norm in every functional Banach lattice ~ on ~0,i~ (see ~ for a definition). Associate with every continuous linear operator T : E the number

and let

~cE):{TcZ(E,E):

~'(.T,E):-O } •

PROBLEM. Under what conditions on E empty, i.e. g(T, E ) =

0

fo r

th~ set ~ (E)

i..ss

ever~ linear operator?

This question arose for the first time in ~2~ (for concrete spaces) in connection with the contractibility problem of linear groups in Banach spaces. In particular an isometry T of L~ satisfying • (T, Lz) > 0 was constructed there. On the other hand~ it has been proved [3~ that ~ ( L ~ ) ~ ~ ( L ~) ~ and that

~(tS=~

,

[~.

for 1 < p ~

Recall now the definition of the Lorentz space

[~ let

for their properties.) ~or a m e a ~ b l e 0C*

lb

(see

f~ction ~

denote the non-increasing rearrangement of 4

~'~

on ~O,g

j xJ

. Then

4

0

It is well-known that (LP'~)* :

hp,r¢[

, where

f

~I

,6< . Therefore without loss of generality i% can be assumed that V - - p ~ . It is also known that ~(L P'~) ~ ~ , ~
~ d t ~ t n ~ ( L ~ "~') ~ For the

c~ ,

case p<$

< • oo

l ~ p ~oo r . ~ . the

situation

remains

unclear.

'

28 Nothing is known about the set ~ ( h~' $ ) except $ ~ when ~([,~) = ~(~) =~= ~ • The problem of non-emptiness of ~ ( ~ ) remains open for Orlicz spaces. The operators T ~ T ( ~ ) ~ ~(~P) constructed in ~ ] depend on and this is not a mere occasion. The set ~( l'P1)~~(--P~) is not empty (let ~ "~ ~I "~ ~ < ~ for the definiteness) iff ~ ~ ~ ~, [6]. However it is not clear what conditions provide ~(~4 ~ ~ ( ~ ) ~ ~ in the general case.

REg~NCES I. L i n d e n s t r a u s s

J.,

T z a f r i r i

L.

Classical

Banach Spaces II . Berlin, Springer Verlag, 1979. 2. M ~ T ~ r ~ H B.C. roMo~on~ecza~ cTpyETypa X~He~Ho~ r p y m ~ 6a~axoBa npocTpsaorBa.-YcnexE ~eTeM.HayE,I970,25, }~ 5, 68--I06. 3. E d e 1 s t e i n I., M i t y a g i n B., S e m e n o v E. The Linear Groups of ~ and ~4 are Contractible. - Bull.Acad. Polon.Sci., Set.Math., 1970, 18, N 1. 4. C e M e H O B E.M., ~ ~ p e a ~ c o H B.C. 88~a~a o MaaOCTH onepaTopm~x 62IOEOB B HpOCTRaHCTBaX ~.-- Zeit.Anal. und ihre Anwend., 1983, 2, N 4. 5. K r e i n S.G., P e t u n i n Ju.I., S e m • n ~ v E.M. Interpolation of Linear Operators. AMS Providence, 1982. 6.

C e M e H o B E.M., ~ T e 2 H 6 e p r A.M. 0nepaTopm~e daOEE B L~pOCTpaHCTBaX ~p,$ • - ~ O F ~ . A H CCCP. 1983, to appear.

E.M.S (E. M. CE

OV 0B)

CCCP, 394693 BopoHem, BopoHemcEE~ rOCy~&pCTBeHH~I~ yHE~BepCETeT

2g

1.10. old

SPACES OF ANALYTIC FUNCTIONS (ISOMORPHISMS, BASES)

0(~) wil~ denote the space of all functions analytic in the domain ~ , ~ C ~ . A domain ~ is called standard if ~(~) is isomorphic (as a linear topological space) to one of three (mutually non-isomorphic) spaces

In [1] the class ~ of all standard domains was completely described. Moreover in [ 1] the properties of ~ ~ were found out determining to which particular one of these three spaces the space ~(~) is isomorphic. These properties involve the structure of the set of all irregular points of the boundary $ ~ (see [2] for notions from the potential theory). The isomorphic classification of spaces ~(D) for ~ not in ~ remains unknown.

Any domain ~(~ ~ ) ~ \ ( ~ ~

(~i ~i) U {0}),

where ~(@,~)={~:l~-~l<~I, Sec0,~), ~= C~i)i,,4 i s a monotone sequence o f p o s i t i v e numbers w i t h

does not belong to

~

(and ~ ( $ , ~ ) ~

whenever the series i n ( l )

diverges). CONJECTURE. There exists a continuum of mutu~ll,7 non-isomorphic

spaces ~(~(~))

.

This conjecture is stated also in [6] (problem 63). Let us mention - in connection with the open question on the existence of a basis in the space ~(~) of all functions analytic on a compact set K , K C ~, that this question is open for K=C\ ~(~,~) as well (under condition (1))~though it was proved [7] for such K that ~(K) has no basis in common with ~(~), ~ being any regular (in the potential - theoretic sense) neighbourhood of ~ . ~rom this fact it follows that ~C ~) has no basis of the f ~=Q, Let

~

be

~ -dimensional open Riemann surface.

30 (a) We say that ~ tion ~(~,E)

with

is regular iff there exists the Green func~

G(~, Z~) = O,

~ e ~,

for any

sequence ( ~ ) with no limit point in ~ . Under additional restrictions (for example if ~ is a relatively compact subdomain of another Riemann surface ~4 ) it has been proved that ~(I~) and ~ are isomorphic if ~ is regul~r (cf.[8] and references therein). Is this true in the general case? The necessity of this condition follows from general results for Stein manifolds ([3] and references therein)• (b) Let ~ be a Riemann surface with the ideal boundary of capacity zero. Is then ~(~) isomorphic to, ~ ? The condition is necessary even in the multidimensional case (unpublished). (c) The QUESTION about the existence of a basis in ~(/i) is solved only under some additional restrictions (even for surfaces satisfying (a) and (b) above [8], ~9~). Clearly ~ (~ ) and ~ (K) are non-isomorphic whenever is open and K is a compact set (~, K c ~ ) . QUESTION. Which other d%fferences in topological properties of sets

~, E£ ( c C)

imply that

~ (~4)

and

~(Ez)

are non-

isomorphic? Here ~(E) denotes the inductive limit of the net {~(V)} of countably normed spaces ~(V) , V running through the set of all open neighbourhoods of E • V.P.Erofeev proved (unpublished) and ~ being an open and a that ~(.~ U & ) 7@- f~(]~ uj~') ,j closed subarcs of the unit circle ~ D • It is not known whether are isomorphic if ~ is a closed non-degenerate arc of S D • In [4] a method was proposed to construct common bases for

,(.9(~)

and

~¢,

K)

,

K c ~

. ~his method uses a special ortho-

gonal basis common for a pair of Hilbert spaces ~, , ~4 and for the Hilbert scale x) ~ generated by ~0 and ~4 essentially genereJ

,,,,,

,,,

,

X) The notion of a Hilbert scale introduced by S.G.Krein has a number of important applications to problems of the isomorphic classification of linear spaces and to the theory of bases• We refer to the paper by B.C.M~THD~H, r.M.XeHEEH , " ~ H e ~ e s a ~ a ~ EOMIRIeECHOPO aHax~3a", Ycnex~ MaTeM. HayE. 1971, 26, N 4, 93-152 containing many results concerning spaces" of analytic functions, a list of unsolved problems and an extensive bibliography. - Ed.

31 lizing well-known results of V.P.Erohin about common bases (see,e.g.

[4], [3], [7],[8], [9] ), THEOREM

([4],[12],[8]). Let

V~ C ~ ( £ ) }

KCO, K : { £ £ Q ; 11(£)1~'~1, is a regular domain in C

and suppose ~ \ K

(or a relativel~ compact domain on a Riemann surf~ce~t Then there exist Hilbert spaces ~0 , ~ with

H ~ (~(~) c_,.O(K)c_, FI, and, for a,,~l spaces

~

(2)

of the,,,,,,,correspondin~scal,e

(O(c/x~ ~),~) c_, I-I~'c_,. 0(£~), ~=~E~;~(~,K,~) <~

where

,

UK

(3)

~(~, K,z)

is the harmonic measure of ~ with respect to ~ \ K ([3], p.299). All embeddings in (2) and (3) are continuous. The common orthogonal basis ~6~) ~ 0 of the spaces N0, H 4 is a common basic A QUESTION arises: hOT "far" is it possible to "move apart" the sDaces

~0 , ~

Let tic in ~

sat isfyin~ (2) with0u ~ breakin~ (~)?

~w(O) be a Banach space of all bounded functions analy. We consider Hilbert spaces ~4 with

The well-known Kolmogorov's problem about the validity totic relation

for the

~

widths

%~ (~)

of the compact

is the Green s capacity of the compact set K can be reduced to the following PROBLEM. Describe all domains suitable

Ho

~

) ( [8], see also ~11]).

with

of the asymp-

set

with respect (4)

>

to ~))

(3) (for a

32 RE~ERENCES

I.

3 a x a p ~ T a

B.II. IlpOCTpaHCTBa ~ y H E ~

o~oro

nepeMe~oro,

aHSJH~TI~qecE~X B OTEpNTRX MHo~eCTBaX ~ Ha KOMIIaETaX. -- " ~ T . C 6 . " ~

2. 3.

I970, 82, J~ I, 84-98. ~ a H ~ ~ O ~ H.C. 0CHOB~ coBpeMeHHo~ T e o p ~ noTeHn~a~a. M., "HayEa", I966. 8 a x a p ~ T a B.II. 3EcTpeMa~H~e n~op~cy6rapMoH~ecEHe ~yH~-L~L~, r ~ B 6 e p T O B ~

4.

5.

6.

7.

8.

9.

mE~

~ ESoMop~EsM npocTpaHCTB

SHS~T~eCENX

~ MHOE~X nepeMeHH~x, I, H. - B c 6 . T e o p M ~ y ~ L ~ , ~yHE~. s~ax~3 ~ ~x npy~oz., Xap]s~OB, 1974, ~ 19, 133-157, ~ 21, 65-83. 8 a x a p ~ T a B.H. 0 npo~oxmaeMHx 0a3Hcax B npocTpaHcTBax a ~ a ~ T ~ e c E m x ~yHE~Z~ O~HOZO ~ MHOr~X nepeMeHR~x. - C~6~pcE.~aTeM.~., 1967, 8, ~ 2, 277--292. ~ p a r ~ e B M.M., 3 axap ~Ta B.H., Xan~aH 0 B M.F. 0 HeEoTop~x npo6~eMax 6aszca aHS~mT~ecE~x ~ y H E ~ . -- B C6. : " A E T y a ~ e npo6~e~m HayE~", POCTOB--Ha--~OHy, I967, 9I--I02. Unsolved problems. Proceedings of the International Colloqium on Nuclear Spaces and Ideals in Operator Algebras, Warsaw, 1969. W a r s z a w a - Wroc~aw, 1970, 467-483. 3 ax ap ~ T a B.H., E a ~ a M n a T T a C.H. 0 cy~ecTBoBarite npo~oJ~aeM~x 6as~coB B npocTpaHCTBaX ~ y H E L ~ , a H a ~ T ~ e c K~X Ha Eo~a~TaX. -- ~T.saMeTE~, I980, 27, ~ 5, 701-713. S ax ap ~ T a B.H., C E ~ ~ a H.M. 0neHE~ ~-nonepe~HI~EOB HeEOTOpNX E~aCCOB ~yHE~E~, aHS~I~TEqecE~X Ha p ~ O B R X HOBepxHocT~X. --MaT.saberEd, 1976, I9, ~ 6, 899-9II. C e M E r y E 0.C° 0 c y ~ e c T B O B ~ O6m~X 6as~COB B npocTpa~cTBe a H ~ T E e C E E X ~ Ha EOMIIaETHO~ p ~ a H O B O ~ HoBepxHOCTE, POCTOB.yH--~, POCTOB-Ha--~oRy, I0 C, 6~0X.7 HaSB. (PyEon~c~ hen. B BMHHTM I5 ~eBp. I977 ~ 620-77 ~en.) P~MaT I977, 6 B I38 ~en.).

I0. W i d o m H. Rational approximation and ~ - d i m e n s i o n a l meter. - J . A p p r o x i m a t i o n Theory, 1972, 5, N 2, 343-361.

dia-

II. C E H 6 a H.H. 06 o~eHEe cBepxY ~-nonepe~EoB o ~ o ~ o F~acca r O ~ O M O p ~ X ~ y H E ~ . -- B c 6 . " T p y ~ m o ~ o ~ x y~eH~X Ea~e~pN BMcme~ MaTeMaTHEE", P ~ M , POCTOB--Ha--~OHy, I978. ~ e n o ~ p o B a ~ o B BMHMTM, ~ 1593-78 ~en. 12. N g u e n T h a n h V an. Bases de Schauder darts oertains espaces de fonctions holomorphes. - Ann.lust.Fourler (Grenoble), 1972, 22, N 2, 169-253.

33 V, P ZAHARIUTA

CCCP 344711

(B. If.SAXAHOTA) o, s. SEMIGUK

POCTOB-Ha-AOHy P O C T O B C E ~ PocyAapCTBeHH~ yHEBepCHTeT

N. I SKIBA

P O C T O B C E ~ Mm~eHep HO--CTpOHT ea~m~2 EHCTMTyT

(H.H.CEHBA)

* * * COMMENTARY BY THE AUTHORS The problems a), b), c) (in a more general situation~namely for Stein manifolds) have been solved in [13] by a synthesis of results on Hilbert scales of spaces of analytic functions [3] and last results on characterization of power series spaces of finite or infinite type [14], [15]. We formulate one of this results as an example. Let ~ be a connected Stein manifold. ~ is said to be P-regular if there exists a plurisubharmonic function ~(E) such that ~(~) < 0 in 0 and ~ ( Z ~ ) - * 0 for any sequence ( ~ ) without limit points in I~I --a THEOREM.

A(~)-~ ~ ( D

)

if an d only if

ii

is P-regular.

REFERENCES 13. 8 a x a p ~ T a

B.H.

ESOMOp~MBM npocTpaHCTB 8aSZ~TM~ecEEx ~yHE-

sm~. - ~ o K a . A H CCCP, 1980, 255, ~ I, 11-14. 14. v o g t D. Eine Charakterisierung der Potenzreihen~'ume vom endlichen Typ und ihre ~olgerungen, preprint (to appear in Studia Math. ). 15. V o g t D., W a g n e r M.J. Char~kterisierung der Unterr~ume der nuk!earen stabilen Potenzreihenr~ume vom unendlichen Typ, preprint (to appear in Studia Math. ).

34 o~ z s o ~ o ~ c

~.~.

OZaSSZ?ZOA~O~ o~

F -s~oss

K(%p)

~e a

I. For a given family of positive sequences {~p~ let ~Sthe space i.e. F is the space of a l l sequences ~ = { ~ 4

satisfying

l~Ip~---~, l ~ t ~ p <+~, p=~,~,....

(1)

The space ~(@{p) is endowed with the topology defined by the family of semi-norms (I). It is called a p o w e r ( K o t h e ) s p a c e if (i) ~ where -o0 <~p(~) ~
~p-kp

kp(~)-k~(h.
E & (~) is said to be of finite are power spaces in our sense, ~ < + ~ (~= +~). (infinite) type if Consider two classes of power spaces: of power spaces of the first kind [!], [2]: I) the class S

2) the class

~

of power spaces of the second kind:

p in both oases. If we consider isomorphic spaces as identical, then ~ n consists of spaces (2) and also of their cartesian products; contains spaces E0 (~) @ E~ (~) [2] and ~ contains spaces of a l l analytic functions on unbounded n-circular domains in C ~ [3].

35

PROBLEM 1. Give criteria of isomorphisms:

'-" E (,p, 6) F( in terms of

(.3)

k, ,,, FI ,

(4)

(k,&) , (J~,~) .

Articles [I], [2] co~tain a criterion of isomorphism (3) but under an ~ddition~l requirement on ~(~,~) (note that in A~S translation of [I] in Lemma 4 the important chain of quantifiers ~p/ ~ p ~ B~/ ~v / ~ ~5 ~5'~ C ~ , Z , 6"~0' has been omitted). Let us formulate one result on isomorphism (4), which somewhat generalizes the result of [3]. Denote by ~ the set of all sequences A=(~%), 0<~%~<4 such that there exist limits

and

~(~)

is strongly increasing in ~

TEEORE~ I. Let 1) -,~

4,

~

~ ~

and

~ i ~" ~

. .0Then (4) implies

,

Note for a comparison that isomorphism (3) takes place for arbitrary ~ , ~ A whenever the condition hi ~ ~ is fulfilled. Class ~ is also of a great interest because the following conJecture seems plausible. CONJECTURE 1. There exists a nuclear powe r space of the' second k indwithout the bases quasiequivalence property ~). 2. Let X be an F -space with the topology defined by a system of semi-norms [ l'Ip, pE ~I and ~(~) be a convex increasing function on [~, ~) . Denote by ~ the class of all spaces X such that J ~ ~¢ 3 M'I,, ~ , C : ,J

,,

*) The definition see e.g. in the article M~T~rBH B.C. AnUpoEcmmBTNBHSM 1085MSIOHOOTT= ~ (~85,~C 1~ B 8~SpHHX HIDOCTI08HCTSSX- - Yonexa UaTeM. HayE, 1961, I6, ~ 4, 63-I32.

36

I,ll , (:f(t,D lllp

C,

IIl, , -t,

Classes ~ ~ being invariant with respect to isomorphisms, are a modified generalization of Dragilev's class ~ [4] (see also ~5]); similar dual classes ~ were considered in Vogt'Wagnet [6S . Classes ~ have been used in [7] to give a positive answer to a question of Zerner. Consider the family ~ of all domains with a single cusp:

being a non-decreasing C1-function on [0,{] , ~(0) =0 . Then there exists a continuum of mutually non-isomorphic spaces C~Q~) with ~ E ~ . The following theorem clarifies the role of classes ~ in this problem.

THEOREM 2. eatisfyin~

C (~.)

~,0~ ~/~(~)%C~ (~,)) J~

PROBT.~ 2. Are all spaces ~

i f f there exist .... for 0(0~ <~o C'( ~ )

~ , ~ >0 •

from the same class

isomorphic or there exists a more subtle (than in Theorem ~)

classification of these s~aces? CONJECTURE 2. There exists a modification (apparently very essential one) of VoEt-Wagner's 01asses

~

which allows to prov e the

oon.~ecture on the exist~noe of a continuum of mulually non-isomorph~o~ spaces

~ (~ (~,$)) (see this Oollection, Problem ~ 10) REFERENCES

I. S a x a p B~COB ~ 772-774. 2. 8 a x a p s~COB ~ s TeM.mEox~ S. S a x a p

~ T a B.H. 06 ~SOMOp~sMe ~ EBaSEaEB~Ba~eRTHOOTE 68-c T e n e ~ x HpOCTpaHOTB E~Te.'~AH CCCP, 1975, 221, ~ 4, ~ T a B.H. 06 EBOMO~I~BMe E EBaSEgEBEBSJIeRTHOCTE 6acTe~eBR~x npocTpaRCTB E~Te.-Tpy~ 7-~ ~porod~cEo~ Mano ~yB~.aRax~sy, M., 1974. ~ T a B.H. 06o6~ea~He ~H~ap~aaTN M~T~Z~Ha ~ EORT~--

37 RyyM gonspRo ReEBoMop~RHX npOCTpaHCTB aHa~ETE~ecEEx ~yREL~2.-

~yREL~EOHa~H~ a r a b 3 4.

5.

E ero n p ~ o ~ e R ~ ,

1977, I I , ~ 3, 24-30.

3 a x a p m T a B.H. HeEoTopNe x~He~RNe T O n O ~ O r ~ e c ~ e ~mBapEaRTH E I~BOMOp~ESMN TeHBOpHNX npOESBe~eR~L~ HeRTpOB mEa~.-HsBecTES CeBepo-KaBEascEoro Hay~Roro ~eHTpa BHcme~ m e o w , 1974, 4, 62-64 V o g t D. Charakterisierung der Unterraume yon S.-Math.Z.,

6.

1977, 155, 109-117. V o g t D., W a g n e r M.J. Charakterisierung der Quotienter~raume von S und eine Vermutung yon Martineau.-Studia Math.

7.

1980, 67, 225-240. r o H ~ a p o B A.H., 3 a x a p m T a B.H. HpOCTpHHCTBO 6ec~oRe~Ro ~ e p e H n ~ p y e M H x ~ y R m m ~ Ra o6nacT~x c yr~aME (to appear)

V.P.ZAHARIUTA

(B.H. 3AXAPIOTA)

CCCP, 844 711, POCTOB--Ha-~oHy, POCTOBCEEI] rocy~apcTBeRH~ yH~BepCETeT

38 1.12.

WEIGHTED SPACES OF ENTIRE PUNCTIONS Le¢ p: C - ~

where ~ E ~ + that

~

be a continuous function and define

. We suppose that

[ ' ~ % II'lls

for

is not t r i v i a l . I t i s e a s i l y seen that

~

~>$>~

and

i s a ~r~chet

space, the ~opology of which is strictly stronger than the topology of uniform convergence on the compact subsets of C . With the help of the Riesz representation theorem the dual space of ~ can be identified with the space of all complex valued measures ~ on C such that

for

an %>~ (see ~5~). As an example c o n s i d e r p(z)=l~l ~ , for t h e space o f a l l e n t i r e f u n c t i o n s o f o r d e r ~

this

=onomials

fZ"t

0

%>0 , then and t y p e ~

constitute

= So

is (see

uder

basis in ~ and ~ is topologically isomorphic to the space of all holomorphic functions on the disc ~ if ~ > 0 and to the space ~(0) of all entire functions if ~ = 0 , both spaces ~ ) and ~L,(.~)endowed with %he topology of compact convergence. Here, ~ is also a nuclear Pr~chet space (see ~7~) and the dual space can be identified with a space of germs of holomorphic functions (Kothe duality [43 ). All this can be used to find a solution for interpolation problems such~as for instance I (~) (~ • ) = ~ • , ~ = 0,4, ~ ,... in the space ~ by means of methods from functional analysis (see~3], [I] ). If of I~I ) nomials Ano%her

we take ~(~)=[~zI = ex , ~ = X + t ~ (here p is net a function , then the corresponding spaces ~ do not contain the polyand properties s~m!lar to the above example are not known,, example o f in%erest is due to Gel'land and Shilov ~2~:

z~

K,OI,"O,4,~,,.,.,where &,J~,A,~>0

and ot+}~4.

39

Tu_fact, each function

~ ~:~ can be extended to coincides with the space of all entire functions

~,~ ~,A

and such that

l where 0< / < ~ ,

>~ >0

(see ~2]).

The following problems are of special interest if the weight is not a function of IEI . PROBLEM I. Is it possible to find a representation of th e dual space

of

as a space of certain ho!omorphic functions or

~erms of holomorphic functions, analogous to the so called "K0thed~lit~" ~4] for the space

~C~)

o_~r ~(~) ?

PROBLEM 2: For which weights

? is the space ~

nuclear?

Nityagin ~6] proved the nuclearity of the spaces ~ ' ~

.

PROBLEM 3. Existence of Schauder bases in ~R . This problem seems to be quite difficult. If ~ is nuclear and has a Schauder basis, then ~ can be identified with a Kothe sequence space (see ~7]). If the monomials {E~I~w0 constitute a Schauder basis in ~ , as in the first example, then ~ is a so called powe~series-spaoe(see ~ 7~). Let ~ ~ ~ be an entire function , which is not of the form ~ + ~ 6b~ ~ , then$~@~(~|)~ W~

: by our assumption on ~ , there exist two points Z~ , Z~ ~ , Z ~ E~

~

with

~Z~)= ~Z~)

denotes the Dir~c measures

/~ ) ~ , ~ ) = 0

~(~:~>~4)~

~

E~

, now set ~ = ~Z~- 4 ~

(~=.~ ~)

; then

~ ~ ~J~

, where and

, t h e r e f o r e , by the Hahn-Banach theorem,

. SO, i f ~

does not contain the monomials, ~

cannot have a Schauder basis of the form { ~ } ~ . B.A.Taylor [8] constructed an example of a weighted space of entire functions containing the polynomials and the function ~(E) , but where ~(~) cannot be approximated by polynomials REFERENCES I. B e r e n s t e i n

C . A . and

T a y I o r

B.A.

A new look

40 at interpolation theory for entire functions of one variable. Adv. of Math.,1979, 33, 109-143. 2. G e I ' f a n d I.M. and S h i 1 o v

G.E.

Verallgemeinerte

Funktionen II, III. VEB Deutscher Verlag der Wissenschaften, Berlin 1962. 3. H a s i i n g e r

~z

P.

and

M e y e r

approv~mAtion and interpolation. 4. K o t h e

M.

Abel - Goncarov

- Preprint.

G. Topologische lineare Raume. Berlin, Heidelberg~

New York, Springer Verlag, 1966. 5. M a r t i n e a u A. Equations diff~rentielles d'ordre infini. - B u l l . S o c . M a t h . 6. M Z T ~ r Z H THHa

S

B.C.

de France,

1967, 95, 109-154.

H~epRocT~ ~ ~ p ~ e

. --Tp.MocEB.~mTeM.O-Ba,

C~O~CTBa

npocTpa~cTB

I960, 9, 817--328.

(Amer.Nath.

Soc.Transl., 1970, 93, 45-60). 7. R o 1 e w i c z S. Metric linear spaces. Warsaw, Monografie ~atematyczne, 56, 1972. 8. T a y I o r B.A. On weighted polynomial approximation of entire functions. F. HASLINGER

-Pac.J.Math.,

1971, 36, 523-539. Institut f~r ~ t h e m a t i k Universit~t Wien Strudlhofgasse 4 A-1090 W i e n AUSTRIA

41 LINEAR ~UNCTIONALS ON SPACES OF ANALYTIC FUNCTIONS AND THE LINEAR CONVEXITY IN ~ ~

1.1.3. old

A domain ~ in ~ is called 1 i n e a r 1 y c o n v e x (1.c.) if for each point ~ of its boundary ~ there exists an analytic p l a n e { ~ C ~ : ~ 1 + ° ° . +~+~=0~passing through ~ and not intersecting ~ . A set E is said to be a p p r e x i m a bI e from inside (from out s i d e ) by a sequence cf domains ~ K , K =~, ~,..,if ~ ~ k ~ ~K+~ (resp., ~K+4 c ~ O K ) and ~ = ~ K ~ K (resp., ~ = ~IK ~ K ). A compact set ~ is called 1 i n e a r 1 y c o n v e x (1.c.) if there exists a sequence of 1.c. domains approximating ~ from the outside. Applications of these notions to a number of problems of Complex Analysis, similar concepts introduced by A.~Tartineau and references may be found in [1]-[5]. If ~ is a bounded 1.c. domain with ~-boundary then every function continuous in ~ 0 ~ ~ and holomorphic in ~ has a simple integral representation in terms of its boundary values. The representation follows from the Cauchy-Pantapple formula [7] and is written explicitly in ~8], [I] ,[2]. It leads to a description of the conjugate space of the space 0 ( ~ ) (resp. 0 ( M ) ) of all functions holomorphic in a 1.c. domain ~ (resp. on a compact set M ) which can be approximated from inside (from the outside) by bounded 1.c. domains with ~2-boundary (see [9] for convex domains and compacta and [2] for linear convex sets ; the additional condition on approximating domains imposed in [2] can be removed). Such an approximation is not always possible [6]. This description of the conjugate space is a generalization of well-known results by G.Kothe, A.Grotendieck, Sebasti~o e Silva, C.L. da Silva Dias and H.G.Tillman for the

case

.

.

henE=

for every ~ ~ F ~is Called the c o n j u g a t e s e t plays the role of '~he exterior" in this description. Let

be the approximating domains specified above, on ~ . Consider a differential form H K='I

~ C ~

and

42

A 8u,,,, A . . . < . , ~> : 9 ~ ~K ('(I)) : CI)~I<<

where where

tional

F

on 0 ( 9 )

A gu,, A ~%~ A ... A %~,,, +,.. + .,. ~ . . ~,et ~r(4)):(~((I)), ...,'%@)) ~ ~ ('~)' :~ >-'1. Every linear cemtinuous func(on 0 ( M )

) has a representation

F(J~)--I ~c~)~c~:c~.~))~(~%~,.(~), ~),

(~)

where ~ 0 ( ~ ) (respectively, ~ e 0 ( M ) ), ~$ depends omly on ~ . Formula (I) establishes an isomorphism between the linear topolo~gical spaces 0' (~) and 0 (~) (respectively and

0I<M)

OLM)

).

PROBLEM I. Describe !.c. domains and compact sets which can be approximated from inside ,(from

C ~ -boundary.

with ~e

Let C n

~ ~a~, 0 C ~ and let F(~) denote the set of such that the plane { ~" < $~, ~ > : ~} passes through and does not intersect ~ .

CONJECTURE I.

A bounded l,c, domain ~

piecewise smooth boundary ~

~

.

Let

F~0

, 0 ~

with the

a~nits the approximation incicated

and only if the set s ~(~)

inPROBLEM I if

~

outside) by bounded l,c. domains

!

are connected for all l

(~)

[~ [(~-y~]

(respectively, is called the

F~ 0 (,M) Fantappi~

).

The function indi-

c a t o r;

here ~ , ~/C~ , 0C~ (respectively, ~ M , t O G M , 0 c M )- The function ~ in (I) is the Fantappi~ indicator of the functional F . A 1.c. domain ~ (a compact M ) is called s t r e n g I y 1 i n e a r 1 y c o n v e x , if the mapping which e s t a b l i s h e s t h e c o r r e s p o n d e n c e between fumctionals and their Fantappi6 indicators is an isomorphism of spaces 01(~) and

0(~) (respectively 0'<M) and 0 ( M ) ). similar definition has been introduced by A.Martineau (see references in [9] ). Every convex domain or compact is strongly l.c. (see, for example, [9] ). At last,the result from [2] discussed ~bove means that the existence of approximation indicated in PROBLEM I is sufficient for the strong linear convexity. Strongly l.c. sets have applications in such prob-

43

lems of multidimensional complex analysis, as decompositions of holomorphic functions into series of simplest fractions or into generalized Laurent series, the separation of singularities D~,E2~,E5~. That is why the following problem is of interest. PROBLE~ II. Give a ~eometrica! description of strongly linearly convex domains and compact sets• CONJECTURE 2. A domain (a compact set~ is a strongly 1.c. set if an d only if there exists an approximation of this set indicated in PROBleM I It was shown in E5] that under some additional conditions, the intersection of any strongly 1.c. compact with any analytic line contains only simply connected components. The next conjecture arose in Krasnoyarsk Town Seminar on the Theory of Functions of Several Complex Variables. CONJECTURE 3. A domain (a compac,t set~ is a strongly l~c. set if and only if the intersection of this set with any analytic line is connected and simpl 2 connected. Let ~ be a baunded 1.c. domain with the piecewise smooth boundary ~ . The set ~ = { C ~ ) ~C2~ : ~ ~ , t~ c ~ ( ~ ) } is called the L e r a y b o u n d a r y of ~ . Suppose that is a cycle. In this case it can be shown that for any function ~ holomorphic in ~ and continuous in C ~ , we have

I

(2}

This representation generalizes the integral formula indicated at the beginning of the note to the case of 1.c. domains with non-smooth boundaries. If a 1.c. domain ~ (a compact set M ) can be approximated from inside (from the outside) by 1.c. domains whose Leray boundaries are cycles then every linear continuous functional on 0 ~ ) (0CMD can be desc bed by a fo ula analogous to (I) with instead of ~ . Note that such a domain ~ (a compactum M ) is strongly 1.c. Therefore the following problem is closely connected with PROBLEI~ II. PROBLEM III. Describe bonnded 1.c. domains whose Lera 2 b o u n d a ~

44 is a cycle. This problem is important not only in connection with the description of linear continuous functionals on spaces of functions holomorphic in 1.c. domains (on compacta). ~ormula (2) would have other interesting consequences (cf° D] , ~ )CONJECTURE IV. The classes of domains in problem~ I I-Ill coincide. REFERENCES

I.

2.

S.

4.

5.

6.

7. 8.

9.

A ~ s e H 6 e p r ~.Ao 0 pasxo~eHH rOXOM0p~H~X ~ y H E ~ MHO-r~x EOM~XeEc~x nepeMeHH~x Ha n p o c T e ~ e ~po6H. - C~6.MaT.x. I967, 8, ~5, II24-II42. A ~ s e H 6 e p r ~.A. JI~He~HaH B H ~ O O T B B C ~ ~ passe,fet e OCO6eHHOCTe~ rO~OMOp~H~X ~y~EL~. - Bull.Acad.Polon.Sci., Ser.mat., 1967, 15, N 7, 487-495. A ~ s e H 6 e p r ~.A., T P Y T H e B B.Mo 0d O~HOM Me-TO~e C ~ o B a H ~ nO Bopam~ ~--KpaTH~X C T e H e ~ p ~ O B . -- CE6. MaT°Z. 1971, 12, ~ 6, 1898-1404. A ~ s e H 6 e p ~ ~.A., ~ y 6 a H O B a A.C. 06 O6~aCT~X IDJ~OMOp~HOCT~ ~ C ~e~OTB~Te~H~m~ ~ HeoTp~aTe~ Te~~OpOBCE~ EOS~eHTa~. -- T e o p . ~ , ~yHE.S2aX~S ~ ~X np~xo~., 1972, 15, 50-55. T p y T H e B B.M. 0 CBO~CTBaX ~ , ID~OMOp~H~X Ha c ~ o Jl~He~n~o B ~ MHo~eCTBaX. -- B C6."HeEOTOp.CBO~OTBa rO~o~op~. ~yHE.MHO~.EOMI~.HepeM.", Kpa0Ho~pcE, 1978, 18~--155. A2 s e~ 6 ep ~ ~.A., D~aEOB A.H., MaEapoB a ~oH. 0 ~ H e ~ H o ~ B~UyF~OCT~ B C ~ . -- CH6.MaT.~. I968, 9, ~ 4, 7~I-746. ~ e p e ~. ~ e p e ~ s ~ _ B H o e ~ NHTeI?pS~Hoe ~ C ~ C ~ e H ~ Ha EOMn-~e~CHOM a H ~ T ~ e c E o M M~O~OO6pas~. M., ~ , 1961. A ~ s e ~ 6 e p r ~.A. ~ T e ~ p s ~ o e npe~cTam~eH~e ~ y ~ , IY~OMOp~X B B~FJ~X o6~aOT2X npocTpaHoTBa C ~ • -- ~AH CCCP, 196S, 151, 1247-1249. A ~ s e H 6 e p ~ ~.A. 0 6 ~ B~ ~G~He~HOPO He~pepHBHOrO SyHE-E~oHs~Ia B HpOOTpaHCTB~X ~ y H E ~ , I~O~O~Dp~K~X B B ~ H ~ O6~aCT~X C~ . - ~ H CCCP, IR66, I66, IOIS-I018o L.A.AIZENBERG

(~.A.A~SEHBEPr)

CCCP, 660086, KpacKo~pcE AF~eMropo~oE MHCTeTrr ~ e S ~ CO AH CCCP

45 COMMENTARY BY THE AUTHOR A solution of Problem II given in ~ 0 ~ , ~ I ] shows that Conjecture III is true. The definition of a strong linear convexity (s,1 c ) due to Martineau differs from the definition in the text only by the power (-I) (instead of (--~)) in the indicatrix f o r m u l a The two definitions turned out to be equivalent Yu.B.Zelinsky has shown in ~2], D 3 ] that the second conditions of Conjecture III and Conjecture I are equivalent They mean the acyclicity of all sections of the domain by analytic planes of a fixed dimension ~, ~ K < ~ , and coincide with the s,l.c- ~] These conditions form a precise "complex analogue" of the usual convexity ~I] , [12]. But the standard convex machinery cannot be generalized to this context. In particular, an &-contraction of a s.l.c, domain is in general no more s.l.c. Using the results of ~0] one can show that the sections of s.l,c. domains are not too tortuous This observation yields examples of unbounded s.l.c, domains non-approximable by bounded 1.c- domains with smooth boundaries,

REFERENCES

I0. 3 H a M e H C K ~ ~ C.B. FeOMeTpHMecKH~ K p ~ T e p ~ C~X~HO~ aHHe~HO~ B~nym~ocT~. - ~yH~.aHaa. H e r o npH~., 1979, T.I3, ~% 3, 83-84. II. 3 H a M e H C ~ H ~ C.B. BKBMBa~eHTHOCTB p a S ~ H ~ X onpe~eaeHH~ CM.~BHO~ ~I4He~Ho~ BB~IyI~OCTM. Mex~yHapo~Ha~ EoH~epeH~H2 no ~OMn~eNCHOMy aHaaHsy H np~omeHHm~. BapHa, 20-27 CeHT26p~ 1981 r.,

30. 12. z e i i n s k y

Y.B. On the strongly linear convexity

tional conference on complex analysis and applications September 20-27, 1981,

InternaVarna,

198

IS. 3 e ~ ~ H C ~ Z ~ D.B. 0 reoMeTp~ecEHx E p H T e p ~ x C H ~ H O ~ m~He~HOB B~yK~OCTM. ~ o ~ a ~ M A E a ~ e ~ M Hay~ CCCP, 1981, T.261, ~ I,

II-13.

46 I. 14.

ON THE UNIQUENESS OF THE SUPPORT OF AN ANALYTIC PUNCTIONAL

old The symbol H(E) will denote the space of all functions analytic on the (open or compact) set ~ , ~ C C ~ , endowed with the usual topology. Elements of the dual space ~(~) (here and below stands for an o p e n set) are called analytic functionals (= a.f.). A.Martineau has introduced the notions of the carrier ~orteur) and of the support of an a.f. A compact set ~ , ~ C ~ , is called a c a r r i e r of an a.f. ~ if T admits a continuous extension onto ~(~) , or equivalently [2] if T is continuously extendable onto ~(~) for an arbitrary open 0J , ~ D ~ J D ~ . Every a.f. has at least one carrier. Let ~ be a family of compact subsets of ~ such that if ~ A ) is a subfamily of % linearly ordered by inclusion then ~ ~ A C ' ~ ". A compact set ~ , ~ ~ is called an ~ -s u p p o r t of the analytic functional T if ~ is a carrier of T and ~ is minimal (with respect to the inclusion relation) among all carriers of T in ~.If D has a fundamental sequence of compact sets from then any analytic functional has an ~ -support but in general the -support is not unique. It is possible to consider various families of compact subsets of ~, e.g. the family of all compact subsets of ~ , the family of ~(~) -convex compact sets, the family of all convex compact sets (in this case an ~ -support is called a convex support or a C -support). Any analytic functional T on C 4 has a unique C - s u p p o r t but can have many polynomially convex ( = e~ ) supports. (If for examp I •

0

then any simple arc connecting 0 and I is a PROBLEM. Describe convex compact sets K

i s the unique

pC -support of T K (C C ~ )

~-support of any a nal2tic functional

.)

such tha t ~-su~ort-

K. C.O.Kiselman [3] has obtained for ~=~ necessary and sufficient conditions for a compact set to be a unique pC -support. For ~> ~ a compact set with a C~-boundary is a unique pC -support ~4]. Kiselman

has proved in ~4] that a convex com-

47

pact set with a smooth boundary is a unique 0 -support. A stronger result is due to Nartineau [5]: a convex compact set K is a unique -support if any extreme point ~ of ~ (~ ~ ~ ~) belongs to a unique complex supporting hyperplane (:with respect to the complex affine manifold V(K) generated by K ). Our problem is stated for the above two families of compscts only, though it is interesting for other families as well. Using the ideas of Martina~u one can prove the following THEOREM. A convex compact set

K~c

C ~)

is a unique

~,sup-

port if the set of all its s u p p o r t i ~ hyperplanes is the closure of the set of hyperplanes

~

with the fq!!owin~ oro~erty: ~

KN

contains a point lyin~ in a unique complex supporting hyperplane ' wit h respect t O

V(K)

•

It is probable that the sufficient condition of the theorem is also necessary. REFERENCES 1. M a r t i n e a n A. Sur lee fonctionneles analytiques et la transformation de Fourier-Borel. - J.Analyse Math., 1963, 9, 1-I 64. 2. B j S r k J~E. Every compact set in C ~ is a good compact set. -Ann.Inst.Fourier, 1970, 20, 1, 493-498. 3. K i s e 1 m a n C.0. Compact d'unlclte pour lee fonctionnelles analytiques en une variable. - C.R.Acad.Sci°, Paris, 1969, 266, 13, A661-A663. 4. K i s e 1 m a n C.0. On unique supports of analytic functiorials. - A r k i v for Math. 1965, 16, 6, 307-318. 5. M a r t i n e a u A. Unlcite du support d'une fonctionnelle analytique: tun th~oreme de C.O.Kiselman. -Bull.Soc.Math.France, 1968, 92, 131-141. .

V. M. TRUTNEV

(B.M.TPYTHEB)

•

r

CCCP, 660075, KpaoHo~poE, y~e Maep~aEa 6, EpaoHo~p0EH~ IDcy~ap O T B e H ~ yH~BepcHTeT

CHAPTER

2

BANACH ALGEBRAS

Thirteen sections of this Chapter can be conventionally divided into three groups. General theory of Banach algebras is represented by Problems 2.1-2.5. This group of problems is connected mainly with the spectral structure of elements of an abstract Banach algebra. The second group consists of a couple of problems concerning Convolution Measure Algebra followed by a problem on harmonic synthesis in group algebras. The convolution algebra M ( $ )

of all finite

Borel measures on a locally compact abelian group

is interesting

G

from many points of view and among them from the spectral one. The subject originates in the classical paper by Wiener and Pitt (Duke Math.J.

1938, 4, N 2, 420-436) and has been intensively studied so

far. However, the bulk of all publications on the theme has revealed only different pathologies in the structure of M ( ~ )

, and the num-

ber of "positive" achievements here is not large. J.L.Taylor has calculated the cohomologies of the maximal ideal space of M(~) (see [2~ in references of Problem 2.6). G.Brown and W.Moran have described the structural semi-groups of important su~algebras of M(~)

(Acta ~ t h . ,

1974, 132, N 1-2, 77-109).

Some years ago B.Host

and M.Parreau solved a problem of I. Glicksberg (C.R.Ac. sc. Paris, 1977, 285, 15-17 and Ann.Inst.Fourier, described all measures ~

1978, 28, N 3, 143-164). They

whose i d e a l s ,

~(~)

is closed in M($)

.

49

The question of description of Shilov boundary for ~ Q$) is the subject of Problem 2.6, undoubtedly,

, which

is the core problem of

the theory. It has been posed by J.L.Taylor and still remains unsolved. Problem 2.7 considers a description of homomorphisms of L-subalgebras of ~ (6)

in the spirit of the well-known Cohen-Rudin

theorem. The last problem of the second group, Problem 2.8, deals with the structure of ideals in group algebras. The third series of questions concerns more visible, but not less mysterious algebras such as the algebra H ~ ( V ) and analytic functions in a domain

Vc C

of all bounded

. Note that Problem 2.9

contains an interesting conjecture about the axiomatic description of H ~ ( ~ )

in the category of all uniform algebras. Eleven problems

are formulated in 2.10 and among them the Corona Problem for H ~ ( V ) which remained unsolved untill now. We would like to complete the list of references to 2.10 by the following ones. M.F.Behrens Amer.~ath.Soc.,

(Trans.

1971, 161, 359-379) has shown that it is sufficient

to solve the Corona Problem for a special class of domains. It is also known that for some of these V

the algebra Ha(V)

does not

have a corona. New progress has been obtained in a recent paper by L.Oarleson (Proc.Conf.Harm.Anal.in Honour of A.Zygmund, Wadsworth Inc., Belmont, California, gebra ~ - ~ - ~ ( ~ )

1983, 349-372). The classical Hardy al-

does not have a corona but nevertheless the

structure of its maximal ideal space remains puzzling. Problem 2.11 is important for the understanding of this structure. The last two problems of the third group concern the disk algebra, though the question posed in Problem 2.13 is considered in a more general setting.

50 2. I. old

THE SPECTRAL RADIUS F O ~

IN QUOTIENT ALGEBRAS

If A is a complex Banach algebra a n d ~ e A let @ ( X ) denote the spectral radius of ~ . If l is a proper closed two-sided ideal of A , ~ + I denotes the coset in the quotient algebra containing ~O . Clearly, by spectral A is called an S R - a I g e b r a if equ~l~t;lholds in this formula for each so , ~ A , and eac~ closed two-sided ideal I of A . The algebra A ( ~ ) of all continuous functions on the disk analytic on its interior is not SR [1]. The following algebras are

inclusion,@Cx÷I)~~

1-31, E l l

C*

g,b

s

Zgebras

algeb

s

.(x~)

oo=paot or

Riesz operators, semi-simple dural algebras, semi-simple a n n ~ l a t o r algebras and algebras with a dense socle. If A is commutative and has a discrete structure space then ~ is ~ . QUESTION. Is this true for n o n c o ~ t a t i v e

A

?

Let ~ be commutative and let A be the Gelfand transform algebz'~, of A and let denote the spectrum of A • Then [2] if ~ is dense in $o(~(A)) , A is an ~ - a l g e b r a , Conversely, if A is a regular ~ - a l g e ' b r a , ~, iS dense an Co(~(~)) .

Z(A)

QUESTION. Can the condition of re&~larSt~ be omitted frgm this ~V~othesis? Let A be a ~-algebra and let ~ let I b e any closed %we-sided ideal of

be any element in A •

QUESTION. ,!,s it true that the,r~,, ,~lwa.ys exists

(dependS- K

on o0

),

such t ~ t

A

and

~j ~ 6 1

~(~÷I)-~-~(X+~)?

This result is true if @ ( X + I ) = ~ 0 and it is a corollary of ~] Theorem 3.8. The case @ ( ~ + I ) ~ - 0 is opera. REFERENCES I. S m y t h M.R.P., W e s t T.T. The spectral radius formula in quotiemt algebras. - Math. Zeit. 1975,145, 157-I 61. 2. M u r p h y G.J., W e s % T.T. Spectral radius formulae. Proc.Edimburgh Math.Soc.(2), 1979, 22, N 3, 272-275. 3. P e d e r s e n G.K. Spectral formulas in quotient algebras. Math. Zeit. 148. 4. A k e r m a n n O.A., P e d e r s e n G.K. Ideal perturbations of elements in - Math.Scand.1977,41, 137-139 G.J.~ 39 Trinity college M.R.P.SMYTH T.T.WEST Dublin 2 Ireland

0*-algebras.

51 EXTREMU~ PROBLEMS

2.2.

1. THE GENERAL M A X I I ~

A

, a compact set ' F

in a nei~hbourhoodpf

PROBLEM. Give n a unital Banaoh algebra

in the plane and a function F

~

holomorphic

, to find

su@{l ~(o~)I: I~I..
[1,2] in the case

of 2. THE SPECIAL MAXi~,~J~ PROBLEM. Let

f~

be a natural number,

a positive number .!ess than one, to find, amon~ all contractions

T

o_~n ~-dimensional Hilbert space whose spectral radius does not exceed ~ those (it turns out that there is essentially only one) for which the norm

ITal

assumes its m a x ~ u m .

This is a particular case of the general problem for

F={~'l~l~t]

A--

,

~(~)=~.

The solution of problem 2 was divided into two stages. The first step consists in replacing the awkward constraint that the spectral radius be ~ ~ by a more restrictive one which makes the problem considerably easier: the operator is to be annihilated by a given polynomial. This is 3. THE FIRST MAXIMU~ PROBLEM. Le t ~ be a natural number, p pn]v~omial of d~aT~ e ~ with all roots inside the unit disc. Let A(p) be the set of a!l contractions T ~ such that (T) = 0 . To find the maximum of ITml , more ~enerally of

a

B(H.)

l CT)I

as T

ranges

A cp)

•

We call this maximum C(.p,~) Having solved the first maximum problem we have te solve •

4. THE PROBLEM OF THE WORST POLYNOMIAL. To find, amon~ all ~ol,ynomials with roots

~ ~

i n modulus that one for which

~(p, ~)

is maximal. Per the case ~ ( ~ ) = ~ the result of [2] shows that the worst polynomial is p(~)-- ( ~ - t ) ~ . The method used in F2] is based on some fairly complicated algebraic considerations and does

52

not extend to functions other than worst polynomial for

0~~

~ other than ~

. It is not known whether the

has a root of multiplicity

whether the roots have to be concentrated on the boundary of the

disc I~1 ~ ~

tion

. Thus it seems useful to study

of the roots of

p

C

; a recent contribution

CP,~)

as a

func-

to Problem 4 i s E ~ .

A list of references up to 1979 is contained in the survey paper [ 4 RETERENCES I. P t ~ k

V.

Norms and the spectral radius of matrices. - Cze-

chosl.Math.J. 1962, 87, 553-557. 2. P t ~ k

V.

Spectral radius, norms of iterates and the criti-

cal exponent. - Lin.Alg.Appl. 1968, 1, 245-260. 3. P t ~ k

V.,

Y o u n g

N.J.

F~nctions of operators and the

spectral radius.- Lin.Alg.Appl. 1980, 29, 357-392. 4. Y o u n g

N.J.

A maximum principle for interpolation in H ~ ,

Acta Sci.~th.Szeged. VLASTIMIL PT/~K

Institute of Mathematics Czechoslovak Academy of Sciences

~itn~

25

11567 Praha I Czechoslovakia

53

2.3.

M~B~JM

PRINCIPLES FOR QUOTIENT NORMS IN H '~'

A surprising variety of starting points can lead one to study norms in quotient algebras of H = b y a closed ideal. Well known examples are classical complex interpolation [4], canonical models of operators [2] and some problems of optimal circuit design [I], It also arises from a maximum problem for matrices to which V.Pt~k was led by considerations relating to numerical analysis. The problem is to estimate the maximum value of ~ ~ (~)~ , where ~ a ~ , over all contractions A of spectral radius at most % < ~ on ~-dimensional Hilbert space (Ptak was mainly concer~ed with the case ~ ( A ) = ~ , some ~ ). An account of this problem is given in [3]. The only known way of handling the spectral constraint is to replace it by the condition ~ ( A ) = o , for some polynomial p , and then to vary ~ among the polynomials of ~egree ~ having all their zeros in % ( 6~S ~ ) ctional

• After some calculations one is led to study the fun-

F(p)-

ti

pH'tl H'/pH

as a function of ~ ~ for fixed ~ . In particular, as p varies over the class of polynomials described above, at what ~ does attain its maximum? The result we should like to prove is that attains its maximum at a polynomial of the form p(~)= for some ~ ~ C with I~] =~4 . Pt~k proved this was so (~) = ~ , and I proved that if ~ is a Blaschke ree ~ then all the zeros of an extremal polynomial : in fact, F is then the composition of a strictly ction and a plurisubharmonic function of the zeros of is known, however, about the most interesting case, (~) = E ~

with

in the case product of deghave modulus increasing fun-

p

[5].

oth ,

~>~

To formulate the problem concisely, let us say that the m a x imum p r i n c i p 1 e h o 1 d s f o r ~" I l C C - ~ if for any compact set K C ~ , the supremum of on K is attained at some point of the boundary of K relative to ~ , And if M is a complex manifold, we shall say that the maximum principle holds for F : M --~ ~ if, for any open set i ~ c C and any analytic function ~ : ~ -'=" ~ , the maximum principle holds for F o ~ .

54

He:@

PROBLEM. Let ~ ~

F

and l e t

F" 0 ~-'~" R

...,oct)-:

* FI='II

be defined by

H~/gH ~

where

9(z}

=

F] (~-~0

Does the maximum principle hold for

F

?

REFERENCES I. H e I t o n

JoW. Non-Euclidean functional analysis amd electro-

nics. - Bull.Amer.Math.Soc.,

2. H z K o m ~ c K ~ ~ Hanna, 1980. 3, P t a k

V,,

1982, 7, 1-64

H.K. ~ e ~

Y o u n g

o~ onepaTope c~zra. U o c ~ a ,

N.J. Functions of operators and the

spectral radius. - Linear Algebra and its Appl,, 4- S a r a s o n

D. Generalized interpolation in

1980, 29, 357-392 H ~ . - Trans.Amer~

Math. Soc., 1967, 127, 179-203. 5o Y o u n g

N.J. A maximum principle for interpolation in H ~

Acta Sci.Math°,

N. J. YOUNG

1981, 43, N I-2, 147-152~

Mathematics

department

University Gardens Glasgow GI28QW Great Britain

-

55

2.4

OPEN SEMIGROUPS IN BANACH ALGEBRAS

Let A be a complex Banach algebra with identity, not necessarily commutative. Let S be some open multiplicative semigroup in A . For an element ~ in A let ~(~) denote the distance from the point ~ to the closed set A k S (in other words, it is the radius of the largest open ball centred at ~ and contained in S ), and let $(~) be the supremum of all ~ 0 such that the elements Cb-~'~ belong to ~ for I~I < ~ (that is the radius of the largest open disk centred at ~ and contained in the intersection of with the subspace spanned by ~ and I). So we clearly have $ ( ~ ) ~ ~C~). For a ~ r i e t 2 of particular semigroups S we know that the formula

)~

is valid for every ~ i_~n A - We list below the most important cases. ~irst of all, if S = G ( A ) , the group of invertible elements, the result follows from the spectral radius formula. Second, the formula is true when S is the semigroup of left (or right) invertible elements of A , cf. E6]. Third, it also holds when S is the complement of the set of left (right) topological divisors of zero in A , cf.~]. Next, the formula is true for various semigroups of the algebra A = B(X) of bounded linear operators on a Banach space. In the case when S is the semigroup of surjective (or bounded from below) operators on X it was obtained in [1] by an analytic argument (in fact, this is equivalent to the third case mentioned before). Using an additional geometric device these results were applied in [6] to prove the above formula for the semigroup S of (upper or lower) semi-Fredholm operators on X , and hence it follows for the semigroup of Fredholm operators as well. In these cases the distance ~CT) admits other natural interpretations, namely, it coincides with (or is related to) certain geometric characteristics of the operator T (like the ~urjection modulus, the injection modulus, the essential minimum modulus [ ~ , etc.). In each of the cases listed above an individual approach was needed to find a proof. The difficult steps are of an analytic character, based on the theorems of G.R.Allan (1967) and J.Leiterer (1978) on analytic vector-valued solutions of linear equations de-

86

pending analytically on a ~rameter. The main idea is well demonstrated in [1] though in that case a combimatorial argument is also available [2]. So there seems to be some motivation for investi~atin~ the problem in ~eneral to seek a theorem which would contain all these oarticula r results. Let us give some warnings. Let S = G ~ CA) be the principal component of the set of invertible elements in A . There are (noncommutative) Banach algebras A for which the group G C A ) / G 4 ( A ) is finite, but not trivial, cf.[4],[3]. In such a situation for every invertible element ~ not in G 4 C A ) one can find a positive integer k such that G)k is in G 4 ~ A ) . Then we have 5 ( 0 ~ ) = 0 but ~ k) > 0 so that the formula cannot be true for this ~ and all 6~ in A . ~oreover, it is easy to see that ~ ~ S ~ ) 4/n cannot exist for these G) . Let A be a commutative Banach algebra and let ~ be the open semigroup of all elements whose spectra are contained in the open unit disk. In Ibis case we have ~ ( ~ ) = ~ ( ~ ) for every ~ in ~ but ~I.i--

, and the

We a r r i v e

a t t h e same c o n c l u s i o n i f

preceding definition

by a

last

inequality

may be

we r e p l a c e t h e u n i t

%-multiple

of it,

with

strict.

d i s k i n the

0 < ~ <

T h i s s u g g e s t s some a n a l o g y between our problem and t h e c l a s s i c a l mula f o r

t h e r a d i u s o f convergence o f a power s e r i e s

for-

(the formula

does n o t g i v e t h e r a d i u s o f t h e d i s k where we a r e c o n s i d e r i n g t h e

function but the radius of the disk where the given function can naturally be defined). Thus some additional conditions should be imposed on 5 in general. ~or instance, the property "if ~ = ~ belongs to ~ then both ~ and 6 are in S " is shared by most of the semigroups for which the problem is solved in the affirmative. This condition ensures, by the way, that 5 ( ~ ) = 5(S))~ ~ for all ~ in A and ~= ~, ~,... (we note that ~ ( ~ ) ~ ~SO) is always true). I s it important that the identity element be in nected, or "maximal"? Does arbitrary open semi~roup

~ S

?

~(~)[~

~

, or that

~

exist for ~

be conin an

%Vhat is th,,e,m,eani,n~ of it in ~ene-

,,ra~? I should like to thank Tom Ransford for a valuable discussion on this topic.

57 REFERENCES I. M a k a i

E., Jr.,

Z e m ~ n e k

J.

dius, packing numbers and boundedness - Integr.Eq.Oper.Theory, 2. M u 1 1 e r

V.

1983,

The inverse

The surjectivity

ra-

below of linear operators.

6. spectral radius formula and remov-

ability of spectrum. 3. P a u 1 s e n algebra.

V.

The group of invertible

- Collo~.Math.,

4. Y u e n

Y.

Groups

of invertible

Bull.Amer.~th.Soc.,1973, 5. Z e m ~ n e k nimum modulus. p.225-227.

J.

6. Z e m ~ n e k

J.

in a Banach

elements

of Banach algebras.

interpretation

Verlag,

Advances

Basel,

of the essential m i -

and Applications,

vol.6

1982.

The semi-Fred_holm radius

of a linear opera-

tor. - To appear.

JAROSLAV ZEN~NEK

-

79.

Geometric

- Operator Theory:

Birkh~user

elements

1982, 4 7

Institute

of Nathematics

Polish Academy

of Sciences

00-950 Warszaws, Poland

P.O. Box 137

58 2.5.

HOMOMORPHISMS ~ O M

C*-ALGEBRAS

Let A and ~ be Banach algebras. A basic AUTOMATIC CONTINUITY PROBLEM is to give algebraic conditions on ~ and ~ which ensure that each hemomorphism from A into ~ is necessarily continuous. An important tool in investigations of this problem is the separating space: if ~ : A - - ~ is a homomorphism, then the s e p a r a t i n g s p a c e of ~ is

~(@) =

~E~

there is a sequence ( ~ ) C A

:

and

with

Of course, 8 is continuous if and only if ~(@) = [0~ properties of ~(8) are described in [6]. Consider the GENERAL QUESTION: i_~f ~ £~(8) abOUt

6"(~0)

, %he spectrum of

~

. The basic

, what can one sa x

in the Banaoh alsebra ~

?

Pirst let us note that, if ~ 65(@) , then ~ ek~tq for each character q on ~ . For such a character q is necessarily continuous, the character ~ o 8 is continuous on ~ , and

commutative, it follows that ~(~) 40} . Is the same result true in the non-commutative case? An element of a Barmch algebra is a q u a s i - n i I p o t e n t if ~(~) ={0} , and so our question is the following. QUESTION I. Let £ % (@ )

o I.~s

@ :~ ~

~~

be a homomorphism , and let

necessarily a quasi-~ilpotent element of ~ ?

It can be shown that ~(~) is always a connected subset of containing the origin (see [6 , 6.16]), but nothing further seems to be known in general. The question was raised as Question 5' in [3], and it is shown there that the question is equivalent to the following. Let @ : A - ~ be a homomorphism, and suppose that ~ is semi-simple. ~ necQssarily continuous? It is shown by Aupetit in [2] that, if ~ and 5 are unital Banach algebras, if ~ ~ ~ B is a homomorphism, and if ~ ( @ ) then 9($@) .4~(~ + ~@) for all ~ A (Here, $ denotes the spectral radius.) Thus, if ~ 6 ~ ( @ ) ~ @ ( ~ ) , t h e n 9(6)=0 and ~ is a

quasi-nilpotent. However, it is not in general true that the set of quasi-nilpotentsin a Banach algebra is closed (see ~I~), and so we cannot immediately conclude from this result that each ~ ~ ~(~) is quasi-nilpotent. Quite probably, there is a counter-example to Question S. However, let us concentrate on the case in which both ~ and B are ~* -a~gebras QUESTION 2. If, in ~uestiQn I, both can we then0onclude

that

~

~

and ~

are

~-al~ebras,

i s necessaril E quasi~nilpotent?

The c o n t i n u i t y ~ : ~ - - ~ B is the set

i d e a i

of a homomorphism

It w~s proved by Johnson (~4~, see~6, 12.2~) that, if ~ is a C*-algebra, then ~(8) is a two-sided ideal in ~ and that its closure has finite condimension in ~ . Next, Sinclair (K5 , Theorem 4.1]) showed that, if ~ and B are both C*-algebras, and if e:~ ~ is a homomorphism with ~ ) = B then ~ I ~ can be decomposed as ~ + ~ , where ~ is a continuous homomorphism and :(~ ~(e)is a discontinuous homomorphism (or ~ = 0 ). Now ~(~----~ and ~(e) are closed ideals in ~ and B , respectively~ and so both are C~-algebras. Moreover, the range ~ ( ~ ) is a dense subalgebra of ~(~) and so, by our above remarks, consists of quasi-nilpotent elements. Thus, ~(~) is a C~-algebra with a dense subalgebra consisting of quasi-nilpotents. No such C* -algebra is known, and I would like it to be true that no such C~-algebra exists. So we come to the sharpest form of our original question. QUESTION 3. Is there a

C~-al~ebra (other than

~0~ )which ha S

a d~nse subalEebra consistin~ of quasi-hi!potent elements? If no such C* -algebra exists, then the homomorphism ~ in Sinclair's theorem must be zero, and so the element ~ in Question 2 must indeed be quasi-nilpotent. REFERENCES S. A u p e t i t B. Proprletes spectrales des algebres - Loot.Notes Math.~ 1979, 735, Springer-Verlag.

de Banach.

60

2. A u p e t i t

B.

The uniqueness of the complete norm topology

in Banach algebras and Banach-Jerdan algebras. - J.Functior~l Analysis, 1982, 47, I-6. 3. D a I e s

H.G.

Automatic continuity: a survey. - Bull.London

Math.Soc., 1978, 10, 129-183. 4, J o h n s e n B.E. Continuity of homomorphisms of algebras of operators II. - J.London ~th.Soc~2)~1969, 5. S i n c 1 a i r

A.M.

Homomorphisms from

I, 81-84. C* -algebras. - Prec.

London ~ath.Soc., (3), 1974, 29, 435-452; Corrigendum 1976, 32~ 322 6. S i n c 1 a i r A.~. Automatic continuity of linear operators. -

London Math.Soc.Lecture Note Series, 21, C.U.P., Cambridge,1976.

H. G. DALES

School of Mathematics, University of Leeds, Leeds LS2 9JT. Great Britain

61 2.6.

ANALYTICITY IN THE GEL~AND SPACE

old

oP

THE ALGEBra

OF ~CRJ

mmTIPLIERS

We shall be concerned with spectral properties of the Banach algebra of those bounded linear operators on ~(~) which commute with translations. However, it is convenient to represent the action of each operator by convolution so that the object of study becomes the algebra M ( ~ ) of bounded regular Borel measures o n ~ . The general problem to be considered is the classification of the analytic structure of the Gelfand space, A , of M ( ~ ) despite the fact that A is sometimes regarded as the canonical example of a "horrible" maximal ideal space from the point of view of complex analysis (cf.[1] p.9). Some encouraging progress has been made in recent years and it will be possible to pose some specific questions which should be tractable. We refer to Taylor's monograph, ~], for a survey of work up to 1973 (Miller's conjectured characterization of the Gleason parts of A has since been verified in [3]) and for further details concerning general theory of convolution measure algebras. In particular, we follow Taylor in representing A as the semigroup of continuous characters on a compact semigroup ~ (the so-called s t r u c t u r e s e m i g r o u p of ~(~)) and in transferring measures in M ( ~ ) to measures on % . In this formulation an element ~ of A acts as a homcmorphism according to the rule

Every member ~

of A

t h e n has a c a n o n i c a l p o l a r d e c o m p o s i t i o n ,

, where and ~ has idempotent modulus. If ~ itself does not have idempotent modulus (a possibility which corresponds to the Wiener-Pitt phenomenon and was first noted by ~reider [4]) then the m a p ~ - ~ l ~ l ~ , for ~ ¢ ~ ) > 0 , demonstrates analyticity in A . ~rom that observation Taylor showed that the ~ilov boundary of ~ (~)

is contained in clos ~

, where

1 S A: I I=

}.

He posed the converse question which is still unresolved. Subsequent work tends to suggest a negative answer so that we propose

It should be noted that the result ~ clos 8 remains valid for abstract convolution measure algebras and that it is very easy to find convolution measure algebras for which ~\ 8=#=~ . It is also

62

possible to find natural ~ -subalgebras of M(~) , itself, for which the corresponding conjecture is true. (An ~ -subalgebra is a closed subalgebra A which contains all measures absolutely continuous with respect to any measure in ~ ). Thus, a disproof would depend on not only a new phenomenon peculiar to M ( ~ ) but one which is specific to the full algebra. In addition we established a weak form of the conjecture in E5] by showing that a certain idempotent ~ (see below) fails to be a strong boundary point for ~ ( ~ ) (although it is a strong boundary point for the ~ -subalgebra of discrete measures). It should also be noted that Johnosn, E6], proved t ~ t A \ ~ = # = ~ but the techniques used to prove this result and its subsequent refinements depend essentially on the use of elements lying outside e . A natural strategy is to embed M ( ~ ) in a suitable super-algebra and prove the impossibility of extension of an appropriate homomorphism, It appears to be almost as difficult to exhibit, in the opposite direction, large numbers of elements of ~ which DO belong to . Before we describe some progress in this direction let us introduce the notation ~ for the unit function in A and the notation ~ for the homomorphism given by

where # ~ is the discrete p~art of ~ . ( ~ plays the role of the unit function for the subalgebra of discrete measures, which can be regarded as M ( ~ ) , where ~ is the discrete real line). Let us define a partial order on A by saying that ~ $ if

We have shown in E7~ that maximal elements are members of the ~ilov boundary. THEOREM ].(i) if ~ is maximal in A then ~ is a strong boundarypoint. (ii) If I~I i s maximal i n A \ ~ then ~ belongs to S . if, moreover, ~he

~ -subalgebra

is countabl~ ~enel~teld then ~

is a stron~ boundary point.

It is obvious that maximal elements belong to 9 but not entirely trivial that there are many examples other than those homomorphisms induced by continuous characters of ~ (viz. extensions of non-zero homomorphisms of ~ ( ~ ) ). To see that this is the case con-

63 sider in connexion with (i), homomorphisms which are induced on the discrete measures by discontinuous characters, and in connexion with (ii), homomorphisms which annihilate some fixed member of ~(~)

.

The additional hypothesis in (ii) does not correspond to a specific obstruction and merely reflects the constructive nature of our proof. We have avoided a similar difficulty in (i) by an appeal to Rossi's local peak set theorem and it seems plausible that a similar device should be available here. A proof which reduced the uncountably generated case to the countably generated case by pure measure algebra techniques would be particularly interesting since this species of difficulty often arises. In any event we propose CONJECTURE 2. If ~

is maximal in A \ I~I

then ~

is a stron~

boundary point. It would be useful to determine for specific subclasses of whether or not the elements are strong boundary points. The result that ~i

is the centre of an analytic disc v~s extended in

[8] to

cover the case of the idempotent corresponding to any single generator Raikov system. On the other hand we show in ~ ]

that ~

is acces-

sible in the sense that it is the infimum of those maximal elements of A \ ~ }

below which it lies. It is natural to expect that both re-

sults extend, although we feel that present techniques would require substantial development to prove CONJECTURE 3. The idempotents corresponding to proper Raikov systems are accessible but fail to be stron~ boundary points. We have chosen to present these problems from the standpoint of the development of the general theory. Prom a practical position the most useful results are those which exhibit classes of homomorphisms which belong to the ~ilov boundary of ~ -subalgebras of ~ ( ~ ) --because such results give information on spectral extension. In fact, THEOREM I is of this type because it remains valid for arbitrary convolution measure algebras(provided the technical hypothesis that is a critical point is added to part (ii)). Variants of that theorem v with the weaker conclusion that ~ belongs to the Silov boundary but valid for a larger class of ~

would be of considerable interest. REFERENCES

I. G a m e l i n

T.W.

Uniform Algebras. New Jersey, Prentice-Hall,

1969. 2. T a y l o r

J.L.

Measure Algebras. CBMS Regional confer, ser.

64 math.,

16, Providence,

3. B r o w n ras.

G.,

W.

- Nath.Proc.Camb.Phil.Soc.,

4. m p e ~ ~ e p ~TeM.c6., 5. B r o w n

D.A.

I95I, G.,

~th.Soc., 7. B r o w n space

1968, G.,

1973.

G l e a s o n parts 1976,

05 O ~ H O Z npm~epe

oSodm§m{oro derivations

1976, 8, 57-64. The ~ilov b o u n d a r y

134, 289-296. M o r a n W.

of a m e a s u r e

algebra.

for m e a s u r e

algeb-

79, 321-327.

29, ~ 2, 419-426. M o r a n W. Point

Bull.Lond.Math.Soc., 6. J o h n s o n B.E.

ideal

Amer.Math. Soc.,

M c r a n

Maximal

xapazTepa.

-

on M ( G )

. -

of M ( G )

. - Trans.Amer.

elements

of the m a x i m a l

- ~th.Ann.,

1979/80,

246, N 2,

131-140. 8. B r o w n ideal

space

G.,

~ o r a n

of M(~)

GAVIN BROWN

W.

Analytic

. - Pacif. J.~ath., University

discs

1978,

75, N I, 45-57.

of N e w South W a l e s

Sydney, WILLIA~ MORAN

in the m a x i m a l

Australia

University Adelaide,

of A d e l a i d e Australia

65

ON THE COHEN-RUDIN CHARACTERISATION OF HO~O~OEPHIS~S OF ~EASURE ALGEBRAS

2.7.

old

Let I.(~) be the Lebesgue space and ~ ( T ) the set of all bounded regular Borel measures on the unit circle ~ . ~ ( T ) is a commutative Banach algebra with the convolution product and the norm of total variation, and ~(T) is embedded in M ( ~ ) as a closed ideal. A subalgebra N of M ( ~ ) is said to be L - s u b a 1 g e b r a if it is a closed subalgebra o f ~ ( ~ a n d ~ ) < < ~ , that is, # is absolutely continuous with respect to ~ , implies I)~ N . Let Af(N)be the set of all homomorphisms of ~ to the complex v numbers (which might be trivial). Then, by Yu.Sreider [I] , for every , ~ A r ( N ) , there corresponds a unique generalized character I ~ / ~ : ~ N ~ or zero system such that

and#~N

In the following we shall use the same notation ~ g e n e r a 1 i z e d c h a r a c t e r~=I~:~Nl

for

. A satisfies,

by definition, (i) ~ I , = ~ ( [ ~ I )

(ii) ~:

~ ~-a. e.

and ~-- ~6~ ~i~p I ~jal ~ 0

i f ~<<~

Let ~ be a homomorphism of N to N Q~) . Then the mapping ~ (~)^(I~), 9 ~ N ~ defines a homomorphism for every integer i~ , where ,,A" denotes the Fourier-Stieltjes transform

? Thus there exists a generalized character ~ ( , ) = I ~ , ( M , ~ ) : ~ E N }

or

zero system such that

= I Let{~.}~

0

'

be a sequence of integers such t h a t ~ > ~

for infinitely many ~

. Put

rl Let

4

~:4

and~>~

66 be a Bernoulli convolution product, where ~ (~) is a Dirac measure concentrated on a point 0~ . We fix such a ~ and denote by N {~) the smallest L-subalgebra containing ~ . "HEOREM ([2],[3]). Let M be I P s u ~ ! ~ e b ~ L ( T ) c r N I ~ ) s ~ a d ~

(,)

for a l l tb i n

Z and ~

in.

M.

.mh~n w,e,~ v e (a) a positive integer ~

and a finite

subsetR=l~4,Bv~+~, ,Bt~

o~

(o)~'#eAIN}with [~}1~=I~;}I (}=%Z,..., ~)

t

~.,

~=n}~

such that

B (2)

wher# 0 E d#notes the characteristic function of the set E Converse l F if {~(~)} is a sequence in A/(~) (a), (b) and (c~, then the mappin~ ~ 9hism Of

M

.

satisfyina (A),

given by (I) is a homomor-

t~o M£~)-

When ~=-h(~) , t h e n A q ~ ) = { 6 {~:Me~}U{0~ and the condition (A) is obviously satisfied. For this case the theorem is due to W. Rudin. The other case, when ~ = N ( ~ ) , the theorem is proved by S. Igari - Y.Kanjin. Since h(~) is an ideal, our theorem holds good for

M=b(]h • NV~). We remark that we cannot expect the conditions (a), (b) and (c) without the hypothesis (A) (cf. [2~. PROBLEm I ° For what kind of L-subal~ebra M

does th e above

theorem hol d good? P R O B I ~ 2. Let M = M ( ~ )

~(~

. Let ~ ( ~ ) }

ho~d for {%(~)}.

b e a hqmomorphism of ~ (~)

be a sequence..o.f Af(M(~))

assume that { ~ ( M ) } Then~ characterize

and ~

~iven by (1) and

satisfies the condition ,(A) for a measure 9 ~

t_£o

such that the conditions (a), (b) and (0)

.

67 REFERENCES

I. ~ p • ~ ~ e p D.A. CTpoe~e M a K C ~ R ~ X ~eaxom ~ ~ox~ax Mep co cBepTKoI. - M a T e M . c 6 . , I950, 27, 297-818. 2. R u d i n

W.

The automorphisms and the endomorphisms of the

group algebra of the unit circle. - Acta 5~ath., 1976, 95, 39-56. 3. I g a r i

S.,

K a n j i n

Y.

The homomorphisms of the mea-

sure algebras on the unit circle. - J.5~th.Soc.Japan,

1979, 31,

N 3, 503-512. SATORU IGARI

Mathematical Institute Toh~ku University Sendal 980, Japan

68 TWO PROBLENS CONCERNING SEPARATION OF IDEALS IN GROUP ALGEBRAS

2.8.

All algebras in this paper are commutative complex regular B~nach algebras. In such algebras all ideals consist of joint topological divisors of zero, i.e. if I is a (not necessarily closed) ideal in a regular Banach algebra, then there is a net (~@) of elements of the algebra in question, which does not tend to zero, but ~@ ~ =0 for all ~ in I (cf.[1]). In this case we say that the net (Z@) annihilates the ideal I. We say that an ideal I = A has the separation property if for each S~ in A \ I there is a net (Z~)cA annihilating ~ and such that the net ~ does not tend to zero. It can be shown that in this case there exists one net ( ~ ) which works for all elements ~ in ~ \ I , and, in fact, I = ={~A:~-~0] (cf.[~). In case when there exists such a bounded net we say that the ideal I has the bounded separation property. An ideal with bounded separation property is necessarily closed. If A is a regular Banach algebra and F is a closed non-void subset of its maximal ideal space, then both the maximal and the minimal (non-closed) ideal with the hull F have the separation property. However the bounded separation property may fail for the minimal closed ideal with the given hull, even if it possesses the separation property. It is also possible to construct a closed ideal in a regular Banach algebra which has bounded separation property and it is different from the intersection of all maximal ideals containing it (the question whether it is possible was stated as a problem in the paper [2], but the construction of suitable example is rather easy: we take as the algebra A the algebra of all continuous functions on the unit interval possesing the derivative at O, and provide it with the norm II~II = l~l~ + I~I(0)I . The ideal in question is then I 0 = { ~ ~ A: ~(o)= ~ / ( o ) = o ] ) . Thus the nets provide a tool for separation and description of ideals. It is particularly interesting whether this tool works for the group algebras. In this context we pose the following problems. PROBLEM 1. Let I be a closed ideal in ~ ( ~ )

fRr an LCA ~roup

, Does I possess the separation property? PROBLEM 2. Does there exist an ~C& group ~

and a closed ideal

I in m4(~) which has the bounded separation property and is not of

the form Q*)

I= 0 {ME CA): I = M}?

69 In fact we do not know any example algebra which has separation

of a closed ideal in a group

property and is not of the form

(*) .

REFERENCES 1. Z e i a z k o

W.

Banach algebras, 2. Z e 1 a z k o mutative

On a certain class of non-removable

- StudiaMath. W.

On domination

Banach algebras.

WIES~AW ZELAZKO

1972,

ideals in

44, 87-92.

and separation

- Studia Math.

1981,

Math.Inst.Polish Warszawa,

of ideals in com-

71, 179-189. Acad°Sc.,O0-950

~niadeckich

POLAND

8

70 2.9. old

POLYNOMIAL APPROXINATION

Let A be a uniformly closed algebra of continuous functions on a compact Hausdorff space ~ . Assume that A contains constants, that ever~ continuous linear multiplicative functional on ~ is of the form ~--~($) for a unique element $ o f S , and that every element of ~ whose reciprocal belongs to ~ is of the form ~ for an element ~ of A • Let ~ be a positive measure on the Borel subsets of 5 ,whose support contains more than one point, such that the closure of A in ~ ( ~ ) , considered in its weak topology induced by ~(~) ,contains no nonconstant real element. Assume that the functions of the form ~ + ~ with ~ and ~ in ~ are dense in ~ ( ~ ) in the same topology. It is CONJECTURED that the closure of

A

i~n ~ ( ~ )

is isomorphic to the al~ebra of functions wh%chare bounded and analytio in the unit disk. Positive measures on the weak topology induced positive measures ~ and (with respect to "~ ) if

the Borel subsets of S are considered in by the continuous functions on ~ . Two ~ are said to be e q u i v a I e n t the identity

holds for every element ~ of A . The closure of the set of measures which are absolutely continuous with respect to ~ and equivalent to ~ is a compact convex set, which is the closed convex span of its extreme points. Extremal measures are characterized by the density of the functions of the form ~ ~ , with ~ and ~ in A , in

Let

be an extremal meas

closure in ~(~)/ and

~

in A

L• (~) / ~

e and let B

of the functions of the form

. It is

~+~

be the wea

with

CONJECTURED that the quotient Banach space

is reflexive.

Pot equivalent positive measures ~ and ~ , define ~ to be 1 e s s t h a n o r e q u a 1 t o Q if the inequality

holds for every element

~

of

A

. If ~

is an extremal measure,

71 it is CONJECTURED that a sreatest element

~

exists in (the closure

of~ the set of measures which are abselutel,7'continuous with respect t_~o ~

and equivalent to ~

of the form

~+~

with

. It is CONJECTURED that the functions ~

and

@

i__~n ~

are weakl,y dense in

REFERENCES I. d e B r a n g e s L., T r u t t D. Quantum Ces~ro operators. - In: Topics in functional analysis (essays dedicated to M.G.Krein on the occasion of his 70th birthday), Advances in Math., Suppl.Studies, 3, Academic Press, New York, 1978, pp.I-24. 2. d e B r a n g e s L. The Riemann mapping theorem. - J.Math. Anal.Appl., 1978, 66, N 1, 60-81. L. DE BRANGES

Purdue University Department of Math. Lafayette, Indiana 47907 USA

72 2.10. old

P R O B L ~ 3 PERTAINING TO THE ALGEBRA OF BOUNDED ANALYTIC ~JNCTIONS

Here is a list of problems concerning my favorite algebra, the algebra H°°~V) of bounded analytic functions on a bounded open subset V of the complex plane. Some of the problems are old and wellknown, while some have arisen recently. We will restrict our discussion of each problem to the bares% essentials. For references and more details, the reader is referred to the expository account [1], where a number of these same problems are discussed. The maximal ideal space of H ~ ( V ) will be denoted by ~ (V) , and V will be regarded as an open subset of ~ ( V ) . The grandfather of problems concerning H°°(V) is the following. PROBLEM 1 (CORONA PROBLEM). I ss V

dense in ~ ( V )

?

The Corona Theorem of L.Carleson gives an affirmative answer when V is the open U~'it disc ~ . In the cases in which ' ~ ( V ) has been described reasonably completely, there are always analytic discs in ~ ( V ) \ V , but never a higher dimensional analytic structure. PROBLEM 2. !s there' alwa,ys an a ~ l ~ i c there ~ver an a n a l ~ i c bidisc in ~ ( V ) The 3hilov boundary of Ho°(V)

disc ~

~(V)\ V ? Is

?

will be denoted by ~ (V)

There is a plethora of inner functions in ~ ( V ) ing question remains unanswered.

. A

, but the follow-

PROBLEM 3. Do ~he !n~er functions s e p a ~ t e th~ points of ~ ( V )

?

An affirmative answer in the case of the unit disc was obtained by K.Hoffman, R.G.Douglas, and W.Rudin ~2, p.316~. The Shilov bemndary ~ ( V ) is extremely disconnected. Its Dixmier decomposition takes the form ~(V)-----T U Q , where ~ ana are closed disjoint sets, O(T) ~--- L~(~?) for a normal measure on T , and Q carries no nonzero normal measures. The next problem is to identify the normal measure ~ . There is a natural candidate at hand. Let ~ be the "harmonic mearures" on ~ ( V ) . These are certain naturally-defined probability measures on ~ (V) \ V that satisfy for

.

73 PROBLEM 4. Can the normal measure restriction of harmonic measure to ~

~

on T

be taken to be ~he

?

There are a number of problems related to the linear structure of H°°(V) o It is not known, for instance~whether H~@(V) has the approximation property, even when V is the unit disc. As a weak-

~t~r olosed s . ~ l g e b ~

of L ~ t V )

,

H~°( V~

i. a ~1

space.

The following problem ought to be accessible by the same methods u s e d t o s%~dy

~V)

°

pROBTRu 5. Does ~ ( V )

have a unique predual?

T.Ando [3] and P.WoJtaszczyk [4] have shown that any Banach space ~ with dual isometric to H°@(~) is unique (up to isometry), However, Wo~taszczyk shows that various nonisomorphlc B 's have duals isomorphic to H@a(~). An extension of the uniqueness result is obtained by JoChaumat [5]. The weak-star continuous homomorphisms in ~ ( V ) are called d i s t i n g u i s h e d h c m o m o r p h i s m s, The evaluatio~Is at points of V are distinguished homomorphisms, and there may be other distinguished homomorphisms. Related to Problem 2 is the following. PROBLEM 6. Does each distinMuished homomorPhlsm lie on an anal~-

The c o o r d i n a t e f u n c t i o n ~ extends t o a map ~ : ~ ( V ) = V . If ~ ~V , then the f i b e r ~ ~ ( ~ } ) c o n t a i n s a t most one d i s tinguished homomorphism. PROBLE~ 7- Suppose there is a distinA~ished homomorphism

~ ,e ~

an~ Sup~o~ e

~

is ~

~,~H~v~as ~ L m i t alon~

aroil n

V tle~inatl,in~

at

~.

~

,

If

V , dge,~ that limit coino~de ~!th ~ (~

?

J.Garnett [6] has obtained an affirmative answer when ~ is appropriately smooth. The next problem is related to Iversen's Theorem on cluster values, and to the work in [7]. Define ~ - - - - ~ ( V ) N ~ , Denote by ~(~,~) the range of ~ , ~ ~e°(,V) , at ~ , ~ ~V , consisting of those values assumed by # on a sequence in V tending to ~ . An abstract version of Iversen's Theorem asserts that ~ ( ~ ) includes the topological boundary of ~ ( ~ ) , so t h a t ~ ( ~ ) \ ~ ( ~ ) is open in . The problem involves estimating the defect of ~(I~ ~) in

74 PROBT.~ 8. I f e v e ~ it~ for some

point of ~ V

f~CtiiOnii,ii ~ ~ ( VJ

l

have zero l o ~ a r i t h m ~ c c a D a o ~ t ~ f o r

is an essential singula rdoe..._~s~ ( ~ ) \ ~ $ ( ~ ) U ~ ( ~ ) ~

each

~

, ~

~(

V~

?

The remaining problems pertain to the algebra H ~ )

, where

is a Riemann surface. We assume that H ~ ( ~ ) separates %he points of ~ . Then there is a natural embedding of ~ Into~(~J PROBLEM 9. ~s the natural embeddin~ ~ p hlsm of ~

and an QDen subset of ~ ( ~ )

PROBLEM 10. If ~ te s ~

, doe_~s ~

r ~/~(~)

~(~)

a home om.or-

?

is a simple closed curve in ~

separate

.

that separa-

?

The p r e e e d i ~ p r o b l e m a r l s e s in the work of M.Hayashi Ea], who has treated Widom surfaces in some detail. For this special class of surfaces, Hayashl obtains an affirmative a n g e r to the followlng proble~ PROBLEM 11. I_~S ~II(~)

extremel~ disconnected ?

REPERENOES I. G a m e 1 i n

T.W.

The algebra of bounded analytic functions.

Bull.Amer.Math.Soc., 1973, 79, 1095-1108. 2. D o u g 1 a s R.G., E u d i n W. Approximation

by inner

functions. - Pacif.J.~ath., 1969, 31, 313-320. 3. A n d o T. On the predual of H ~° . - Special issue dedicated to W~adis~aw Orlicz on the occasion of his seventy-fifth

birth-

day. Comment.Nath.Special Issue, 1978, I, 33-40. 4. W c j t a s z c z y k P. On projections in spaces of bounded analytic functions with applications. - Studia Math., 1979, 65, N 2, 147-173. 5. C h a u m a t

J.

Unicit~

du pr~dual. - C.R.Acad.Sci.Paris,

S~r A-B, 1979, 288, N 7, A411-A414o 6. G a r n e t t J. A n estimate for line integrals and an application to distinguished homcmcrphisms. - I l l . J o ~ t h . , 1975, 19, 537-541. ~. G a m e 1 i n T.W. Cluster values of bounded analytic functions. - Trans.Amer.Math.Soc., 1977, 225, 295-306. 8. H a y a s h i M. Linear extremal problems on Riemann surfaces, preprint.

Dept.

of Nath.UCLA,

T.W.GAMELIN

LOS Angeles,

CA 90024, USA

-

75 2.1 I. old

SETS OP ANTISYMMETRY AND SUPPORT SETS FOR ~ +

G.

Let X be a compact Hausdorff space and A a closed subalgebra of 6(X) which contains the constants and separates the points of X . A subset ~ of X is called a set of antisymmetry for A if any function in A which is real valued on % is constant on ~ . This no~ion was introduced by E.Bishop EI~ x) (see also E2~,), who established the following fundamental results: (i) X can be written as the disjoint union of the maximal sets of antisymmetry for ~ ; the latter sets are closed. (ii) If ~ is a maximal set of antisymmetry for ~ then the restriction algebra A IS is closed. (lii) If ~ is in ~(~) and ~ I ~ is in A I ~ for every maximal set of antisymmetry ~ for ~ , then ~ is in ~ . A closed subset of ~ is called a s u p p o r t s e % f o r A if it is the support of a representing measure for (i.e., a Borel probability measure on ~ which is multlplicative on ). It is trivial to verify that every support set for ~ is a set of antisymme%ry for ~ . However, there is in general no closer connection between these two classes of sets. This is illustrated by B.Cole's counterexample %o the peak point conjecture E3, Appendlx~, which is an algebra ~=~0(X) such that X is the maximal ideal space of ~ and such that every point of X is a peak point of ~ . For such an algebra, the only supper% sets are the singletons, but not every set of antisymmetry is a singleton (by (iii)). The present problem concerns a naturally arising algebra for which there does seem to be a close connection between maximal sets of antisymmetry and support sets. However, the evidence at this point is circumstantial and the precise connection remains to be elucidated. L e t ~oo d e n o t e t h e I,°° - s p a c e o f Lebesgue measure on T . L e t ~oo be t h e s p a c e of boundary' functions on ~ f o r bounded holomorphic functions in ~ , and let ~ denote C(T) . It is well known that ~oo+ C is a closed subalgebra of ~eo E4], so we may identify it, under the Gelfand transformation, with a closed subalgebra of C(~(~,°°)) , where ~(~oo) denotes the maximal ideal space of ~oo (with its Gelfand topology). In what follows, by a set of antisim~etry o~ a support set, we shall mean these notions for the case X--~(]oo) and A----- ( t h e G e l f a n d t r a n s f o r m o f ) ~ 0 o C . A l s o , we s h a l l i d e n t i f y the functions in ~.o with their Gelfand transforms.

See the note at the end of %he section. - Ed.

78 The first piece of evidence fo~ the connection alluded to above

the fonowi

[5]"

is in l," and

is in

(H°°+C) I Z for each support set S , then ~ is in ~ ' + 0 , This is an ostensible improvement of part (iii) of Bishop's theorem in the present special situation. It is natural %0 ask whether i% is an actual improvement, or whether it might not be a corollary to Bishop's theorem via some hidden connection between mav~mal sets of amtisymmetry and support sets. The proof of the result is basically classical analysis and so offers no clues about the latter question. The question is motivated, in part, by a desire to understand the result from the viewpoint of abstract function algebras. A second piece of evidence comes from [6], where a sufficient condition is obtained for the semi-commutator of two Toeplitz operators %o be compact. The condition can be formulated in terms of support sets, and it is ostensible weaker than an earlier sufficient condition of Axler [7] involving maximal sets of an%isymmetry. Again, i% is natural to ask whether the newer result is really an improvement of the older one, or whether the two are actually equivalent in virtue of a hidden connection between maximal sets of an%isymme%ry and support sets. As before, the proof offers no clues. As a final bit of evidence one can add the following unpublished results of K.Heffman: (I) If two support sets for H °°+ O intersect, %hen one of them is contained in %he other; (2) There exist maximal support sets f o r ~ O. All of the above makes me suspect that each maximal set of antisymmetry for Hoe+ 6 can be built up in a "nice" way from support sets. It would no% even surprise me greatly to learn %hat e a c h maximal set of antisymme try is a s u p p o r t s e t. At any rate, there is certainly a connection worth investiEa%ing. RE~CES I. B i s h o p

E.

A generalization of the Stone-Welerstress

theo-

rem. - Pacif.J.Math., 1961, 11, 777-783. 2. G I i c k s b e r g I. Measures or%hogonal %o algebras and sets of antisymmetry. - Trans.Amer.Math. Soc., 1962, 105, 415-435. 3. B r o w d e r A. Introduction to Punction Algebras. New York, W.A.Benjamin, Inc., 1969. 4. S a r a s o n D. Algebras of functions on the unit circle. Bull.Amer.Math.Soc.~1973, 79, 286-299.

77 5. S a r a s o n

D.

Functions of vanishing mean oscillation.

-

Trans.Amer.Math.Soc.,1975, 207, 391-405. 6. A x 1 e r S., C h a n g S.-Y., S a r a s o n D. Products of Teeplitm operators. - Int.Eqt~at.Oper.Theory, 1978, I, N 3, 285-309. 7. A x I e r Berkeley,

S.

Doctoral Disseration. University of California,

1975.

DONALD SARASON

University of California, Dept.Math., Berkeley, California, 94720, USA

EDITORS' NOTE: The notion of a set of antisymmetry was introduced by G.E.Shilov as early as in 1951. He has proved the first theorem about representation of a maximal ideal space of a uniform algebra as a union of sets of antisymmetry (see Chapter 8 of the monograph M.M.rex~a~, r.E,~HXOB, ~ . A . P a ~ o B , "KOMMyTaT~BHNe H O p M Z p o B a ~ H e Eo~a", M., ~ESMaTFEB, 1958).

COB~ENTARY BY THE AUTHOR The structure of the maximal sets of antisymmetry for H ~ t C remains mysterious, although a little progress has occurredt P.M.Gorkin in her dissertation [8] has the very nice result that M ( ~ ) contains singletons which are maximal sets of antisymmetry for H ~ + C . Such singletons are of course also maximal support sets. The author's paper [9] contains a result which is probably relevant to the problem. REFERENCES 8. G o r k i n P.M. Decompositions of the maximal ideal space of L~ , Doctoral Dissertation, Michigan State University, East Lansing, 1982. 9. S a r a s o n D. The Shilov and Bishop decompositions of H ~ + C - conference on Harmonic Analysis in Honour of Antoni Zygmund, vol.II, pp.46]-474, Wadsworth, Belmont, CA, 1983.

78 2.12. old

SUBALGEBRAS OF THE DISK ALGEBRA

Let A denote the disk algebra, i.e. the algebra of all functions continuous on ¢~@~ ~ ard analytic on ~ . Fix functions and ~ in A • We denote by [~,~] the closed subalgebra of A generated by I and ~ , i,e. the closure in A of the set of all functions N

'i~,H1,=O We ask: w h e n d o e s[~,~]= A ? Necessary conditions are I) ~ , ~ together separate points of 0 ~ 2) For each @ in ~ , either~@)=~ 0

or ~Q~)=~= 0

and 2) together are not sufficient

. . some regu-

larity condition must be imposed on the boundary. We assume

3) ~ are smooth on ~ , i.e. the derivatives ~/ and ~/ extend continuously to ~ . I), 2), 3) are not yet sufficient conditions. We add , e i t h e r ~ ) =~= 0 or ~I(@)=t=0 . 4) For each ~ on ~ In [I] R.Blumenthal showed THEORE~ 1. I), 2), 3)

and 4) to~ether are sufficient for

Related results are due to J.-E.Bjork, [2], and to Sibony and the author, [3] • Condition 4) is, however, not necessary, since for instance

[(~_~)% ,(~_~)3]= A and conditions 1), 2), 3) hold here while 4) is not satisfied. The problem arises to give a condition that replaces 4) which is both necessary and sufficient f o r ~ . q ] ~ A . In the special case ~.__(~_~)3 this problem has been solved~'°~ by J.Jones in [4] and his result is the following: let ~ + and W - be the two subregions of clos~ whioh are

identified

by t h e map ( E - { ) 5

~*~

. Put

'I + ¢-X-C~ - ' / ) .

Then for ~ in W +, ~ lies in W and (~--I) 3 identifies and ~* . Let % be an inner function on W + whose only singularity is at ~ ° Then for some t , t > 0 ,

79

TH~ORE~ 2 ( [ 5 ] ) .

Let ~ be a f u n c t i o n in, A such that ~ = [ 2 - { ) ~

to~ethe= satis~.~ ~), ~). ~). Then IT, ~] ~ A

a~d ~

if for some %

i f and onl,y

of the form ~i),

(2) for all ~

__ in W +

, where

K

is some constant.

We propose two problems. PROBLEM I. Prove an analo6ue o~ Theore m 2 for the case when is an arbitrarEfunctionanalytic C~

~

in an open set which contains

bE findin~ a conditio n to replace (2)whac k t0getherwith

1), 2}, 3} ~neoessar~and sufficient ~or [ & ~ ] = ~ A • Furthermore, condition (2) implies that the Gleason distance from ~ to ~ , computed relative to the algebra [ & ~ ] , approaches O rapidly as ~ - - ~ , and so is inequivalent to the Gleason distance computed relative to the algebra ~ . Let ~ denote a closed subalgebra of ~ which separates the points of C ~ ~ and contains the constants. Let ~B denote the Gleason distance induced o n C ~ by ~ , i.e.

I1~1t-- I Let ~

denote the Gleason distance on 6~X>5~

PROBLEM 2. Assume that (a) The maximal ideal space of B (b) There exists a constant ~

Show that then

~

A

•

induced by A

is the disk 6~O~

, ~>0

, such that

.

80 RE~EEENCES 1. B I u m e n t h a I

R.

Holomorphically closed algebras of ana-

lytic functions. - Math.Scand.,1974, 34, 84-90. 2. B j o r k J.-E. Holomorphic convexity and analytic structures in Banach algebras. - Arkiv for 3. S i b o n y

N., W e r m e r

Trans.Amer.Math.Soc.~1974, 4. J o n e s

J.

9, 39-54.

Generators for

• -

J.

June, 1977. Subalgebras of the disk algebra. - Colloque

d'Analyse Harmonique

et Complexe, Univ.Narseille I, Marseille,

1977. J . ~

A(~)

194, 103-114.

Generators of the disc algebra (Dissertation),

Brown University, 5. W e r m e r

Mat.,1971, J.

Brown University Department of ~ath. Providence, R.I., 02912

USA

81

2.13.

A QUESTION INVOLVING ANALYTIC FAMILIES OF OPERATORS

Let A be a uniform algebra with Shilov boundary ~ . (The case when A is the disk algebra and ~ - - T is an interesting example for this purpose. ) Suppose we are given a linear operator which maps A into C ( ~ ) and has small norm. Suppose further that the image(I+S)(A)~ g ( ~ ) is a subalgebra (here I is the inclusion of

A

into

).

QUESTION: Is there an analytic family of linear operators ~ (~) defined for ~ for each %

i_~n ~

i_nn D

so that ( I + $ ( ~

and so that

(A) is a subal6ebra of C ( ~ )

~(~)=~

?

The hypotheses are related to questions of deformation of the structure of A , see [2] for details and examples. In cases where (~) can be obtained the differential analysis of ~ (~) connects the deformation !theory of A with the oohomology of A . (See [I]). For instance, ~ (0) would be a continuous derivation of the algebra A into the ~ -module C ( ~ ) / A • Such considerations lead rapidly to questions about operators on spaces such as V~lO(=C(T)/disk algebra). (See [3] for an example.) REFERENCES I, J o h n s o n B.E. Low Dimensional Cohomology of Banach Algebras. - Proc.Symp.Pure ~/ath. 38, 1982, part 2, 253-259. 2. R o c h b e r g R. Deformation of Uniform Algebras. - Proc. Lond.Math.Soc. (3) 1979, 39, 93-118. 3. R o c h b e r g R. A Hankel Type Operator Arising in Deformation Theory. - Proc.Symp.Pure Math. 35,1979, Part I, ¢57-458, RICHARD ROCHBERG

Washington University, Box 1146 St.Louis, M0 63130 USA

CHAPTER

3

PROBABILISTIC

The p r o b l e m s a s s e m b l e d i n t h i s

PROBLEMS

Chapter are of probabilistio

ori-

g i n b u t a r e more o r l e s s c l o s e l y c o n n e c t e d w i t h S p e c t r a l F u n c t i o n T h e o r y . Nowadays, p r o b a b i l i s t i c

methods a r e i n c r e a s i n g l y

applied in

Harmonic A n a l y s i s . Many such examples can be f o u n d , f o r i n s t a n c e , t h e book by J . - P . K a h a n e "Some random s e r i e s The t h e o r y o f s t a t i o n a r y

s u p p l i e s F u n c t i o n T h e o r y w i t h many

p r o b l e m s . T h i s does n o t e x h a u s t , o f c o u r s e , a l l

ons b e t w e e n t h e two t h e o r i e s . application

nianmotion

connecti-

R e c a l l , f o r example, t h e t r a d i t i o n a l

of Fourier integrals

distributions,

sical

of f u n c t i o n s " .

Gaussian processes b r i d g e s P r o b a b i l i t y

and l ~ a n c t i o n T h e o r y . M o r e o v e r i t interesting

in

to the investigation

of probability

o r t h e m a r t i n g a l e t h e o r y o f Hardy c l a s s e s ,

which i s a t r a d i t i o n a l

o r t h e Brow-

source of counterexamples in clas-

F o u r i e r a n a l y s i s a s w e l l as a p o w e r f u l t o o l f o r t h e s t u d y o f

b o u n d a r y p r o b l e m s . However, we would l i k e "spectral"

ideas arising

t o emphasize t h e c i r c l e

in the theory of stationary

of

Gaussian p r o c e s -

s e s ( s e e 3 . 1 , 3 . 2 , 3 . 3 , 3 . 4 b e l o w ) . These p r o c e s s e s a r e l i n k e d w i t h S p e c t r a l F u n c t i o n T h e o r y by t h e c o n c e p t o f f i l t e r tion operator applied this

~ ~

K*~

(i.e.

) . N.Wiener s y s t e m a t i c a l l y

concept, originating

the convolu-

and s u c c e s s f u l l y

in engineering, to various purely

m a t h e m a t i c a l p r o b l e m s . Almost any s t a t i o n a r y

G a u s s i a n p r o c e s s can be

considered as a r e s p o n s e of some filter to a "white noise".

All sta-

83

tistical

information

being contained

lem is reduced to the study of its

in the "white noise",

redistribution

the prob-

under the action

of a given filter. This is, maybe, one of reasons why Hard~ classes, entire functions, etc. - in other words almost all the tools used now in Function Theory - are so important for some purely probabilistic papers. The questions posed in 3.1 can hardly be considered as concrete problems. Rather, they indicate possible directions of investigation and the interested reader may consult the excellent book

[I~ (see

References in 3.1). In contradistinction t o Problem 3.1 the "old" l>roblem 3.2 contained a series of analytic questions of which one is solved (see the Commentary). The first part of Problem 3-7 has also been solved, whereas, all aspects of Problem 3.~ remain open. In this edition this Chapter has been enlarged by three Problems. Problem

3.3 deals with the Hilbert space geometry of Past and Futu-

re. The questions posed there concern also the theory of Hankel operators. Problem 3.5 outlines a new field for dilation theory in the theory of Markov processes.

Problem 3.4 deals with limit theorems~

We conclude by indicating some Problems from the remaining ("deterministic") part of the Collection.

In 8.4 approximation by trigo-

nometrical polynomials with bounded spectra is discussed. The probabilistic interpretation is well-know~ (see e.g. the above metnioned book by Dym and McKean,

~IJ in 3.1). Chapter 6 has some relations

with Probability as mentioned above~

Sarason's result cited as Theo-

rem I in 3,2 has very much in common with the contents of that Chap~ ter. The years since the first edition

have been marked by closer

and clearer connections among the elements of the triad "Punctlon Theory - Operator Theory - Probability".

This tendency (partly ref-

84 lected

papers

in the operator-theoretic

[5]3 , [9] L ~

t

item 3°5) is well illustrated

cited in the Com,,entary to Problem 3.2

by the

85 3. I. old

SOME QUESTIONS ABOUT HARDY FUNCTIONS

The theory of Gaussian-distrib,ated noise leads to a v~riety of substantial mathematical questions about Hardy functions. I will put the questions in a purely mathematical way; the reader is referred to [I] for the statistical interpretation and/or additional information. I. Let A , A ~ 0 , be summable on the line and let

~ X ~ i×~----~ .

Then the exponentials , but how is ~ x T

for fixed T > 0

e f f i c i e n % I Z

approximated (by these func-

tions)~ See [1], § 4.2.

2. ~et

~

,

~ ~

, be ~ter,

let T

, T >0

, be fi~ed, = d

let

~($) being the inverse transform ~

w.at ,c~. be s a i a about

aloe, ~

K

can be e n o = c ~ l y

3. Let ~

, ~6~

I1z

?

s~ar;

cannot be
for all small T

;

see Ell- ~4.4.

, be outer. The QUESTION is to explai n ~ t

makes ,the phase function ~ * / ~

the ratio" of two inner functions

or the reciprocal of an inner function; see [I], §4.6. 65~T~*/~ is itself an inner function if and only if ~ ral of exponential type ~<~ . 4. Let ~

,~

span, of ~i~$H °°, ~ 0

, be outer. When does ~ * / ~

, .-.in ~oo ? see El],

belon~ %o the

~4.12.

5. The following conditions are equivalent for outer ~

~ H~" ~ c~ ~ / ~ is H~ and a function of class H~

the ~tio ; b)

is integ-

,

e~ ~ ~ u n o t i o n o f c1~ss

;RI ~ l ~ l ~ m

< oo

for

86 of exponential type ~
some integral function ~

~

~

for some ~, ~ 0 , such function s ~

see [1], §4o13o ~ f

~6~;

cad be said about

? Note that b) is a problem of "multiplying down"

the function 4 / ~ in the style of [2]. What outer funct$cn satisfy a), b), c) for every T , T ~ 0 ? for no T , T > 0 ? Note that cazmot satisfy c) for T , T = O .

6. ~he phase ~-~ctlo~

~/~

is ublq~tous. What c ~ be ~ i d

abOUt it for the ~eneral outer f%mction ~

,~

~

?

REI~ENCES

I. D y m H., M c K e a n H.P. Gaussian Processes, l~anction Theory, and the Inverse Spectral Problem. New York, Academic Press, 1976. 2. B e u r 1 i n g A., M a 1 1 i a v i n P. On Fourier transforms of measures with compact support. - Acta Math. 1962, 107,

291-302. H.P.MCEEAN Co~t

New York University, Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012, USA

COMMENTARY In connection with section 4 see the COMMENTARY to problem 3.2. The question discussed in section 5 is related to the paper [3]. REFERENCE 3- K o o s i s

P. Weighted quadratic means of Hilbert transforms. -

Duke Math.J., 1971, 38, N 3,609-634.

87 SO~E ANALYTICAL PROBLEMS IN TI~ THEORY OF STATIONARY STOCHASTIC PROCESSES

3,2. old

I. Let ~ (~) be a stationary Gaussian process with discrete or continuous time (see [I] for the definitions of basic notions of stochastic processes used here). Denote by ~(~) the spectral density of ~ (in case of discrete time ~ is a non-negative integrable function on the unit circle ~ ; in case of continuous time is replaced by the real line ~ ). Let ~(1) be a Hilbert space of functions on T (or ~ ) with the inner product

For

~>-0 , let ~+~(~) (resp. [,-~(~) ) be the subspaces of (~) genetated by exponentials ~ t ~ with ~ >/~ (resp. ~.<-g ). Let ~ j and ~be the orthogonal projections onto ~+~(~)

and ~-~(1)

. Consider the operators

These positive selfadjoint operators were introduced into the theory of stochastic processes in [2]. Many characteristics of processes can be expressed in their terms. In particular, important classes of Gaussian processes correspond to the following conditions on ~ : a) B~ is compact for all (sufficiently large in case of continuous time) ~ ; b) ~ is nuclear for all (sufficiently large in case of continuous time) ~ . Since the finite-dimensional distributions of a Gaussian process are completely determined by its spectral density ~ , i t would be desirable to describe properties of ~ in terms of ~ . 2. Processes with discrete time. THEOREM I ( ~ ) . the spectral density

The operators ~

~g

are compact if and onl 2 if

can be reoresented in the form

88

Here

is a pol,Tnomial with roots on the unit circle and the fu,'~c-

P

tions

~

and ~

are continuous.

THEOREM 2 (I.A.Ibragimov, V.N.Solev, cf.[I] ). The operators B ~ belon~ to the trace class if and only if

I(x) : Here

~

IP(e ~x )1 ~ e°'(~),

stands for a polynomial with roots on T

~(~)

T ~,ie ~'ix

and

}]l~,il~lil <~

PROBLEM 1. Under what conditions on spectral density operators

~

belon~ to the class

T~

' ~ ~ ~ ~ co

~ do the

, i.e. for

what

~ ~,I~ < co, where

~i~ are ei6en-values of

~

?

Theorems I and 2 deal with the extreme cases

~=co

,I.

THEOREM 3 (l.A.Ibragimov, see [I]). The estimate

I:B I- 0 (% is an integer,

0 < o~ < ~

for

) holds if and only if

~(x) =lP(e~x) I~ e~ (x~ , where

P

tion and

is m polynomial, ~(~)

~ is an

-time differentiable func-

satisfies the Lipschitz condition of order

•he val~e 15~I

is expon~ntiall7 sm~ll for

9n12 if the spectral density

~

C

~

A

if and

is an analytic function.

3. Processes with continuous time.

•

Nothing similar to

89

theorem 3 is known in that case. PROBLEM 2. Under what conditions does the value with power or exponential rate as THEOREM 4 (I.A.Ibragimov,

~

IB~I

decrease

~+ co ?

[I~). Let

~ C~) = I~(~)I -z

is an entire function of exponential type with roots

E~

, where

El~..,

.

"~ -.--~+ oo

2. _co<~
~


PROBLEM 3. Investigate the case when .

an_~d ~

, ~

.

.

.

.

.

are entire functions0f

0~(~)= I~ (A)I~

exponential t~pe.

This problem is essential for the analysis of the operator ~ in the multivariate case ~4~. Note in conclusion that Problem I can be easily reformulated for continuous time.

RE PEREN CE S I. H 6 p a r ~ M o B M.A., P o S a H o B D.A. I~ayCCOBCE~e c ~ ~ e nposeccH, M., "Ha~-Ka", 1970. 2. r e x ~ ~ a H ~ E.M., H r x o M A.M. 0 B ~ H C X e H ~ Eox~eCTBa ~ m ~ o p M ~ O c ~ y ~ a ~ o ~ ~ y H E n ~ , co~epma~e~c~ B ~pyro~ TaEo~ ~ . --YcnexH MaTeM.HayE, I957, XH, I, 3-52. 3. S a r a s o n D. An addendum to Past and Future. - Math.Scand., 1972, 30, 62-64. 4. M 6 p a r ~ M o B E.A. 0 no~Eo~ peryxapHOCTH MHOroMepH~x cTan~oHapH~x npo~eccoB. - ~ o E x . A H CCCP, 1962, 162, ~ 5. I.A.IBRAGIMOV V.N, SOLEV

(H.A.HHPAI~0B) (B.H.C0~EB)

CCCP, I9IOII, ~ e H ~ H ~ a ~ , ~OHTaHEa 27, ~ 0 ~ AH CCCP

90

CON~ENTARY

Problem I has been solved by V.VoPeller in ~5S : B T e ~p

=1~1~'~ ~ where ~ the Besov class

is a p ol,ynomial with roots on T

if f ~ =

an__d_d~ belongs to

~l~,p ..~,p .

Things are more complicated in case of continuous time Let ~ be the outer function in ~+ satisfying ~ ~ I~I ~ on ~ and let AI~-~- ~ ( ~ } (U~0~-~" ~ ) Then it is easy to show that

I

I%1

tjI =

.H

stands for the usual Hardy algebra in

] ,

where ~@o

~T , and therefore a process

satisfies the strong mixing condition iff ~ / ~ ~ A I . The structure of the Douglas algebra A 4 in contrast with that of H~ C(T) is very complicated, This is the main reason of troubles arising in the investigation of the continuous time case~. Many results valid for processes with discrete time could be easily extended to processes with continuous time, if the following factorization were true

~=~-I, where ~ is an invertible unimodular function in A I and I is an inner function in ~ o Being valid in ~ + C , see ~ ] , this faotorization, unfortunately, does not hold in AI ~].. The factorization theorem in ~ + ~ can be applied to a description of ~-r e g u 1 a r G a u s s i a n p r o c e s s e s with continuous time, i.e. the processes with compact B ~ , ~ ~ > 0 . Let C denote the space of all continuous functions on ~

having a finite limit at infinity.

THEOREM (Hru~8~v - P e l l e r ) . A stationary. Gaussian proces..s. [X~} ~ 6 ~

with spec.tral measure

where $ ~ , v 6 C , that

I ~6

t~+~E~4

A

is

~

-regular iff

is an e.ntire function .of exponential, type such

91 PROOF (SKETCH). Let

~

be an outer function i n ~

satisfying

(see e~g. [9])~ which in turn by the factorization theore~ of T,Wolff is equivalent t o ~ 6 $ ~ - 6 ~(v~). B , where ~ ~ 6 ~ and is a Blaschke product in ~ . Consider an auxiliary outer function ~@ defined b~ ~ . I~o(~)1~-J-~{~(3) -- ~(~)}, ~ e ~ . Let~=~. Then clearly ~ 6 5 ~ = B , which implies that

is a restriction to ~ of an entire function of exponential type° Conversely, s ppose { w th entire function ~ of exponential t y p e Clearly, ~ belongs to the Cartwright class and it can be replaced by an entire function of exponential type being an outer function in C $ of the same modulus on ~ . It follows that on

Since ~ r is of exponential type, Therefore~ ~ ~ ~+ C . •

~V 6 ~

~@@

for some ~

0

Some sufficient conditions for the strong mixing were obtained in [8]. See also [9] for a brief introduction to the subject

REFERENCES 5. fI e ~ a e p B.B. 0nepaTop~ F a H K e ~ E ~ a c c a ~p H ~x npH~o~eHH2 (pa~HoHa~bHa~ a n n p o x c H M a ~ , rayccoBcK~e npo~eccH, n p o ~ e M a Ma~opa~HH onepaTopoB). - MaTeM.c6opH., 1980, I13, ~ 4; 588-581. 6. W o I f f T. Two algebras of bounded functions~ - Duke Math J , 1982, 49, N 2, 321-328, 7. S U n d b e r g C. A counterexample in H@°+ B ~ C - 1983 (May-Jtme), preprint. 8. H a y a s h i E. The spectral density of the strong mixing stationary Gaussian process. - 1981, preprint. 9. H e 2 ~ e p B.B., X p y A 6" B C.B. 0nepaTopM F a H K e ~ , H a ~ e H p H 5 2 ~ e H H 2 H CTa~HoHapHMe rayCCOBCEHe npo~ecou. - YcnexH MaTeM. Hays, 1982, 37, ~ I, 53-124.

92 MODULI OP HA/~KEL OPERATORS, PAST AND FUTURE

3.3.

A discrete, zero-mean, stationary Gaussian process is a sequence {X~}~e ~ in the real U~ space of a probability measure i), such that ~ X ~ O, i.e. depends only on ~ - - ~ ; every function in the linear span of the functions X~ has a Gaussian distribution. In prediction theory the Past is associated with the closed linear span (over ~ ) ~p of X k , ~ < 0 , and the Future with the span ~ of X k ,k>~O. These closed subspaces are usually considered as subspaces of the complex Hilbert space ~ spanned by the whole sequence { Xk ~ keZ • Our problem concerns the description of all possible positions in ~ of the Future ~ with respect to the Past ~p . The sequence { Q ( ~ ) ~ 6 Z being positive definite, there exists a finite positive Borel measure /~ on ~ satisfying ~ ( ~ ) ~ A ~(~), ~Z . The measure ~ is called t h e s p e c t r a 1 m e a s u r e of ~ X ~ } ~ 6 ~ . Clearly the mapping ~ defined by ~)X~l,-~-~ ~ , ~ 14~ ~ can be extended to a unitary operator from to ~(~) . To avoid technical difficulties we consider henceforth all stationary sequences I ~ e Z in ~ , not necessarily real. A stationary sequence is unitarily equivalent to a Gaussian process iff the spectral measure /~ is invariant under the transform ~ ~ of T. Consider the set of all triples (~ ,~, ~ ) where A and are closed subspaces of the complex separable infinite-dimensional Hilbert space ~ such that ~ 0 ~ (A + ~.] ~ ~ . The triples (~,~,~i) and (A$, ~ , ~ ) are said to be equivalent if there exist isometr V o to tis ying = V~ ~- ~ . Let ~ be the set of all equivalence c l a s s e s w i , h respect to the introduced equivalence relation. PROBLEM I. Which classes in ~ (~,

~

~)

contain at least one element

corresponding to a stationar,y sequence ~ X ~ } ~

The class ~ admits a more explicit description. Let ~A denote the orthogonal projection onto the subspace A . Each triple ~ ~-(A,~) defines the selfadjoint operator ~A ~ ~A and the numbers

?

93

L]~I:NA. % triple ~ = unitaril 2 equivalen~and

(A~,B i, ~ ) ~± (~) =

is equivalent to ~ = ~±(}~).

SKETCH OF THE PROOP. We may assume without loss of generality (note that ~ i = ~ under the assumption of the lemma). Given a subspace C in ~ let 6 I = ~@ C Consider the partial isometrics V~,Vz determined by the polar decompositions

Let ~

be an operator on ~

defined by

A nB

~I ~in~ 1 is an arbitrary unitary operator from onto A~N~I • It is easy to see that ~ is defined correctly (if ~ ~A~n B then V~ V ~ z ). Also clearly ~ maps isometrically A~ onto A~ • it remains to verify that ~ is a unitary operator on ~ . Clearly it suffices to show that (~'~,~)---~(~,}) for • sider only vectors Z from (I) that

, this is evident• Now we can conof the form ~---J~A E , ~ ~ ~ It follows i

Pot every ~ ~+ m {~J and f o r every s e l f a d j o i n t operator T on a Hilbert space T~ such that ~ T ~ I ~$~T~{~ there exists ~ (A, ~)~ ~ - ( ~ ) = ~ and such that ~B ~A ~B I ~ e A 1 satisfying ~+(~)=~ is unitarily equivalent to~ Indeed, without loss of generality ~ ± (~] ~ 0 By the wellknown Naimark theorem in ~i ~ ~4 there exists a projection ~A defined by

T

?{(z-T)}) ¢

T (I-

I- T

g4 PutB=~le

{@} a

then ~ and c ~p ~ ~

s/Id

~

and ~ = 6 6 6 ~

ternative

e

ther

(A+B)

. Then clearly

%=

or C, where ~ is an outer function in is a singular measure on ~ o We have cP'~(h~(~o) and therefore only the case ~ ~ 0 is interes-

ting. Remall that for a bounded function %0 on ~ the Hankel opera, where ~+ is tor H~ on ~ is defined by ~ ~ - C I- P÷) ~ H ~ the orthogonal projection from ~ onto ii with the unimodular symConsider the Hankel operator r1~,/h' bol ~---6/~ • It is easy to see that ~G~ ~ p ~ @ ~ I ~$ is unitarily equivalent to ~%1% H~/~ (see [I], Lemma 2.6). The modulus of an operator ~ on Hilbert space is the selfadjoint non-negative operator (T*~)4/~ . Problem I is therefore intimately connected with the problem of description of the moduli of Hankel operators up to the unitary equivalence. PROBLEM 2. Which operator

can be the modulus of a Hankel ooera-

tor? There are two n e c e s s a r y c o n d i t i o n s for an operator to be the modulus of a Hankel operator which imply evident restrictions on triples equivalent to (~p, ~ ~). For any ~ the operator (H~ ~ /~ iLS not invert ible because ~ II~ ~II = 0 It follows that ~ does contain an orthogonal basis { 8~ } ~ o with ~ 6~ ~ = 0 (i.e. an orthogonal sequence "almost independent" with respect to the Past) provided Gp =9= $& . • ~/~ The kernel ~e$(H~ ~ ) is either trivial or infinite dime n.s.ional. Indeed, being invariant under multiplication by ~ by Beurling's theorem, it is either trivial or equal to ~H ~ for am inner function 0. Note that for ~ === (~p, ~ CT) we always have ~(~)-~--~-[~). Indeed passing to the spectral representation ~ we see that Y: i ~ * ~ is an isometry of ~ (over ~ ) with ~ @p=== G~ Y~= ~ . SO we have either ~÷C"~) ~ - ( ~ ) ~--- 0 or ~+(~)~-~_(~)=~ . Under the a priori assumption that the angle between ~p and ~ is positive Problem I can be in fact reduced to Problem 2. Indeed, if the angle between ~rp and ~ is positive then by the Helson-Szeg~ theorem the spectral measure ~ is absolutely con-

95

tinuous and ~ =

I

~&

I

~@~

is unitarily equivalent to *

H~~

H~/k~t HK/~, ..........

• On the other hand if ~ = and II~ ~ < then there exists an outer function ~ in ~ with H ~ - H ~ / ~ (see [2]). Put ~==[,~(I~l~ , B = :~pa~,L~(t~,l~) ~ 2 ~ : ~ 0 } , A~__~h:(I~I:)I ~ : ~ < o} . Clearly [ ~ is a stationary sequence with the Puture B and the Past A . It follows from the above considerations that in the case of nonzero angle between A and ~ the problem of the existence of a stationary sequence with the ~uture ~ and the Past A can be reduced to the existence of a Har~kel operator whose modulus is unitarily equivalent to ~ B ~A ~B • In connection with Problems I and 2 we can propose two conjectures. CONJECTURE I. Let

~ ~ }~ o

positive numbers and let

be a non-increasin~ sequence of

~,~ ~ =

O.

operator whose singular numbers ~) ~ ( ~

Q~cH~) =

CONJECTURE 2. Let that

~

T

,

~

Then there exists a Hankel )

satisf,y

0

be a compact selfad~oint operator such

~eg T

is either trivial or infinite dimensional. Then there * )~ exists a Hankel opera to r U @ satisfyina T ~ (~ ~ •

It can be shown that the last conjecture is equivalent to the following one. CONJECTURE 2'. Given a triPl e ~ (~,B,~)6 ~ such that ~B J~A

is compact

and

~+(~)-~- ~-(~)

there exists a stationary sequence Future is

B

and Past is

~ X~ } ~ 6 ~

is either 0 i__nn ~

o_~r~

whose

~.

We can also propose the following qualitative version of Conjecture I. THEOREm. Let

$> 0

and ~ ~ } ~

o

be a non-increasir~ sequence

of positive numbers. Then there exists a Hankel operator H ~ fyin~ 4

*) See def. of singular numbers in [3~,

satis-

08 PROOE. Let ~ be an interpolating Blaschke product having zeros ~} ~o with the Carleson constant ~ (see e.g, ~ ) . Consider the Hankel operators of the form ~i~ , ~ H ~ Then we have (see [4], ~h. VIII)

where ~ is the compression of the shift operator ~ to ~ d¢__j~ = H~e~, ~ ~PG~ ~ 6 ~ , P~ is the orthogonal projection onto ~ , ~ is multiplication by ~, ~ ( ~ ) &e~ 2~ ~ # ~ ~ . Since ~ is an interpolating Blaschke product, there exists a function ~ in H @0 satisfying $ ( ~ ) ~ ~ 1,1,~0. It follows from (~) that ~ ( ~ ) ~ ~(~(~)), ~0. Consider the ~ectors6 ~ . t I~ (~ IZ;-I~)~t~" We have~(~. )6~,.--~(~ )e_,~ (see L~J, Ch. VI) and there exists an invertible operator V on ~6 such that the sequence I V¢~} ~ 0 is an orthogonal basis of 6 , moreover if ~-- I is small enough then we can choose V so that I]V If' IIV411~ ~ +8 (see ~], Ch. VII). The result follows from t h e obvious estimates

¢

NvlI.Hv"II

llv II. llv"ll

•

Conjecture I can ~e interpreted in terms of rational approximation. It follows from the theorems of Nehari and Adamian-Arov-Krein (see [I]) that for a function ~ in ~ 0 A ~P# ~$ I.°° we have

~ ^ where ~@ is the operator on with the matrix{~(~+~)}~,k$ 0 , ~ is the set of rational functions with at most ~ poles outside ~0~ ~ (including possible poles at co ) counting multiplicities, Conjecture S is equivalent to the following one. CONJECTURE I'. Let ~4~%~-0

I~}¢~0

. Then there exists

~

be a non-increasin~ s.e.quence, in B ~ O A

such that ~ ( S ) ~

~,

~0. If the conjecture is true then it would give an analogue of the well-known Bernstein theorem [5] for polynomial approximation. Note

97 in this connection that Jackson-Bernstein type theorems for rational approximation in the norm B ~ 0 ~ were obtained in ~ ] , K7], ~8~. We are grateful to T.Wolff for valuable discussions.

REPERENCES

I. H • z ~ e p B.B., X p y ~ ~ B C.B. 0 n e p a T o p ~ r a H x e ~ , ~ a ~ y ~ e mpM6xHxeHM2 M CTaUMoHapHMe rayccoBcKMe npoKeccN. - Yc~exH MaTeM. Hays, 1982, 37, ~ I, 53-124. 2. A ~ a M ~ H B.g., A p o B ~.3., K p e ~ H g.F. BecKoHe~H~e ra~KeaeBH m a T p M ~ H O606~eHH~le 8 a ~ a ~ KapaTeo~op~-~e~epa ~ M.mypa. ~y~K:~. aHa.~. M ero npH~., 1968, 2, ~ 4, 1-17. 3. F o x 6 e p r H.L~., h p e ~ H M.F. BBeAeHHe B TeopMI0 XHHeF~-Ib~X HecaMoconp~e~ onepaTopoB. M., "HayKa", 1965. 4. H M K o x ~ c x ~ ~ H.K. ~ l e ~ o6 oHepaTope C~B~Pa, ~., "HayKa", 1980. 5. B e p H m T • ~ H C.H. 06 06paTHO~ sa~aue ~eop~H Ha~nnyumero ~p~6x ~ m e ~ Henpep~H~x ¢ y ~ . - C o 6 p a ~ e c o ~ H . , T.2, I4~-BO AH

CCCP, I954, 292-294. 6. I I e s x e p B.B. 0Hepa~op~ I~a~ex~ ~ a c c a $'~ , ~ Hx npv~omeH~ (pa~MoHax~Ha~ a n n p o x c ~ a ~ , PayccoBcK~e r~po~eccN, npo6xema ~a~o p a ~ , onepa~opoB).-ga~eM, c 6 o p ~ , I980, I I 3 , ~ 4, 5 3 8 - 5 8 I . 7. P e I I e r V.V. Hankel operators of the S c h a t t e n - v o n Neumann class % , 0~ ~< ~ . - LO~! Preprints, E-6-82, Leningrad, 1982. 8. S e m m e s Preprint,

S. Trace ideal criteria for Hankel operators,

1982.

VVoo

S. V. HRUSCEV

(C.B.XPm R) V. V. PELLER

(B. B.nF.JL'IEP)

CCCP, 191011, JleHHHrpa~ ~OHTaHI
0~p~.

98 SOME PROBLF~S RELATED TO THE STRONG LAW OE LARGE

3.4.

N~ERS

nary

F~

STATIONARY PROCESSES

Let in the (~E : E~Z) beige pro~ess and in 11='11: IL~(n'~'P)=d2>O. wide sense with ._~---L.o~, ~ Denote the correlation function of the process by

ff,

static-

=

and let

be its spectral representation. Here Z ( ~ A ) stands for the stochastic spectral measure of the process ( ~ ) ; Z(~) is a process with orthogonal increments. It is well-known that the strong law of large numbers (h e r ea f t e r a b b r e v i a t e d a s SLLN) holds for all processes stationary in the strict sense, that is, the limit of the means , . ~ = ~ 4 ~:~.g~ exists a.e. But there exist processes stationary in the wide sense such that the means ~ converge in L~(I~) and diverge a.e. (see[1],[2]). SLLN criteria are given in [2]. THEOREM (Gaposhkin [~ ). In the above notation

(1)

Thus SLLN holds iff the limit

111,"'oo I -.$ Z(d,?,)

exists a.e.

(2)

99

T~e theorem implies the following: if

R.(~)=o((.to~ bqM,) then SLLN holds provided ~ > 0 hold in general. In all knov~ counterexamples

) ~,_oo

, while for

(.3)

6=0

it does not

E I~,lP=o0 (P>~,) PROBLEM I. Is the condition

P

]p>2: s,~pIF I~1
~(~)=

0

) sufficient

for the SLLN? PROBLEM 2. Is the condition

I1~11oo< oo (may be with the above supplementary c qndition~ sufficient for the

SLI~? PROBLEM 3. If the a n s w e r s t o

problems 1 and 2 ~re negative,

we ma 2 ask: are there stationar2 processes

(~)

satisfyin~

while SLLN does not hold? 0r .............condition ..... (~) can be relaxed for oo L -bouuded processes? All processes stationary in the strict sense obey SLLN, and so the Theorem implies the existence of limit (2) as well.

100 PROBLEM 4.

W h ,7

does limit (2) exist for stationary (in the

strict sense~ processes? Analogous problems are of interest not only for unitary operators determining stationary processes but also for normal operators in ~ ( X ) (see an ergodic theorem of this kind in [3]). one of possible problems in this direction.

mo~ ~eln~ a

5. ~et T

be a no=al o~erato~ ~= ~

Here is

(×, 9)

~-fAnite measure. Su~ose ~TII=~, ~e~(X), ~

IT

(~)I ~
Does

exist a.e.? REFERENCES 1. B 1 a n c - L a p i e r r e loi forte des gran~nombres. 740-743. 2o r a n o m E ~ H

B.~.

A., T o r t r a t A. Sur la - C.r.Acad.sci. Paris, 1968, 267 A,

KpHTep~

ycHae~oro

8aKoHa 6oa~mmx ~ c e a

EaaccoB CTan~oHapHNx B mHpoEoM CMNc~e npo~eccoB ~ o~Hopo~m~x

cay~a~HHX noae~ - Teop.BepOaTH. ~ ee npzM., I977, 22, ~ 2, 295-319. 3. r a n o m E ~ H B.~. 06 m % E ~ y a a ~ H o ~ ~ p z D ~ q e c K o ~ TeopeMe HOpMaa~m~X onepaTopoB B I 2 . - ~a~m/.a~aa~8 ~ ero n p ~ . , 198I, I5, ~ I, I8-22. V. F. GAPOSHKIN

CCCP, 103055, MocEBa

(B.~.rAnommm)

y~.06pasnoBa, 15, M O C E O B C E ~ HHCT~TyT m~zeHepoB zeaesHo~opo~soDo TpaHcnopTa

101 THE THEORY OP NARKOV PROCESSES FROM THE STANDPOINT OF THE THEORY OF CONTRACTIONS

3.5-

Let (X,~) be a Lebesgue space. A contraction P on ~ (X,~) is called a M a r k o v o p e r a t o r if P is order-positive and preserves the constants. In other words ~ is Narkov if = , P = and P positive. The integral representation of such an operator is given by a bistochastic measure

Marker operators form a convex semigroup with a zero (=projection onto the constants) and wlth a unit. This is a functional equivalent of the semigroup of multivalued maps, admitting an invariant measure, of (X,~) onto itself. A detailed account of an analogous view point see in [1]. A ~ r k o v operator gives rise in a natural way to a stationary Markov process with the state space X , the initial measure and the two-dimensional distribution ~ (see above). In the space 2 = ~ X~ measure

~X~-~) M~= ~

of realizstions of the process a ~ r k o v

appears. The left shift

a unitary operator UT on (non-minimal in general). The main problem

~(~,M)

T

~ .

(3, ~)

generates

of the theory of Markov processes is the in-

vestigation of metric properties of the shift ~arkov operator

in

, a unitary dilation of

T

in terms of the

The classical theory virtually used spectral

properties of ~ only. This is insufficient for metric problems, being nonselfadJoint. Modern tools of the theory of contractions seem not have been used for this aim and we want to draw attention to this point (see also [1] ). The connection between the contractions theory, their dilations, the scattering theory on the one hand and N~rkcv processes theory on the other can be usefully applied in both directions. I. PROBLEMS ABOUT PAST. It is easy to check that a Markov process is forward (back) mixing in the sense of Kolmogorov iff

P

be-

102

longs to the class ~0. (resp. C.o), see notation in [2]. The opposite class C4 includes two subcases. The first one is of no interest and corresponds to an isometric P and to deterministic processes. The second one, namely, the case of a completely non-isometric contraction, is very interesting.Its very existence is far from being obvious (for Markov operators), an example was given by M.Rosenblatt [3]. An important theorem (see [2]) asserts that the corresponding process~being non-deterministic~is quasisimilar to a diterministic one. Our PROBLE~ is as follows: ~ e the technique of the theory of contractions to study mixing k%nds, d e t e ~ i s t i c bernoull!ty etc.

criteria of various

and quasideterministic , the exactness ~1], th e

A powerful tool for these topics is the characte-

ristic function of a Markov contraction. No adequate metric analogue of this notion seems to be found (e.g. how can one connect this function with the bistochastic measure ~ ) 2. NON-LINEAR DILATIONS. The theory of Markov processes imlicitly includes some constructions unfamiliar in the theory of contractions. We have already mentioned that a unitary operator UT acting in ~ ) is not the minimal dilation of the Markov operator . The minimal dilation can be easily described in these terms. It coincides with the restriction of UT to ~ = $ ~ { ~ ) : ~

~

,

~g(~i)

being the subspace of

~,~)

consis-

ting of functions depending on the ~ -th coordinate of { ~ I ~ only. The subspace ~ is the subspace of all linear function~ls of realizations ("one-particle" subspace). Thus the theory of minimal dilations corresponds to the linear theory of ~ r k o v processes whereas the dilation UT has to be interpreted as a "non-linear" one (clearly O T is a linear operator acting on non-linear function~ls of realizations). The investigation of the pair (P, UT) ("a Markov operator plus a non-linear dilation") is of interest for the theory of contractions connecting it with methods and notions of the metric theory of processes (mixing, bernoullity etc.) E.g. the problem of the isomorphism of two Markov processes is analogous with the problem of existence of the wave operator in scattering theory. The enthropy yields an invariant of the dilation etc. It would be interesting to define the non-linear dilation for an arbitrary (non-positive) contraction.

103

3. C*-ALGEBRA GENERATED BY MARKOV OPERATORS. Let us mention a more special problem: to deacribe the C~-envelopeof the set o T a! ~ Markov ope~tgrs.

This algebra does not coincide with the al-

gebra of all operators. (G.Lozanovsky gave a nice (unpublished) example: the distance between the Fourier transform as an operator in ~£(~) and the set of all regular operators (= differences of positive operators) is one). It seems likely that a direct description ef elements of this algebra can be given in terms of the order. This C * -algebra plays an important r61e in the theory of gruppoids. REFERENCES I.

B e p m E K A.M. I~orosHa~H~e O T o 6 p a ~ e H ~ c HHBapMSHTHO~ Mepo~ (nomm~p~s~) ~ ~ p E O B C K H e onepaTopH. - 8 a n . H ~ . c e M m ~ . ~ 0 ~ , 1977, 72, 26-61. t 2. Sz.-N a g y B., F o i a ~ C. Analyse h~rmonique des operateurs de l'espace de Hilbert. Budapest, Acad.Kiado ~, 1967. 3. R o s e n b I a t t M~tationary Markov Processes. Berlin, 1971. A. M. VERSHIK

(A.M.BEF~)

CCCP, 198904, HeTpo~Bope~, MaTemaT~Eo-Mexasm~e CEm~ ~aKyaBTeT JlePmHl?pa~oEoPo yH~BepO~TeTa

104 3.6. old

EXISTENCE OF MEASURES WITH GIVEN PROJECTIONS

Let F denote a measurable subset of the unit cube ~ c [Pv S~K being the canonical operator of projection onto the k-th axis, k ~ ~ , . . . , ~ , and X K the side of ~ situated on the k-th axis. Consider the linear operator ~ transforming every finite measure ~ on F into the system (M~ 4 , ~ , . . . , ~ ) of its margi~als ,

A

~ A),

of importance to know

A =XK

•

It is often

whether a ~iven s~stem ( ~ , . . .

longs to the 'image under

~

,~)

be_-

of a natural class of probability mea-

sures on F • some partial results are known, mostly for ~ - ~ . E.g. for a class ~ of subsets F of ~ defined in terms of measure spaces (Xk, ~ K ) , k-~ ~, 2 , an existence criterion of a probability measure on F with marginals ($1~ , m ~ ) is as follows: the required measure exists iff no subset of ~ x X~ of the form

F rl ( A x B) , where A c X ~ , B = X ~ a~d ~ A + m~ B > 4 , is a union of subsets N~, N2 with ~ ( ~ N I ) = ~ ( % ~ N 2 ) -- 0 (cecil] ~ a smaller class of closed set was considered in [2]). The class ~ is in particular characterized by the following property: for every F ~ ~ the set of all measures on F with given marginals is compact in the topology of convergence on sets of the form A× ~ . A similar condition (where the non-decomposability of sets F D ( A x B) is replaced by ~ 6 S ~ ( F Q ( A x ~)) > 0 ) is a criterion of existence of a probability measure on F with marginals ( ~4, M$~ ) subordinated to the Lebesgue type (i.e. absolutely continuous with respect to the Lebesgue measure M~65~ ),

[1]. Analogous conditions fail to be sufficient for ~ > ~ , and the corresponding criterion is unknown. For ~--~ it is not known whether the Lebesgue type can be replaced by any other type in the last sentence of the previous paragraph. For ~ > ~ there is no existence criterion for a positive measure on the cube ~ with given marginals whose density function with respect to Lebesgue measure is majorized by the density function of a given probability measure on ~ (see the discussion in ~I] ). REI~ERENCES I.

Cy~aEOB

B.H.

reoMeTpEqecEEe n p o 6 ~ e ~

Teop~J4 6eCEOHe~Ho--

105

2.

Mepm~xBepoaTHocTH~xpacnpe~eaemm~. - Tpy%M MMAH, 141, M.-Z., HayEa, 1976. (Proc. of the Steklov Inst. of Math., 1979, issue 2). S t r a s s e n V. Probability measures with given marginals. - Ann.Math.Stat., 1965, 36, N 2, 423-439. V.N. SUDAKOV (B.H.C~0B)

CCCP, 191011, JIeHHHI~8~, • OHTaHEa 27, .E01v~

COMMENTARY BY THE AUTHOR Consider a finite or countable family of probability distributionsI~K,kEK} The answer to the question as to whether there exists such a familyI~K, K E K I of random variables, each ~K being distributed according to ~ K , that for every pair (~i~ k~) the equality

holds, depends on existance of a probability measure ~ ~ on the set

Here ~ is the Kantorovich distance and potential function (see e.g. D]):

UKil(2,

with marginals

stands for related

One o~l show that for such a special type of subsets there always exists a measure with given marginals ~ WSK, K ~ K } , so that the family { ~K} under discussion does exist. I am grateful to B.I.Berg for stimulating discussions

106

ON THE FO~T~.R T~L~NSFOP~ OT THE INDICATOR OF A SET IN ~ OP P ~ I T E ~]~SGEE MEAS~J~

3.7.

old

Consider a set E C ~

o< 1 EI <

of finite and non-zero Lebesgue measure:

. The f u n c t i o n %E' ~(E= 1 ~; X~x@~ ~

indicator of E

. Set

^

! ~

the ~ourier transform of ~E 0 "< I EI ~ ~

is called the

such that

~E

. We ask whether there is a set E van lxhes on an

open non-empty

,

set

A , A

Note that if E is bounded then ~ E is analytic in ~ $ and therefore cannot vanish on an open non-empty set. Some other similar cases are considered in the author's paper [I]. If it turns out that there exists such a se% ~ , 0 ~ I ~ ~< co with ~H vanishing on a n open non-empty set, the SECOND PROBLEM will be to describe all sets E with this property. These questions are related to the uniqueness problem for a finite Betel measure ~ (in ~ ) with prescribed values ~ ( ~ o + ~ ) (~e~) , B@ being a given Borel set, 0<JE@J < ° ° . It follows A from [1] that if there is no open non-empty ~ c ~ ~ with~IA~ 0, then such a ~ is unique. And, conversely, it is not unique provided does exist. REFERENCE 1. C a ~ o ~ o B H.A. 0d o ~ o ~ npodxeMe e ~ C T B e a a o c T x ~ ~oHe~HHX Mep B e ~ E y o m H x npOCTpaHcTmaX. - - 3 a n . a a y q B . c e ~ . ~ 0 ~ , I974, 4I, 3-I3.

N.A. SAPOGOV

CCCP, 191011, 2 e ~ I ~

(H.A. CAIIOIDB)

~osTaKKa 27, 3 0 ~

107

COMMENTARY The first problem has been solved by Kargayev E2]. Sets E , A Ec~, with 0 < I E J < o o and ~ E vanishing on an interval DO EXIST. Moreover, it is shown in E ~ that given numbers ~,~, 0 < @ ( ~ < ~ , and an even function k: ~-->-EJ,$oo) increasing on ~0,+ @o) and such that

too

there i,, a se~.~no~ { r~,,,

~'.*~',,,]

0@

} ~,--I

of disjoint seg-

ments ~ t i s f y l ~

^

and

~,EI(~,,6)~O,

F 4~ ,a6,E U REPERENCE

2. E a p r a e ~

H.H. Hpec~pasoBan~e Cyp~e xapaavepHcTzqec~ol ~ f 2 ~ n ~ ~oxecTPa, xcqesam~ee ~a ~ T e p ~ e . - MaTeM. C6., 1982, II7, • 3, 897-4II.

CHAPTER

4

OPRR ATOR THEORY

Due t o f o u n d e r s

rator"

of the Spectral

became a l m o s t i n s e p a r a b l e

connection

was s o t i g h t

that

Operator

Theor~ the word "ope-

from the word "eelfadjoint".

e v e n now i t

is

still

This

cowmon t o s p e a k o n

"NON-selfadjoint operators" as though forgiving general operators for the absence of appreciable intrinsic struotttre. This kind of inferiority complex is being overcome nowadays under pressure from Physics (recall complex poles of resolvent on "the non-phTsioal sheet" in the resonance scattering) and under the influence of the increasing power of Analysis. "Analysis" means here mainly " C ~ l e x

A.AlyslS",

and the above tendency may be well illustrated,ln partlonlar, by operator-theoretic problems of this book (first of all problems in Chapters 4, 5, 7). Almost all of them are ~elated to the spectral theory tending to blend with Complex Analysis and, in any case ,to borrow from it

solething

more significant

Lio~ville

and Stone-Weierstrass

classical

approach.

theorems,

t~

Cauch~ f o r ~ l a

tools

This blend is probably

not exceeded by the

the most characteristic

leafy.re of the present-day

theory,

or at least

this

of this

mutual penetration

b o o k . ~he f i r s t

thirties

and forties

~he new s p e c t r a l tional

steps (Stone,

theory

Wold, P l e s s n e r ,

or

of its

I,Erein,

parts

close

to

w e r e made i n Livshic).

begins with working out convenient

func-

models whereas in classical analysis such a model was often

109

t h e end o f i n v e s t i g a t i o n . to the multiplication ssions

to suitable

Many p r o b l e m s i n t h i s C h a p t e r a r e r e l a t e d

operator ~-~#

whose r e s t r i c t i o n s

and compre-

s u b s p a c e s y i e l d models we have J u s t m e n t i o n e d .

One o f t h e most p o p u l a r models c a n be d e s c r i b e d i n t e r m s o f t h e s o called characteristic -

F o i a ~ model and i t s

f u n c t i o n of the o p e r a t o r (the Sz~'kefalvi-Nagy generalizations).

problems to the i n v e s t i g a t i o n

T h i s model r e d u c e s s p e c t r a l

of boundary properties

functions of the Nevanlinua class.

of vector-valued

The q u e s t i o n s p o s e d i n Problems

4.8-4.20 exhibit distinctly enough the present state of affairs which oanbe

s u w n a r i z e d as f o l l o w s .

Almost a l l

achievements of the

(free interpolation, functions,

H

Pt h e o r y have

the delicate ~tltiplicative

c o r o n a Theorem e t c . )

T h e o r y i s n e e d e d now. I t s

been e x p l o i t e d

structure

of

H P-

and a new " o p e r a t o r - v a l u e d " F u n c t i o n

contents are essentially

this fact is oftendisguisedbyformally

n o n - s c a l a r though

dimension-invariant state-

m e n t s . T h e r e f o r e t h e new p r o g r e s s r e q u i r e s n o t o n l y new e f f o r t s the spirit

of the standard

of special non-co~tative

HLt h e o r y ,

intuition.

but rather creating a kind

V i v i d examples a r e Problem 4.12

and t h e Halmos-Lax t h e o r e m d e s c r i b i n g i n v a r i a n t tiple

in

subspaces o f the mul-

shift. This theorem is deciphered (for a very special situation)

in Problem 4.14. As to Problem 4.12, its seeming simplicity conceals many interesting concrete realizations. Namely, Problem 4.12 includes as partlcular cases the principal question of Problem 4~I0 and the matrix generalization of the Corona Problem discussed in the Commentary. Many problems of the "vector-valued" function theory admit interesting scalar interpretations (as, for instance, Problems 4,4, 4.9, 4.12). Other problems make sense only when the space of values is multidimensional. I% is not easy to classify rapidly growing Operator Theory with its intertwining ramifications. The same is true even for its parts presented here. However, we tried to group the problems in

110

accordance with the intrinsic logic of the subject which we understand approximately as follows. Selfadjoint

f

S p e c t r a l Problems

1

orlginatlng in Differential E q u a t l o n s ~

Schauaer ~,ory I

s~l

~eorem I ~ h e o r y

_~ I I

I

~eory

I

__~_

GENERALIZATIONS

0ono'e'e i I

I

I I I I I I I l I

C H A P T E R

BanaOh Algebra

II

5

LModels

Integral O p e r a t o r s

"

1

with Kernels having

Operator-valued

a k i n d o f Symmetry Hetho4s:

Analytlc (Hankel, T o e p l i t z ,

C*-Algebras,

l~tions,

Hiener-Hopf, Singular Representations

I

their £nte~el

Commutators,

I 1 I I !

Operators)

~Lltiplicative

Calculi

~

Structure

pectral

Theories

module

an

Ideal,

Systems o f a l m o e t Commuting O p e r a t o r s

L

Of course, the scheme leaves aside a greet number of links existing i n O p e r a t o r Theory as w e l l as c o n n e c t i o n s w i t h o t h e r b r a n c h e s o f Mathematics. But t h e B i n

p u r p o s e o f t h e scheme i s t o e x p l a i n t h e

f o l l o w i n g arrangement o f Problems i n s i d e t h e C h a p t e r . I. Selfadjeint

spectral

theory, including Perturbation

Theory

111 and Scattering

~S:teozT~and

problems they

generate

(NN 4.1-4.8, 4.15).

This ~romp represents so well-known domains of Operator Theory that we do not risk commenting It. II. Punctlonal Models, Characteristic Function and other operator analytic functions (4.8-4.20, 4.4). Being of quite different origins, these Problems mostly can be reduced to %he investigation of the multiplicative structure of operator-f~nctions and they need certain development of operator (or matrix) analogues of

HP-tech-

nlques. Problems 4.12, 4.15, 4.20, 4.21, 4.24 are directly related to the themes of the next Chapter 5. IIl. Banach Algebra Hethods are a common feature of Problems 4.22-4.29 and they also play an important role in Problems 4.30-4.36. We mean here various aspects of "algebraizatlon" as an alternative approach to generallsatlons of the Spectral Theorem and to interpretations of spectral nature of non-commutlng objects. These aspects are C*-algebras, calculi, symbols, theories h la Fredholm-RieszSchauder etc. IV. "Near-normal" (i.e. h~po-, semi-, sub-, quasi-,...nQrmal) operators as a particular case of the perturbation theory for families

( . A t , . . . , A I~)

case ~ = ~

of commuting selfad~oint operators

(namely,

the

). Such operators are the subject of problems 4.30-4.36

(and 4.37 - from the technical point of view). An analogy with the classical Perturbation Theory may be drawn as follows: the classical smallness of "the ima~Insry part"

~

A = ( A - f ) / 3 ~ is replaced here

by the .al~ess of the selfoo=utator [ A f] : A A~ the aocretlvity condition ~ A

f A

..<0 by the ~pono=~llty

[A

A].
etc. The new situation is In a sense "two-dimansional", the operator

A b e i ~ viewed as a pair CP~ A, ~

A) . This leads to essential

complications in compemlson with the "one-dimensional" (i.e

the

selfadOolnt) case. Some quite simple "one-dimensional" problems look rather difficult in the "%we-dimensional" settle, as for example

112

the stability of continuous spectrum or the solvability of equation J~A~

for K e ~ ( ~

being a given operator ideal). These problems

likely require a hard analysis of operator algebras and

finding

out of algebraic obstacles to solve the equation [ A , A ~ E ~

in the

form A = N + ~

where[NjN~]~-0, ~ .

Problems of this direction are rep-

resented rather distinctly by items 4.30-4.38. Problems 4.39, 4.40 deal with interesting concrete questions (circular symmetry of spectra of endomorphisms, quasinilpotency of indefinite integration) which can hardly be placed at a definite point of our scheme

On the other

hand, parts of Operator Theory indicated in the scheme, are included in Chapters 5 and 7. Let us enlist some close cross-links between problems of this Chapter and of other Chapters. Chapter 2 (Banach Algebras): Problem 2.13 could be placed among items 4.8-4~20; Problems 2.1, 2.4 are related to 4.39. ~inally, Problems 2.2 and 2-3 deal with a subtle behaviour of the norms

~II~I,

and therefore find a responce in 4.38. Chapter 9 (Uniqueness). Problem 9.4 admits a clear operatortheoretic interpretation in the spirit of Problems 4.31-4.36, and it is recommended to read Problem 9.6 and Problem 4~4 simultaneously. Numerous problems on multipliers dispersed throughout the volume are related by their common origins and intrinsic connections (4.21, 4°25, 4.26, 2.6, 10.3, 9.9, I0.8). Unfortunately (or fortunately - taking into account inevitable volume restrictions) many parts of Operator Theory are not mentioned here. Por example, the invariant subspace problem is not presented explicitly. But it is alluded to, incidentally, in constructions of items 4.8, 4.13, 4°22 whereas other problems of this "descriptive" direction of Operator Theory have to wait for another Problem Book. The same fate (i.e~complete oblivion) is shared by many consumers of Operator Theory. The reader will find in our Book neither "pseudodifferential operator", nor "integral Fourier operator", nor "operator -theory" ....

113 BOUNDEDNESS OF CONTINUUM EIGENFUNCTIONS AND THEIR

4.1.

RELATION TO SPECTRAL PROBLEMS We will describe a set of problems for matrices acting on ~ ) . There are analogous problems for ~ ( 2 v) and for suitable elliptic

ope~tors on L~CR~

.

Let A

be a bounded self~djoint

ope~to~

on ~ ( Z ) whose matrix elements obey 6~ ~- C ~ A~j) ~-0 if I $-j I~ ~ . A fundamental result asserts the existence of a measure ~p(E) , a f~otion #(E) taking the values O, t , . . . , c o (infinity allowed)with }~(E)~{ (~)-a.e. E ~nd #(E)=O if E~p and for each E , Y~(E) l i n e a r l y independent sequences ~ ( E ; ~) ; ~ = 4 ~ . . . ~ ( g ) (not necessarily in ~ ) so that

(a) I%(E;~)t%c(t+t~l) (C) Let ~E)

=

E;

(b) ~ j ~ ( E ; j ) = E % ( E ; ;

having values in

note sequences in ~ ) into ~ i by (U~)@ { E ) =

a unitary ~ p of ~(Z)

i.e. functions, ~ CE)

(where C °O= ~

0 ; , on

£

with

) and let ~

de-

of compact support. Define U taking C0 ~ ~-~ET ~i ~(tr~). Then ~ extends to

o~to ~ ;

(d)

U(Ao)=ECU~)

These continuum eigenfunction expansions are called BGK expansions in [13 in honor of the work of Berezanskii, Browder, G&rding, Gel'fand and Kac, who developed them in the context of elliptic operators. See EI,2,3~ for proofs. These expansions don't really contain much more information than the spectral theorem. The most significant additional information concerns the boundedness properties of ~ ; see E4,5S for applications. Actually, the general proofs show that (~+l~l) in part (a) can be replaced bye(4 + If~l)@ for any ~ > ~ . Indeed, one shows that for any ~ " , one can arrange that for (~)-a.e. ~(') ~ { E ~ ' ) ~ % If one could arrange a set, ~ , of good E ' s where ~ ^ ~ f o r a 1 1 ~ with ~ ( ~ \ ~ ) = 0 , then on ~ ,~ . Thls leaves open: QUESTION I. Is it true that for

(~)-a.e.

E

, each I$~(E~')

is bounded? There is a celebrated counterexamFle of Maslov E6] to the botutdedness in the one dimensional elliptic case. As explained in Eli, Maslov's analysis is wrong, and it is not clear whether his example has bounded u's a,e.

We believe the answer to question I (and all

114 other yes/no questions below) is affirmative, but for what we have to say below, a weaker result would suffice: QUESTION 2. Is it a t least true that for alZ

oD : ~N+~

~m,~N

l~o~(E, ~)1

QUESTION 3. Is it true

N-,-oo exists?

T h e

~

Given a subset

is ~o~dea?

that

I%(E,#)I=- E (a.,E)

~N+~

1fl,14N

w e

M

(~)-a.e. E

will

b y ~(~,E).

denote

{(E,~) : E ~I~ , o~< N}

, of

I

we define

.gE,-)CU)CE)AoCE)

{E:CE,~)~M} where a suitable limit in meam may need to be taken. Define

M~={CE,~): ~CE,') ~ £~}

but (E,~) ~

M~:{CE,~) : ~(~, E) =0

M4}

M3={(E, ~) : i ( ~ , E) ~ 0 } .

Obviously, P(M,) o~ A

i s the p r o j e c t i o n onto the p o i n t spectrum

. QUESTION 4. I s i t

sim~lar

true that

contimuous space of A

P(Mg)

,,is the p,ro,~ection ont,o the

~n d P(Ms)

absolutel~ continuous ~pectrum of a

th,e p r o j e c t i o n onto the

?

Among other things this result would imply that in the Jacobi case (where the number K of the third sentence in this note is 2), the singular spectrum is simple. In higher dimensions, one ca~ see situations where A separates

tie ~( Z~)=~(% v')® ~ CZar)

where A

~d A:A4®I+ I®A~)

has a.c. spectrum with eige~functions decaying in ~2 di-

mensions but of plane

wave form in the remaiming

~ -dimensions.

115

One can also imagine a.c. s p e c t u m from for

and

In either case

combining singular spectrum for lot. of continuum a.c.

eigenfunctions. QUESTION 5. I s t h e r e a s e n s i b ! e ( i , e ~ not obviously false ) verslon of ~uestion 4 in the multidimensional case? There are examples [7] of cases where A has only point spectrm~ but there is an eigenfunction with ~(~, E ) > 0 (since it occurs on a set of ~ -measure zero, it isn't a counterexample to a positive answer to Question 4). Does the second part of Question 4 have a positive converse? QUESTION 6. Is i t true that if function with

~ >0

measure r then A

for a set, ~

A~=~

~s

a bounded ellen-

, of E's of positive Lebesgue

has some a.c. spectrum on

~

?

QUESTION 7. Wha~ is the 'proper analo~ of ~uestion 6 for singular continuous spectrum? REFERENCES le S i m o n B. Schrodinger semigroups. - Bull.Amer.Math.Soc., 1982, 7, 447-526. 2. B e p e B a H C ~ ~ ~ D.M. PasxozeHEe no C O 6 C T ~ e H H ~ S y H E n ~ M ca~oconp~eHH~X onepaTopoB. K~eB, HaYEOBa ~ D ~ a , I965 (Transl.Math. Mono., v. 17, Amer.Math.Soc., Providence, R.I., 1968). 3. K 0 B a ~ e H E 0 B.$., C e M ~ H O B DoA. HeEoTopHe BonpocH pa3~o~eHz~ no 0 6 0 6 ~ e H H ~ C06CTBeHHNM ~yHK~F~K~I onepaTopa mpe~zHrepa c c~Ho c~zT~L~p~ Howe~dEa~2~. - Ycnex~ M&T.HSyE, 1978, 33, B~n.4, I07-I40 (Russian ~ath.Surveys,1978,33, 119-157). 4. P a s t u r L. Spectral properties of disordered systems in onebody approximation. - Comm.I~ath.Phys.,1980, 75, 179. 5. A v r o n J., S i m o n B. Singular continuous spectrum for a class of almost periodic Jacobi matrices. - Bu~l.Amer.Math.Soc., 1982, 6, 81-86. 6. ~ a c ~ o B B.H.06 a C ~ T O T Z E e O~06~eHH~X C O 6 C T B e H H N X ~ ypaBHeH~pe~Hrepa. - Ycnex~ MaT.HayE,1961,16, ~ n . 4 , 253-254. 7. s i m o n B., S p e n c e r T. unpublished. BARRY SIMON

Departments of ~,~thematics and Physics California Institute of Technology Pasadena, California 91125 USA

116

SCATTERING THEORY ~0R COULO~B TYPE PROBLE~S

4.2.

1. Let self-adjoint operators A and Ao act in a Hilbert space H and suppose the spectrum of A o is absolutely continuous. Suppose further that there exists a unitary operator function Wo($) satisfying the conditions:WoC$)WoC%)=WoC$)WoC~)~ W0(~)Ao=AoWo(t)

W~(t+~)Wo(t)=

s -~

E

and there exist l~m~ts

S-f/~ ea~PC¢At)ec#C-'~Aot)WoCt)=U+_ CA,Ao)

(1)

s - ~ , Wo(t~)e/Jr, pC~Aof,)ec, p (-~AD P= U_+(Ao,A)

(2)

t~-_..+_o~

I;--~_+oo

where p is the orthogonal projection onto the absolutely continuous part of ~ . For the generalized wave operators U~ ~-~ the equality A U i ( A , A o)= U ± (A,Ao) Ao holds. The factor W0 C$)

is not uniquely defined. The factor Wo(~)=WoCt)V~ Ct ~ 0)

can be used too when obvious requirements to V~ are fulfilled. Due to this ambiguity the naturally looking definition

SCA, Ao)=U+CA,Ao)U-CA,Ao)

(3)

of the scattering operator becomes senseless since ~ ( A , Ao) = = V+* S CA, A~)Vis in fact an arbitrary unitary operator commuting with A0 PROBLEM I. Find physicall2 motivated normalizatio n of W0C$) when $

=~

, removin~ the non-uniqueness in d e ~ i t i o n of the

scattering operator. This problem has been solved for the scattering with the Coulomb main part, i.e. when

q---B--@ ~+ L~F¢¢¢+Q ~*avC~'~J~'T, Ao~=--~+--~--~ ct~ ~(¢+0

(4)

117 where 0 < ~ < ~

is of

the

form

,

~> 0

~-~]

. For the system (4) the factor W0($)

: Wo(t)= romp [ ~ ( ~ t ) ~ I t l % / { # o ]

•

In [2] it is proved that with such Wo(~) the scattering operator (3) coincides with the results of the stationary scattering theory. This fact suggests that the normalization of ~o($) is physically reasonable. A similar problem has been solved for the Dirac equations with the Coulomb main part [5]. Consider the system

(5)

where

~(OC)~o~±/G), PROBLEM 2. Find

~ ± ~

W0($)

and solv,e Problem 1,,,for the sys,t,em

(5). the system (5) can be reduced to the Coulomb When ~ + = - ~ _ case, when o~+ = ~ _ it is considered in [2]. The case % _ = 0 , ~+~0 is of importance in a number of physical problems [6-~. 2. The Coulomb interaction of ~ particles is described by the system

(6)

where ~K is the radius-vector of the K-th particle, tK,~ = = I~K - $~I . Taking bound states into account leads to the transition from A o ~ to the extended operator Ao ~8]. The operators ~o and Wo(t ) for the system (6) have been constructed in [ 9 ° The construction is effective when ~=3 . In [9] the existenoe

of U± ( A

is proved

PROBLE~g 3. prove the existenc e of

U ± (A0~ A)

for the s2s-

tem (6), i.ue.~ the completeness of the correspondin~ wave operators ,. 3. We shall consider the non-standard inverse PROBLE~ 4. Let operator

Ao

and generalized scattering operator

be knewn~ Recover the operator No(t) Golution.

, find

A

and the correspondin~

classes where the ~rob!em has on e and only one

Consider a model example, namely

(?) i%

oB

where p(S~) belongs to ~he Holder class and p(s~)>O , There is an effective solution of the "direct" problem (the construction of WoC$)

where

and of

~--- C~-,) -- ~

$

~ C ~)+

) for the system

(7) [ ~ :

and (9)

Suppose in addition that

g

Then o p e r a t o r

&

a

has no d i s c r e t e spectrum, The formulae ( 8 ) ,

(9)

give an effective solution of Problem 4 for system (7) by the factorization method, i.e. ~ ( ~ ) and p(D~) are found by S(G~) and hence A and W@ ($) . some problems with non-local potentials and the case (5) with ~ + = o ~ _ can be reduced to the system (7). 4. Now we come to s t a t i o n a r y i n v e r s e p r o b 1 e m s. When ~ > 0 the discrete spectrum of the operator A (see (4)) is described by the Ritz half-empiric formula 2

its proof is given in [10]. The number ~ is called a quantum defect of the discrete spectrum. The same number ~ serves aS a deviation measure of the operator (4) from the case of hydrogen nucleus -

119

PROBLE~ 5. 1~ind a method to recover the potential

~(~)

from

...). Here the representation of ~e by solutions of the Schrodinger equation ~10] and the transformation operator ~1 ~ can be useful. The definition of the quantum defect ~ K ( K = ± ~ , t ~ ...) is introduced in ~I~ for the Dirac radial equation too

& (10)

So problem 5 can be formulated for the case (10) too. Note that the classic inverse problem for the Dirac equation has not yet been solved even when the Coulomb member is missing. The systems of the Dirac type have been thoroughly investigated when (see (10))

~V(~)-m W(~) H( I

L

where V($) , W ( ~ ) are real functions from t (0, oo) . The peculiarity of the system (10) is defined by the fact that the element W(~) is known E W ( ~ - K / ~ ~ L (0, oo)~ • It is therefore perhaps not necessary to use the scattering (or spectrum) data over the whole energy interval--Co < ~ < co . We come to a peculiar half-inverse problem both for the Coulomb and non-Coulomb case. PROBLEM 6. Let

~ , ~ 4 ~ W ( $ ) =-K/~)

(K--- ± ~ , ± ~ , o . .)

be known. Reconstruct the element V ( ~ ) - - - - - ~ ÷ tering (or spectrum)

~(~)

by the scat-

data, be lon~ing to She energy interval

0~<~. The model half-inverse problem has been solved in [13].

120

REFERENCES I. D o i i a r d

J. A s y m p t o t i c

-J.~th.Phys.,1964,

convergence and Coulomb interaction.

5, 729-738.

2. C a x H o B ~ ~ ~.A. 0606~eHHHe BOXHOB~e onepaTopH. - ~laTeM.c6~ I970, 8I, ~ 2, 209-227. 3. C a x H O B E ~ ~.A. 0606mgHHNe BOXHOB~e onepaTop~ ~ peryxapHsany~ p ~ a T e o p ~ BOSMy~eHm~. -- Teop. E MaTeM.~E3HEa, I970, 2, I, 80-86. 4. B y c ~ a e B B.C., M a T B e e B B.B. BOa~OB~e onepaTopH %m~ y p a B H e m ~ m p ~ r e p a c Me~eHHo y6~B~ noTeHn~a~oM. Teop. E MaTeM.~EsEEa, 1970, 2, 367-376. 5. C a x H o B H ~ JI.A. llpmn~Lu m~BapHaH~n~OCTE ~ o6o6~eHH~z( BOJ~HOB~gX ollepaT0pOB. -- ~yHEl~.aHa~H3 E 8tO ITIOE~IO~., 1971, 5, ~ I, 61-68. 6. Tym~ex~H~e ~ e m ~ B T 2 e p ~ x Teaax. MMP, 1973. 7. B p o ~ c E ~ ~ A.~., I~ y p e B ~ ~ A.D. T e o p ~ s~eETpoH-HO~ SM~CC;~H ~13 MeTa.EJIOB, 1973. 8. ~ a ~ ~ e e B ~I.~., MaTeMaTE~ecEEe BonpocN EBaHTOBO~ Teop~E pacce~ ~ C H C T e ~ Tpex ~IaCT~I~. - Tp.MaTeM.EH--Ta ~M. B . A . C T e E XOBa, 1963, T.69. 9. C a x H o B ~ ~ J~.A. 06 y~eTe BCeX EaHaJIOB p a c c e H ~ B sa~a~e Te~ C Ky~OHOBCK~M Bsam~o~e~OTBEeM. -- Teop. ~ maTeM.~Es~k~a, 1972, 13, ~ 3, 421-427. I0. C a x H o B ~ ~ ~.A. 0 ~ p ~ x e P~TSa ~ EBaHTOB~IX ~e~eETax cneETpa p a ~ a x ~ H O r O y p a B H e ~ ~pe%~H~epa. - HsB.AH CCCP, cep.MaTern., I966, 30, .% 6, I297-I310. II. K o c T e H E o H.M. 06 O~HOM onepaTope npeo6paso2a~m~. - ESB. BMcm.y~.saB., MaTeMaT~Ea, I977, 9, 43-47. I2. C a x H 0 B ~ ~ ~.A. 0 CBO~C~BaX ~ c E p e T H O r O ~ Henpep~B~o~o cneE~poB p a ~ E a ~ H o r o ypaBHeH~a ~ p a ~ a . - ~oEa.AH CCCP, 1969, I85, I, 61-64. 13. C a x H o B ~ ~ ~.A. 06 O~H0~ no~yo6paTHO~ sa~a~e. - Ycnex~ MaTeM.HayE, I963, I8, • 3, I99--206. L.A. SAHNOVICH

(~.A.CAXHOBH~)

CCCP, 270000, 0~ecca, 3xeE~poTexK~e c E ~ m~CT~TyT OB~S~ ~.A.C.HonoBa

121 4.3. old

A QUESTION OP POLYNOMIAL APPROXIMATION ARISING IN CONNECTION WITH THE LACUNAE OF THE SPECTRUM OF HILL' S EQUATION Let Q = -

~ ~ + ~(×) ~X

be a Hill's operator with ~

C~oo

,

the class of real infinitely differentiable functions of period I. The spectrum determined by the periodic and anti-periodic solutions Q ~ = ~ , 0-<x < I comprise a simple (periodic) grotmd state ~o followed by separated pairs ~ _ ~ < i ~ ,t$-----~,Z,... of alternately anti-periodic and periodic eigenvalues increasing to + oo , the equality or inequality signifying the dimensionality (=I or 2) of the eigenspace; see [1], The intervals [ ~ . _ i , ~ , $$=~... are the I a c u n a e of the spectrum- "'~-of ~ in ~,~(~) . Hoohstadt [2] proved that the infinite differentiability of ~ is reflected in the rapid vanishing of the lengths ~ of the l a c ~ e [~,-4, ~ ] as ~ + c ~ . Trubowitz [3] proved that the real analyticity of ~ is equivalent 4

to ~W<~@6 - ~ ,

~ $~

. A comparison between

4~

and-"~~(~)~-Jg%w~"~(x)~X O

springs to mind. Interest in sharpening these results arises in connection with the following geometrical problem. Let ~ , ~ c C 1 , be the class of functions giving rise to a fixed periodic and anti-periodic spectrum ~o< ~ ~ ~ < ~ ~<~@ <.. • an~ let ~ , ~ ~ oo , be the number of pairs of simple eigenvalues ~9,tl,-t < ~ . ~ is a compact ~-dimensional ictus identifiable as the real part of the Jacobi variety of the hyperelliptic curve of genus ~ , ~ eo , with branch points over the real spectrum, augmented by the point at oo ; see [4] and [5] for $ < o o , and [6] for ~ = o o . ~ admits a family of transitive commuting (iso-spectral) flows expressible in Hamiltonian form as ~ = X~ with X~--(~%~)', prime signifying - ~ , in which I is~.,~asimple eigenvalue of Q and grad ~ is the functional gradient the square o f the normalized eigenfunotionLI(x)J ~ - -- . The i 0 c a I flows:

dA/d~(X)=

4 0S~.....

(Korte-

weg-de Vries), e t c. are more familiar. The latter belong to the span of the former, but can be expressed in an independent fashion

E)~=(#'t~(~~i,~)'] starting from H_1 ~

via the rule l ; for example,

122

(

•i

o

.(

o

o

THE GEOMETRICAL QUESTION is to decide if the loca I vecto r fields X~ , X%

.ys

• etc. span the tangent space of ~

the case if ~ < ~

at each point. This is

~ see ES] or E~- MoKean-~bowitz

al-

E6J make

the question precise for ~ - c o and prove the following necessary and sufficient condition. Let ~ be the space of sequences ~ ( ~ ) ,

Let ~ be the s u b s p a c e ~ ( ~ Z ~ ) ~ ( ~ ) , ~,~,~,... , with a polynomial. T h e n t h e s p a n n i n g o f t h e local vector fields takes place if a n d o n I y i f ~ s p a n s ~ . The condition is met if is real analytic ( ~ @ ~ - @ ~ , ~ ~ + ~).It is known that ~ , ~ + +Co÷G~'~C~-4~... (~.~), permitting the application of a result of Keosis KaS in the case of purely simple spectrum to verify that the spanning takes place in that circumstance only if E Contz~z~wise,

the

spa~ng

c a n n e t

t

a k e

~

$~

p 1 a c e

if

vanishes on an interval; in fact, if the local vector fields span the tangent space, then the associated gradients ~ H / ~ $ span the normal space of ~ , and the two together (tangent and norreal) fill up the whole of the ambient space, taken to be ~(0,I) , which is impossible since and ~ / ~ are universal polynomials in ~ , Sf, $[~ , etc. without constant term and consequently vanish on the same interval as $ . I t s e e m s I i k • i y that the spanning becomes critical in the vicinity of quasi-analyt ic~ . The same questions arise for ~ on the line with ~ C ~ , the class of infinitely differentiable functions of rapid decay at ± 0o . The rate of vanishing of the lacunae is replaced by the rate of decay of the reflection coefficient 6,1~(,K) o f F a d d e e v F9], e . g . , $ ~ C ~ is reflected in 61~e C ~ , while the analyticity of ~ in a horizontal strip is reflected in I G~(~)I < @~-~1~1 ~ ~ _ ~ + =o;

X~

see DO]. ~ e

to~. ~

is n ~

replaced by ~ - ~

- dimensio=l cy-

123

linder specified by fixing I~% I and finite number of bound states (negative simple eigenvalues) - M = ( ~ = , . , ~) and the vector fields from~ K~(~I,=~,..,~) presumably span the tangent space; see ~1] for preliminary !nformatlon. The local vector fields ~ $ ~ ~/, ~ ¢ $ = 3 ~ - ~ r$ .( f f f operate aS before, and the question is the same as before: d o t h e y s p a n t h e t a n g e n t s p a c • o f ~ ? The technical clarification of the question is a necessary part of any d/scussion. REPERENCES

I. M a g n u s W., W i n k 1 e r. Hill's Equation, New York, Interscience-Wiley, 1966. 2. H o c h s t a d t H. Function-theoretic properties of the discrlmi-A-t of Hill' s equation. - Nath. Zeit., 1963, 82, 237-242. 3. T r u b o w i t z E. The inverse problem for periodic potentials. - Comm.Pure Appl.Math.~ 1977, 30, 321-337.

4. ~ Y 6 p o B ~ H B.A., H 0 B ~ ~ o B C.H. H e p z o ~ e c ~ a ~ sa~a~a y p s 3 n e ~ KopTeBera-~e *p~3a ~ I I T y p M a - ~ y B ~ . HX CB~S~ c a~redpaEecEo~ reoMeTpze~. -~oE~.AH CCCP, 1974, 219, 3, 531- 534. 5. M c K e a n H.P., P. van M o e r b e k e. The spectrum of Hill's equation. - Invent.Math., 1975, 30, 217-274. 6. M c K e a n H.P., T r u b o w i t z E. Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points. - Comm.Pure Appl.Math.~ 1976, 29, 143-226. 7. L a x P. Periodic solutions of the ~ equation. - Comm.Pure Appl.Math.~ 1975, 28, 141-188. 8. K o o s i s P. Weighted polynomial approximation on arithmetic progressions 223-277. 9. ~ a ~ ~ e e mpe~nrepa. IO. D e i f t

of intervals or points. - Acta Nath.~ 1966, 116,

B ~.~. CBO~CTBa ~ - - ~ T p ~ U H o ~ o ~ e p n o r o ypaBneH~ -Tp.MaTeM.~--Tn AH CCCP, I964, 73, 314-336. P., T r u b o w i t z E. Inverse scattering on

the line. - Comm.Pure A p p l . ~ t h . , 1979, 32, N 2, 121-251. 11. M c K e a n H.P. Theta functions, solutions, and singular curves. (Proc.Conf., Park City, Utah., 1977, 237-254), Lecture Notes in Pure and A p p l . ~ t h . , 48, Dekker, New York, 1979. H.P.MCEEAN New York University. Courant Institute of Mathematical Sciences, 25~ Mercer Street, New York, N.Y. 10012, USA

124 ZERO SETS OF OPERATOR FUNCTIONS WITH A POSITIVE I~iAGINARY PART

4-4 old

Let E be a separable Hilbert space, ~ be a function analytic in the unit disc ~ , taking values in the space of bounded operators on ~ and continuous up to the boundary of D . Suppose also that

where CC~) is a compact operator on following properties are satisfied: A) For a modulus of continuity

~olds with B)

. we also assume that the the inequality

~', ~' ~ D .

M

has a positive imaginary part in D

A point

~

in

~0~D

will be called a

r 0 0 t

0 f

M

if

11¢1--4 Since I - M C ~ ) is compact, for any root ~ there exists e e E such that M (~)e = ~ . It is not hard to verify that the roots of a function with a positive imaginary part can lie only on T . Denote the set of all roots of M by A and let ~A~ be the Lebesgue measure of its ~-neighbourhood in CONJECTURE I. Under hypotheses

A. B

the inequality m ~ C w ( ~ )

holds for a ~qsitive cQnstant C. It seems to be natural to weaken hypothesis A and replace it by the following one taking into account the behaviour of ~ only near the set A .

A,}

I M4C~)I"~ .~a(~,cz, A)).

125 CONJECTURE 2. Under hypotheses

hol,d,s,,,for a positive constant

0

~;,B

the inequality

•

Let us note that the validity of any of CONJECTURESS or 2 with ~(~) ~ C ~ ~ would imply that ~ is finite. The above CONJECTURES agree with known results of operator theory and complex analysis. Their proof would permit us to describe the structure of the singular and discrete spectra of perturbed operators in terms of "relative smoothness" of perturbation. To indicate links let us point out a situation of perturbation theory where the questions of such sort arise. Let H be a Hilbert space, ~o , V be self-adjoint operators on H ,

V~

,

A=A°+V E = ~ ~H . It follows immediately from the second resolvent identity that the following relation between the resolvents of the original and perturbed operators holds

, Ira, z :,0,

Here 1{~ :(~-ZIf ~ , 1~° :(A °- zI) -~

o The function ~

(1) defined

by o

Ncz~ = I E + V ~/~~,, V ~1~, N: E -" E has a positive imaginary part. The perturbation V will be called s m o o t h i f f o r some s p e c t r a l r e p r e s e n t a relatively tion ~ of A°

an operator with a smooth kernel corresponds to V . The perturbed operator A in this representation coincides with the so-called Friedrichs's model (see for example [1]),

126

The problem of investigation of the singular spectrum ~$ and the discrete spectrum ~ can be reduced (see [1]) to the investigation of the zero set of the corresponding operator-valued function ~ , since ~s U ~d C ~ according to (I). Just in this w~y in IS] a theorem is proved which clain~s that if the kernel function ~has a good behaviour at infinity and satisfies the smoothness condition i~@LLp& , &~/~ then the discrete spectrum is finite and the singular spectrum is empty . The crucial point of the proof of this result is the fact that ~ has better smoothness properties at the points of in comparison with I~ and so A') is satisfied for m with Investigation of the one-dimensional Friedrichs's model with the kernel of class L ~ , & < ~/~ (see[2]) shows that in this case the same phenomenon takes place.

~mom~ [2 ]. ~ ~'(z,~)=~(o~)c{(~), 0<~,~;<4,

S,(o)=c~(~=O,

~ hLp ~ , ~ < 4/~ , then the func,tion M

Mc~)= t-~ ~Li;, I~o, satisfies A') with

~[~) ~ - C O~

~l"

, the singular and discrete speO-

5

tra of the operator ~

o_~.n E~C-~,~)

defined by

-t

are contained in the set

~

of zeros of

m

and

It is proved also in [2] that the above theorem is precise (in a sense).

The tool of [ 2] is the scalar analogue of CONJECTURE 2 proved in [3]. As to CONJECTURE I in the scalar case, apparently it can be considerably strengthened. See in more details 9.6 of the present volume. We do not venture to formulate so fine conjectures in the multidimensional case.

127

RE~ERENCE S I. 2.

3.

• a ~ ~ e e B ~.2. 0 Moaex~ ~ r ~ x c a B Teopz_~ BOSr,grgeam~. T p y ~ MaTeM.~ra-.Ta AH CCCP ma.B.A.CTez,,aoBa, 1964, 30, 33-75. II a B a O B S.C., H e T p a c C.B. 0 CKHIV/ITpHOM clleETpe caa6o BO@MyI~eHHoPo onepaTopa y~mo~eama. - ~/azn.saaa. ~ ero npm~., 1970, 4, ~ 2, 54-61. II a B ~ 0 B ]3.C. TeopeMa e~]~ICTBeHHOCT~I ~ ~ y H E I ~ C HOJIO~Te-~ a o ~ M m ~ o ~ ~acT~m. - B Ea. : Hpo6aeM~ MaTeM.~SNI4~, 2LI~, 1970, II8-I24. L. D. ~ADDEEV

CCCP, 191011, 21eHEHI~a~ $OHTaHEa, 27, JIOM~I

B. S. PAVLOV

CCCP, 198904, J l e H m ~ a ~ HeTp0~Bopes, SESr~qecEH~ ~8.EyJIBTeT ~eam~pa~cEoro yaazepc~TeTa

(B. C. IIABJIOB)

COMmeNTARY The following progress is obtained in [ 4] : THEOREM. Let C(~) ~ ~ ~ ) and let invertible ~$ some (and then at any) ~oint

~o E ~

MC~o)

b_~e

. Then

c

where

0 A is a constant dependin~ o n only.

&

,

0 < & ~< ~

, and on

M(0)

This theorem implies that CONJECTURES 1 and 2 hold for I-~(~)

E~I

and

~d(~) _~dv.

It is possible to prove an analogous proposition for an arbitrary modulus of continuity

CO

. However, the condition

C(~)6 [4

128

seems to be essential [5]. The above Theorem allowed to describe (see [4]) the structure of the singular spectrum of selfadjoint Friedrichs model that is discussed after the statement of Conjecture 2 RE FERENCE S 4.

5.

H a 6 o E o C.H. TeopeM~ e ~ C T B e H H O C T H ~X~ o n e p a T o p - ~ c nO~O2~Te~BHO~ MRHMO~ ~aCTBm H C E H I ~ J ~ H ~ cneETp B caMOconp~~eHHo~ M o ~ e 2 ~p~p~xca. - ~ o F ~ . A H CCCP, I98S (B HeqaTE). H a 6 o E o C.H. Private communication.

129

POINT SPECTRUM OF PERTURBATIONS OF UNITARY OPERATORS

4.5.

Let ~ be a unitary operator with purely singular spectrum and let an operator K be of trace class. QUESTION. Can the point spectrum of the perturbed operator ~ + ~ be uncountable? If it is not assumed that the spectrum of ~ is singular the answer is YES. A necessary and sufficient condition for a subset of T to be the point spectrum of some trace class perturbation of some (arbitrary) unitary operator was given in [1] : such a subset must be a countable union of Carleson sets (for the definition see, e.g., 9.3 of this "Collection")~ A version of the reasoning in [I] allows to reduce our QUESTION to a question of function theory. PROPOSITION. Let

~

be a subset of

T

. The followin~ are equi-

valent. I)

~

is the point spectrum of some trace class perturbation of

a unitary operator with singular spectrum. 2) There exist two distinct inner functions ~4(~)-~)~_~

e4 and $~

such that

}

Note that if one of the inner functions is constant then the latter set is countable since it is the point spectrum of a unitary operator, namely of a rank one perturbation of a restricted shift (cf

[2]). REFERENCES 1. M a E a p 0 B

H.F. Y H ~ T a p H ~ T o ~ e ~ H ~ cneKTp nO~TZ yH~TapH~X one-

pa~opoB.- 3an.Hay.H.Ce~ZH.20MH, I983, I26, I43-I49. 2~ C 1 a r k D. One dimensional perturbations of restricted shifts~ - J.Analyse Math., 1972, 25, 169-191. N.G.MAKAROV

CCCP, 198904, ~eHZ~I~a~,

(H.F.MAKAPOB)

~eTpo~sope~, J I e ~ H K ~ o ~ ; ~ a p c T ] 3 e ~ yH~Bepc~TeT

roc~-

130 4.6.

RE-EXPANSION

OPERATORS AS OBJECTS

O~

SPECTRAL ANALYSIS

I. Notations, ~Z~ and L~@ are subspaces of even and odd functions in [.~(~) ; ~ is the Fourier transform; Z is the multiplication by , . ~ on ~ (~) ; K ~ ~ * ~ ~ is the Hilbert transform; ~_~L~.= ~*~ ~ ~ . Let ~ ( ~ ) be the following unitary mapping

from

onto

:

%

and ~& are the E?urier cosine and sine transforms on I. ~ (~+); ~ ¢~ % , ~ 2fL ¢ ~. Let ~ denote the multiplication by 6~ = ~ ( ~ + i ) on ~ ( Z ) • Integrals with singular kernels are understood in the sense of principal value. 2. Re-expansion operators appear quite often in scattering theory Namely, the wave operators for a pair of self-adjoint operators H,,~ -

J~.i~± oO

can be obtained as follows. A given function is expanded with respect to the eigen-functions of He and then the inverse transform using the eigen-functio~ of H is taken. Let for example, Hs sad ~@be the operators -i~S on L~(~+) with the domains ~efined by ~(0)=0 and W(0)--0 respectively. Then ~ + ( H @ , ~ ) = + ~ , Indeed, let

Then

I g k oCk) %chc )

k,

e

Simple calculations by the stationary phase method show that ~0(~)~ ~(~) when ~--~+_oo provided ~ i = ~ 0 . The re-expansion operator ~ arises in the polar decomposition of A = - { ~ on ].~(~+) with the boundary condition ~(0)=0 , namely, A = ~ IAI and A = M IA I . Let us verlfy the flrst equallty. Since [AI=( A'A)'I/Z= H~,' , we have (using the above notation @4---~14)

131

hence

JR+

Concret~ re-e~sion o p e r a t o r s a r e a p p a r e n t l y i n t e r e s t i n g from t h e a n a l y t i c a l p o i n t o f v i e w and a s t h e o b j e c t s o f s p e c t r a l ~ n a l y s i s .

In this connection (see Secto5) we propose some problems~ But at first (in Sect.3,4) we use re-expansion operators as specimens for observations. The author thanks N.K.Nikol'skii and M.Z.Solomjak, whose remarks are incorporated into the text. 3, Put 9= ~ ~ ~ , in accordance with the decomposition ~ ( ~ ) = ~~ ~~ • It is easy to see that @

g~= ~, M~ ~,

'S

hence

(,M w)($)= ~

~- $~

(i)

l~+ The operator

V

defined by

V~(~)=~/~ ~(~) maps isometrically ~(~+) onto [~ (~) , Clearly, V~coincides the convolution operator ~-~ ~ ~ , where

~(s)-

with

~I ~e sl~.

Taking the Pourier transform, we obtain *)

(2)

M=V*~*ECV, where E is the multiplication by the function

~C~) on

L~(~) •

It

follows

= co~,h, F~ from

(3)

'

(2) and (3)

that

the spectrum

of

n (res-

: ~) The spectral decomposition of M can be also deduced from (see Oh.IX} but the proof given here is more direct amd simple.

[I]

132 pectively of ~ ) is absolutely continuous and fills out the semicircle T ~ { ~ E ~ , ;~ez>zO} (respectively T ). These imply that is unit ar ily e quival e n t to the s h i f t o p e r a t o r on ~ ( ~ ) . Note that

I ¢~-@~1:1 I - r l l : ~ and that the equality ~)@~ : ¢5 ~

\{ ~}

is impossible

~0(b:~ -II~

, though i t holds f o r

f~r

and i n f a c t

l~eL~('~\

¢~0:

: (~$~,o : I,~ o •

4. A re-expansion operator on [.~(~), A:(-~,~O , with analogous properties appears in connection with the system

I'

]

,

z,

d~:L~(A)-,L~(Z)

(but not with the usual trigonometric system). Let be the ~ourier transform corresponding to this system. Let &+=(O,~)~ ~S and ~ be maps of ~ (A÷) onto ~ ( ~ ÷ ) corresponding to the systems

Purther put

The sense of the following notations is clear by analogy with Sect.3 The operator K acts as follows

,I I ~(~)~5

(4)

A

Changing variables by the formulae

lc r-g'

we red uc@. K to the Hilbert transform- R : that ~ -- ~ Z can be written in the f o r m decompose ~ into two p a r t s ~ v e n and odd): ~ ~ 5 = ~ s where ~ s - - ~ l ~ . S~nce ~$we obtain a unitary equivalence of ~ and ~

~'K ~ . This implies ~ 5 ~ ~ . Further, :~~ & . Then

¢ , namely

where 0 + - G I ~ (~÷) . Note that M describes the non-trivial part of the scattering matrix for the diffraction on a semi-~mfimite screen (this is shown in [ 2], where a unitary equivalence of M and of the

133 multiplication Another (and a following. Let dary condition B~: MI~,|, TO "~ ~ ~'~* (= ar form

by function (5) is presented in an explicit form). more elementary) situa$ion where ~ appears is the ~, (resp~ ~ ) be -~ ~ on ~ (/~+)~with the boun~(0)=0J~resp W(~)=0 ~ Then Bo= M I ~ , writ t ~ in a matrix form we note that the operator ~ ~ [ ~* ) on ~(~) has the following bilineT

5. PROBLEMS. • @ I) equality n 5 implies t at H is bounded On LF(~+) , 4< p<~. What is the norm of I- ~ .~ (If ~=~ see (4)). It is not excluded that the answer can be extracted from the results of [I], Ch.IX. 2) Multi-~mensional analogues of the operator ~ can be described in the following way Let ~ 9 be unimodular functions on ~ satisfying ~ ( ~ ) = X ( ~ ) , 9($~)-- 9(~) for SF0 Let L,~ be multiplications by these func,t~on@ on ~,~(~). If ~,~ is the Fourier transform on ~£(~m)then 9 ~ ~ ~ N is a unitary operator. It would be of interest to investi6ate its spectral properties It might be reasonable to impose some additional conditions on ~ and (e.g. some symmetry conditions). 3) Let ~ be an even positive function on ~ and {p~} be the or-

thogonal family of polynomials i n ~ ~ (-~,~; ~)

, 0<~<~o.

Then an analogue of the operator ~ appears in ~~ ( 0 ~ ) . namely, the re-expansion operator from the even polynomials {p~'~ to the odd ones {P~÷4~ properties.

• It would be of interest to investigate its spectral

4) Consider the following systems in ~£(A~):

The second system has defect I. Let P be the re-expansion operator . from sines to cosines . . This is a semi-unitary operator on ~Z (A+ ) with defect indices (1,0). Is it completely nonunitary? In other words is the orthogonal system 5) The operator

~

{ P~:

~)0}

complete in

~(A+)

?

is connected with the harmonic conjugation.

What does the theory of invariant subspaces of the shift mean in terms_

134

of

~

? What is the role of zeros and poles of the function (3) in

this connection? REFERENCES I. r o x d e 1o r H.II., E p y n m ~ K H.H. BBe~eKze B TeoIx~ o~Ho~epE~x c ~ H ~ 1 1 ~ x HHTerI~JA~HHX onelmTOlOOB, l~m~SeB, mTH~HK~, I973.

2. H ]i L B H E.M. XSpSETelD~0TRI~ ImcCesHms ~ . salaam o ~ m @ I m ~ s B S8 F~L~He ~ H8 SEI~He. - 88n~CK~ HBy,.CeMMH.~0MH. 1982. 107. 198197.

M. S. Birman

(M.m.z~mH)

CCCP, 198904, HSTIDO~80108~I, ~sH~eOl~R~ ~sI~JL7~Te T JleHSHI~S~ONRR y H H B e I ~ T e T

135

4.7.

MAXIMAL NON-NEGATIVE INVARIANT SUBSPACES OF ~ -DISSIPATIVE OPERATORS

Let ~ be a ~ -space (Krein space) i.e. the Hilbert space with an ~_Ter product ( ~ ) and indefinite ~ -form[~,~l=(~,~), ~ = ~*= (more detailed information see, for instance, in [1] or [2]). ~ subspace ~ is called n o n - n e g a t i v e if [~, ~ ] ~0 for ~ ~ , and m a x i m a 1 n o n - n e g a t i v e if it is non-negative and has no proper non-negative extensions. A linear operator ~ on ~ with a domain ~ $ is called d i s s i p a t i v e ( ~- d i s s i p a t i v e ) if ~ ( ~ , ~ ) ~ O (~[~,~ ~0) for all O~ ~ . Such an ~ is called m a ximal dissipative (maximal ~-dis sip a t i v e ) if it has no proper dissipative ( ~ -dissipative) extensions. PROBLEM 1. Does there exist a maximal non-negative invariant subspace for an~ bounded

~-dissipative

o~erator ~

with ~

=~

?

This problem has a positive solution if ~ is a u n i f o r m 1 y ~ -dissipative operator i.e. there is a constant ~Q, ~[~,~] ~ ~llO~l~~, In that case ~(~) ~ = ~ and hence ~ e Riesz projection generated by the set 6~(~) ~ C* gives us the desired subspace. Note that ~-dissipativity of ~ implies the uniform dissipativity of ~ E = ~ * 4 ~ C8~0) . As ~ posseses a maximal non-negative invariant subspace, it is natural to use the "passage to the limit" for ~-*0 . Such a passage - M.G.Krein's method (see [2]) - leads to a positive solution of Problem I if (I+ ~ ) ~ ( I - ~ ) ~ Too ' In the general case Problem I has not yet been solved and therefore subclasses of operators for which it has a positive solution are being considered and on the other hand attempts are being made to construct counterexamples. THEOREM. If = if

~

0 i_~sa

=

~

is a continuous

is a

~ -s~ace,

=

~ -dissipative operator if and only

continuous d issipat%ve operator in ~

has a maxima! ' non-negative (with respect to the riant sub space.

then

; Ain tha ~ case ~-form)

inVa-u

136

PROOF. On~ verifies immediately that a maximal non-negative subspace ~ of ~ is invariant under ~ iff it is a graph of an o p e • rator ~ ) &ere ~ is dissipative in % and ~ = ~ (~=V{<~,-~)~£%). Such an operator ~ does exist and is bounded by the theorem of ~tsaev-Palant[3]. ~tSa~v-Palant's result about the square root of dissipative operator was developed by H.Langer [4]. It was proved there in particular that each maximal dissipative operator posseses a maximal dissipative square root. This result allows to omit the requirement of a continuity of ~ in the above Theorem and replace it by the maximal dissipativity condition,

where ~

is a continu0us

there exist a maximal (in under the

a~-dissipative ~

operator in ~

. Does

) noD-negative subspace invariant

~ -dissipative operator

~

?

REFERENCES I. B o g n a r J. Indefinite inner product spaces. - SpringerVerl~g, 1974, 2. A s ~ s o B T.H., H o x B H ~ O B H.C. ~m~egaHe onepaTOpHB ]IpOOTpaHOTBaX C I~f~e~NB~THO~ MeTpHEo~ E I~X IIpN2Io~eHN~. "MaTeMaT~-~ecEm~ ssaxHs. ToM 17 (HTOr~ H a y ~ ~ TexHzFm)", 1979, MOOEBa, BMH~TM, 105-207. 3. M a s a e B B.H., H a a a H T D.A. 0 cTe~em~x orps~mqe~oro ~cc~naT~m~oro oHepaTopa. - YEp.MaTeM.~pHaa, I962, I4, 829--887. 4. L a n g e r H. Dber die Wurzeln eines maximalen dissipativen Operators. - Acta Math. 1962, XIII, N 3-4, 415-424, T.Ya.AZIZOV (T.H. ASMBOB) I.S.10HVIDOV

(~.C.H0XB~0B)

CCCP, 394693, BopoHex, YH~Bepc~ Te TOKa2 Ha.l, BopoHemcEz~ rocy~apcTBeHH~ yHl4BelOC~ Te T

137

4o 8°

PERTURBATION THEORY AND INVARIANT SUBSPACES

If ~ is a given coefficient Hilbert space, let ~(~) Hilbert space [I] of square summable power series ~ ( ~ ) ~ with coefficients in

be the ~6E~

2-:. If B (~) is a power series whose coefficients are operators on ~ and which represents a function which is bounded by one in the unit disk, then multiplication by ~(~) is contractive in ~(~) , Consider the range ~ ( B ) of multiplication by B(~) in ~(~) in the unique norm such that multiplication by B(~) is a partial isometry of C (~) onto 11~(B) o Define ~ ( B ) to be the complementary space to ~ ( ~ ) in ~(~) . Then the difference-quotient transformation ~(~) into K~(~)-~(0)]/~ in ~ ( B ) is a canonical model of contractive transformations in Hilbert space which has been characterized [ 2 S as a conjugate isometric node with transfer function B(~). If ~ (E) is a power series whose coefficients are operators on ~ a n d which represents a function with positive real part in the unit disk, then

is a power series which represents a function which is bounded by one in the unit disk, Define ~ ( ~ to be the unique Hilbert space of power series with coefficients in C such that multiplication by + B(~) is an isometry of ~(~) onto ~ ( 6 ) ° Then the difference-quotient transformation has an isometric adjoint i n ~ ( ~ ) . The overlapping space ~ of ~ ( B ) is the set of elements ~(~) of C (~) such that B(E) ~ (~) belongs to ~ ( B ) in the norm

The overlapping space ~ is isometrically equal to a space ~ ( 0 ) . A fundamental theorem of perturbation theory [3B states that a partially isometric transformation exists of ~ (~) into ~ (~) which commutes with the difference-quotient transformation., The transformation is a computation of the wave-limit. The wave-limit is isometric on the square summable elements of ~ (~) and annihilates the orthogonal complement of the square sum~able elements of ~ ( ~ ) , if T denotes the adjoint in ~ ( ~ ) of multiplication by B(~) as a trans-

138

formation in C (~) , then the wave limit agrees with ~ ~ 1 - o n square s~unmable elements of ~ ( ~ ) . A fundamental problem is to determine the range of the wave-limit in

~(~)

. It is known [4] that the ran-

ge can be a proper subspace of

~(~)

• The orthogonal complement of

the range of the wave-limit in

~(~)

space ~ ( C )

such that

~(Z) = ~ ( ~ ) C

is contained isometrically in

is the overlapping space of a (E)

for a space

~

(~)

which

~(~).

CONJECTURE. The range of the wave-limit contains ever 7 element of

~(@)

if the self-a~]oint part of the operator

~(0)

is of Ma-

tsaev class. The llatsaev class seems a reasonable candidate because the existence of invariant subspaces is known for contractive transformations~such that

|-T~T

is of~atsaev class. Invariant subspaces exist

which cleave the spectrum of the transformation° An integral representation of the transformation exists in terms of invariant subspaces [5]. For reasons of quasi-analyticity, such results do not hold for any larger class of completely continuous operators, Some recent improvements in the spectral theory of nonunitary transformations limk the Nmtsaev class to the theory of overlapping

spaoes [6]. REFERENCES I~ D e

B r a n g e s

Wesley, to appear. 2. D e B r a n g e s

L. Square Summable Power Series, AddisonL. The model theory for contractive transfor-

mations. - In: Proceedings of the Symposium on the Mathematical Theory of Networks and Systems in Beersheva, Springer Verlag, to appear. 3. D e B r a n g e s

L.,

S h u 1 m a n

tary transformations. - J.Math.Anal.Appl., 4. D e

B r a n g e s

L. Perturbations of uni1968, 23, 294-326-

L. Perturbation theory, - J~Math Anal.Appl.,

1977, 57, 393-415. H.E., K p e ~ H M.r. Teol~s BO~LTepI~O~X OHe~TOI~OB B rM~6elYrOBOM II~OCT~HCTB8 M ee ~ o z e ~ s , M., HSyFm, I967 (Translations of Mathematical Monographs, 24, Amer

5. r o x 6 e p r

Math.Soc., 1970). 6. D • B r a n g • s

L. The expansion theorem for Hilbert spaces

of analytic functions, Proceedings of the Workshop on Operator

139

Theory in Rehovot, Birkhauser Verlag, to appear~

L. DE BRANGES

Department of ~athematics Purdue University West Lafayette, Indiana 47907 USA

140

OPERATORS AND APPROXI~ATION

~-9. old

1. ~nat is a"Blaschke product"? As long as we are concerned with scalar-valued analytic functions in the unit disc ~ , the answer is well-known: this is a function B satisfying one of the following equivalent statements. ~k (i) ~ can be represented as a product ~ = ~ ~(A) of elementary factors on from D

= ~-

to nonnegative

~-~E

(here

~

is a functi-

integers with ~ ~(~)(I-I~l) < +co)

(ii) ~ is inner (in Beurling's sense) and the part of t ~ a~ schift operator ~ * on the invariant subspace ~ , ~ = KB ~8~ i has a complete (in ~ ) family of root subspaces, Here ~ is the standard Hardy class and • , ~ are operators (iii) The same for t to

~I~ (iv) ~

ra

( PK stands for the orthogonal projection onto is inner and

~ ).

~{4 T The spectral interpretation of (i) and (iv) is of importance for studying operators in terms of their characteristic functions and the problems discussed in this section are essentially those of a "correct" choice of the notion of Blaschke product in the general case, when operator-valued inner functions are considered (the equality I~(~)I =~ a.e. is replaced in this case by the requirement that ~(~) be a unitary operator on an auxiliary coefficient space ; ~ is replaced by ~(~) , and so on). Statements (i)-(iv), appropriately modified, are still equivalent for operators ~B having a determinant (i.e. when I- ~ B is nuclear). If this is no longer true, (ii) AND (OR) (iii) prove to be the most natural definitions of a Blaschke product. QUESTION 1 !s it true in case of an arbitrar~ operator valued inner function ,that one of the Conditions ,,(ii,~, (iii~ implies the other one?

The definition under consideration schould presume a metric criterion for a characteristic function ~ to belong to the class

141 of "Blaschke products" (that is a criterion for T B to be complete).

AND (OR) --~'I'~

QUESTION 2. Do the followin~ conditions ~ive such criteria:

If we restrict ourselves to the case when T B = ~ t K ple spectrum, Question 2 reduces to the following one. QUESTION 3. How to describe in terms of B ted b 2 ,,,ellen-functions of ~

,

has a sim-

the subspaces ~enera-

i . e . the subspac,,es

Here { A~ : ~ ~ } is a family of orthogonal projections on It seems important to know when the space (V) coincides with H~(F) i.e. QUESTION 4. For which families ~AA - ~

SeH~(E)

and A~(~)= @, ~ D

If a ~

or i

queness theorem ~

AyO

~ }

the conditions

imply ~ 0

?

, Question 4 clearly reduces to the scalar uni(4-I~I)~ ~

. The last condition remains

necessa-

ry in general case. Perhaps the answer to Question 4 is the following: ~(~-I~l) IIA~ ~----~ for ~ As to question 3, in case ~

belonging to a complete family in E ~

~

the answer can be expressed in

terms of the so called "pseudocontinuation" of functions in (V) (M.~. DJrbashyan, G.C. Tumarkin, R. Douglas, H. Shapiro, A. Shields and others). Possibly the same language fits for ~ E

> ~.

2. Weak ~enerators of the a!~ebra ~(T@) .In this section ~ is a scalar inner function and ~(* ) is the weakly closed algebra of operators generated by the operators, A~(T@)

and I .It is known(D.Sarason) that

iff A---~(T@) for some ~ in H@~ (The operator ~(T@) acts in

~@by the rule ~(TG)~--]pK~I ' I~).

The description of weak generators

142

of ~ ( ~ ) = ~ is also known in a geometrical language, in terms univalent) image ~(~) . Since the are isometrically isomorphic, it is rem should admit "projecting": QUESTION 5. Is it true that if and only if

~ + e H~

(D. Sarason) and can be expressed of properties of (necessarily algebras ~ (T e ) and ~ H ~ plausible that the Sarason theo-

~(~(Te) ) =

(le)

contains a generator of algebra

QUESTION 6. Which operators

~(~)

(I.e. f,o,r which ~ there exists {

have

~?

simple spectrum?

in K~ with span (PK~{:~)O)=

=Ke?) If ~ is a generator of Hoo then cyclic vectors ~ from Question 6 do exist and can be easily described. In the particular ~+~ case ~ =~X~ ~_~ Question6 reduces (for some functions ~ at least) to the question whether lar (or the same question about the operator

~ITO) ~ is unicellu~ I'~(5)K(~-$)~5 v~

on ~(0,~) , G.E.Kisilevskii). Related to this matter are a paper of J.Ginsberg and D.Newman (J.Aprox.T., 1970, 24, N 4) and the problem 7~19 of this Collection. Other references, historical comments and more discussion can be found in two papers of N.K.Nikolskii (in books: ETOI~ HayEH, ~aTeMaT~qecE~ a H ~ 3 , T.12, 1974; Teop~ onepaTopoB B ~yH~mOHaX~HHX npooTpaHCTBaX, HOBOO~6~poE, I977).

N.K.NIK0~SKII

CCCP, IPIOII, ~eHm~rpa~ ~OHTaHEa 27, ~ 0 ~

CO ~ N T A R Y B.H.Solomyak has answered QUESTION 5 in the negative (oral ¢ommuD/cation). Por the sake of convenience we replace here the unit disc by the upper half-plane ~----{~ : I ~ ~ 0 } and consider corresponding spaces ~ , Hn Let %--~£~ ~--, £~ , , ~ > ~ . Clearly ~ n

co

and it is proved in [I] that

~(~(Te))~(T 0)

143

(This fact is just equivalent to the unicellularity of ~ S(~) %~ ~+~(~/~)~+ (~/~)~ ~ g rapidly

(~f)(x)~

in ~ ( ~ I ) ). On the other hand, for any ~ n ' ~I ; ~(~e~)~n , ~>0 Since is nonunivalent i n l l ~ > ~ } and 6 ~ tends to zero

a s I1,~,~----~+oo

nonunivalent. Thus ~

, it

can be easily

cannot be an

Another countere~mple

for

verified

~N -generator.

~---~

that

~

is

also

•

, an interpolation

Blascl~e

product, was constructed by N.G.Makarov. REPERENCE I. Frankfurt R., Rovnyak J. Finite convolution operators~ - J.Math. Anal.Appl., 1975, 49, 347-374.

144 4.10. old

SPECTRAL DECOMPOSITIONS AND THE CARI~SON CONDITION

Completely nonunitary contractions can be included into the framework of the Sz~kefalvi-Nagy-Foia9 model [I]. Especially simple is the case when ~ ~T) does not cover ~he unit closed disc 0 ~ D and ~ , = & , < C ~ where c{,=dAm,(l-T~T)H, d,, = cl,b ~ , ( l - l T * ) H are the defect nt~nbers of tions are satisfied then i.e to the operator

T T

P~Ik

• If S -~4~I,(J)=U and the above condiis umitarily equivalent to its model,

K = H~(De e H~(E),

where E is an auxiliary Hilbert space with ~ E=~ , ~ is a bounded a ~ l y t i c (E-~E)- operator valued function in O whose b o u n ~ ry values are unitary almost everywhere on the unit circle T, H2(E) is the Hardy space of E-valued functions, ~ is the multiplication operator ~ ~ - ~ , p is the orthogonal projection onto K • is called the characteristic function of T . it is connected very closely with the resolvent of T e.g. I ~(X,T)I × (4-IXl) -~x ' l e(X)-~i, X ~ O . In terms of e the operator T can be investigated in details, namely, it is possible to find its spectrt~m, point spectrum , eigenvectors and root vectors, to calculate the angles between maximal spectral subspases etc. (cf.[1-~). In particular operator T is complete (i.e. the linear hull ef its eigenvectors and root vectors is everywhere dense) iff d65 ~ is a Blaschke product. A more detailed spectral analysis should include, however, not only a description of spectral subspaces but also methods of recovering T from its restrictions to spectral subspaces. The strongest method of recovering yields the unconditionally convergent spectral decomposition generated by a given decomposition of the spectrum. For a complete operator T the question is whether its root subspaces {K A :~ ~ gp (T)} form an unconditional basis. In the case of a simple point spectrum necessary and sufficient conditions of such

~p(T)

"spectrality" (i.e. in the case under consideration for the operator to be similar to a normal one) were found in [2],[3]. These conditions are as follows: the v e c % o r i a 1 C a r 1 • s o n condition

" - e X-t

>~'Ae:dpO

(I)

145 holds and the following valid:

imbedding

T_., (HXl)IIA x J~CA)~E
Here ~ and A x

t h e o r e m s

C~-l~i)lt~,~CX)ltE<~,V~{~H(E). (2)

is the orthoprojection from is the orthoprojection from

E

X E&X~ x + (I-~)] the eigenspace KX=

are

E

onto the subspace KPJ~,~C~) onto K ~ e ( X ~ ; e=ex '

is the factorization of 0 corresponding to K~(TIXl) , ~ IXI~( X - g ) ( ~ - ~ ) - i

From geometrica& point of view condition (I) means nothing else as the so called uniform minimality of the family {Kk: ~ c dpJ, Moreover,

1 4 exCx)AxlE =sN, C Kx, Kx), where

cf.[2]. In the case 6 = ~ , = ~ L.Carleson proved that (I) implies (2) (cf.[4])~in the case ~ = ~ , = o o this is no longer true ([3]). PROBI~M 1. Prove or disprove the implication

(I)

;" (2)

in the case, 'I <~ = 4 . < o o The case ~ = ~ . = ~ seems to be an exceptional one, because for an arbitrary family of subspaces the property to be uniformly minimal is very far (in the general case) from the property to form an unconditional basis. However for ~ = ~ , = ~ these conditions coincide not only for eigenspaces but for root subspaces as well and, moreover, for arbitrary families of spectral subspaces of a contraction T [5]. The proofs of this equivalence we are aware of (cf,[5], [4] ) represent some kinds of analytical tricks and depend on the evaluation of the angles between pairs of "complementary" spectral subspaces K ~ and K ~ ~ K~, corresponding to a divisor ~ of ~ . Here ~ and ~/ are left divisors of ~ corresponding to a given . A divisor pair of subspaces ; if ~ = 61,,= ~ then ~ / = e is called spectral divisor if ~ is a spectral subspace. So the main part of the above mentioned trick consists in the following implication K5S ; let ~ E = ~ a~d let ~ ~} be an arbitrary family of speot-

146 ral divisors of

@

, then the condition

(3) t7

6~E ~D 11611=~

implies the following one '

d where

~d

6~E

i s the i n n e r f u n c t i o n corresponding to the subspace

proof uses a lower estimate of

and

e

~KD

Ile(:)ell E

depending on ~g(~)e~ E

II g ' ( O e l l E o n l y . However such an e s t i m a t e i s i m p o s s i b l e E>~ (L.E. Isaev, private communication). PROBLEM 2. Let

(3) ~

~<~=~,
for

Prove or disprove the implication

(4) for an arbitrary family {~}

of spectral divisors of ~ .

REFERENCES I. S z ~ k e f a I v i - N a g y B., F o i a @ C. Harmonic analysis of operators on Hilbert space, North Holland/Akad~miai Kiad~ (Amst erdam/Budapest, 1970 ). 2. H ~ E o a ~ c E n ~ H.K., H a B a O B B.C. Basncw ns COOC~-BeHHHX BeETOpOB BHO~He HeyH~TapHax C ~ a T ~ ~ xapaETep~cT~ecEa~ ~yam/m:. - H s B . A H CCCP, cep.MaTeM., 1970, 34, ~ I, 90--I33. 3. H ~ E O a ~ C E E ~ H.K., H a B a o B B.C. PasaomeHm: no CO6-CTBeHHRM BeETOpaM HeyHHTS~HHX onepaTopoB ~ xapaETepEcTH~eCE&K ~yHEKm:. --8an.Hay~H.CeM~H.ZONE4, I968, II, I50-203. 4. H ~ E o ~ ~ c E ~ 2 H.K. ~eEKnE o6 onepaTope C~B~ra. H. - 8an. H~.ceMHH.~0M~, I974, 47, 90-II9. 5. B a c m H Z H B.E. BesycaoBRO c x o ~ e c a cneETpaaRH~e pasaoaeH~a ~ sa~a~a ~HTepnoamm~. - Tpy~N MaTeM.EH--Ta ~M.B.A.CTeEaoBa AH CCCP, 1978, 130, 5-49. N. K. NIKOL 'SKII CCCP, 191011, ~eHzHzpaA @oHTaREa 27, ~ 0 ~ (R.K. K0m CK ) B. S. PAVLOV CCCP, 198904, HeTpo~Bope~, * ~ s ~ e C F ~ ~Ey~TeT ~eH~HrpaAcEoro yHSBepO~TeTa (B.C. EAB~0B) V. I. VASYUNIN CCCP, I9IOII, ZeH~Hrpa~ @OHTa~Ea 27, JlOMI4 (B.M.BACDHMH)

147 4.11

SIMILARITY PROBLE~ AND THE STRUCTURE OF THE SINGULAR SPECTRUM 0P NON-DISSIPATIVE OPERATORS

•

old

The similarity problems under cosideration are to find necessary and sufficient conditions for a given operator on Hilbert space to be similar to a selfadjoint (or dissipative) operator. ~or the first proBlem an answer was found in terms of the integral growth of the resol-

.ents

[3] (see also [2]):

PROPOSITION 1. An opera.tqr

~.

is similar to a selfad~oint opera-

tor if and only if

s~p~, (I II(L-K ~>0

R

The second problem is not yet solved. Here we discuss an approach based on the notion of the characteristic function of an operator KIBo Fora dissipative operator -" L- -( I n ~ L - = ( 2 ~ ' -( L' -- L - * -) ~ @ ) there is a criterion of similarity to a selfadjoint operator (due to B.Sz.-Nagy and C.Foia~) in terms of its characteristic function ~ , namely:

s=plS -~(~)I <+°° I~,X>O The main tool in the proof of this result was the Sz.-Nagy - Poia~ ftuactional model which)yields, a complete spectral description of a dissipative operator . For a non-dissipative operator L an analogous condition on its characteristic function

(sup IB(~I< +~, ~x~e-'(x)l<+~) is sufficient

for

L

to be

I~>O

similar to a selfadjoint operator (L.A.Sahnovich), but not necessary. It is possible to give counterexamples on finite-dimensional spaces which show that operators whose characteristic function does not satisfy the above condition can be similar to selfadjoint operators (cf [4], where related problems of similarity to a unitary operator and to a contraction are discussed). To be more precise, consider the characteristic function

s~i [ + ~t(ivl)'/~(A-~lvi-A)-'(ivl)'~ i

i

,u

s(X)E-,-E,I~A>O,

,

~) Nevertheless &his result can be obtained without using the functional model [3, s ]

148

oz ~ a ~ l i a r y

dissipative operator

A + ~ i vl

,

where

A ~ ~e L ,

V = I m L , E =c~s Ra~V ~d let V=J-IVl, J= 3 ~ V

be the

polar decomposition of V . The latter operator for the sake of simplicity is assumed to be bounded. The characteristic function ~ of and the function S are connected by a triangular factorization

~)(x) ~=~I • ~;(ivl)~ (2-Z) -' (lvl)W= (X_+%+S(x))(X++ X_S (x))-'

IX+- S()~)X+I_

where Z+_ ---- (I-+~T)/~, [ 6 ] . Note that I ~ > o. Under the additional condition

~ J

with

the above condition of bounded invertibility of ~ is necessary for L to be similar to a selfadjoint operator and the condition @~ I ~(~)I < + o o is necessary and sufficient for the similarity to a aissipative operator [ 7 ] . In this case corresponding selfadjoint and dissipative operators can be constructed explicitly in terms of the Sa- Nagy - Foia~ model for A ÷ ~IVI. In general case (beyond (I)) serious obstacles appear. The reason is that it is difficult to obtain a complete description of the spectral component of L corresponding to the singular real spectrum The solution of the LATTER PROBLEM would be of independent interest. Let us dwell upon this question. The operator L is supposed to act on the model space K which can be defined as follows~ Let be the Hilbert space of pairs (~o' ~) of E - v a l u e d functions on tR square sure.able with respect the matrix weight (~ ~ ) ,

S (K) ~ S (K + ~0) being the b o u n ~ r y values (in the strong topology) of the analytic operator-valued function S . Then K=

e

,

,

where (F) are the Hardy classes of E-valued functio in the upper and lower half-planese The absolutely continuous subspace N e in the model representation of L has the following form [6 ]

Ne = N#,L)- oBs PK (,~e (,o&~s(X_U(E), ~÷ C(E)))),

(2)

149 i s the orthogonal projection from onto I%, U( E),-,-e, L: k L , , , , . I . . The s i ~ l a r subspace ,,~ i s defined by where Ne ~ Ne (.L* L It is natural to distinguish in" N~ ~o subspaoes N L and a~ , the f i r s t one corresponding to the point

P~

where

I/

spectrum in the upper half-plane and a part of the real singular continuous spectrum, the second one to the point spectrum in the lower half-plane and another (in general) part of the real singular continuous spectrum:

N~(L) a~ N~+

(E) e (%_+ S*%+) HI(E),0),

(Hi

NT~(L) ~{ N[ ~fc~s PK(O,H+(E)e(X++£~_) H + ( ' The subspace

~i (N~)

' E)). (3)

is analogous to the subspace correspon-

din& to the singular spectrum of a dissipative (adjoint of a dissipative) operator~ Nevertheless if (I) fails, N~ does not necessarily coincide with 0~$ { N ~ + N~ } , In particular the eigen-vectors and root-vectors of the real isolated spectrum do not belong to Therefore in general it is necessary to introduce a "complementswhich would permit us to ry spectral component" ~ 0 ( N ~ c N~) take into account the real spectrum of k Put

where N~* =~ N +- (L*) • PROBLEM 1. When ~es

between

NC

~ ----O B S and

I N~"+ ~]~,- +

N~± ~ N e

N~0 1

? To estimate

in terms of the cha.ra.c.teristic g

function

@

to that of onto

K

. To dive aniiiei~plioit description of N~+- ~

Me

~

(similar

), ~.£r e~.~.mole as the closure of the projection

some linear manifold in ~

described in termS.of characteris-

tic function. ~ O B L E M 2, Find the factor of

~

correspondin~ to

~

. Investi-

gate ..itsfurther factorizati.on. Describe properties of this...,factor .an.d @

..co.nnection of its roots wit.h t.he .spectrum..of m l N~

o How to sepa-

150 i

of O

PROBLEM 3~ Make. clear the spectral structure of

N~

, i.e, in

termsof the model ~.pace construct spectral projection onto intervals of the real singular, spectrum and onto the root-space correspondin~ ,to the real point spegtrum. PROBLE~ 4. Le.t

L

be similar to a dissipative operator~

Then @

does,

e,q,ual to

[ N? +

cide with the..sub.space of

K

+

oeo

coin-

correspondir~ to the. s , i n ~ u l a r

continu-

ous .~.pectrum plus point spectrum of the se..lfadjoint par.t of this di..ssipative operator? PROBLEm5.

Co=sider concrete examples (Priedrichs m o d e ~ wit.h

rank one perturbation, SchrBdim~er operator on

~

with a ~owerllke

. . . . . . . . . . .and . . . . desqribe No~ f o r , , , s u c h o p e r a t o r s decreasin ~ potential~ @

N~

Besides the real discrete spectrum the space apparently can contain one more spectral component. The elements+of N~ no longer have the "smoothness" properties as those of N~- (namely for a dense set vectors Ue N~ we have Perhaps, the structure of NI is similar to that of the singular continuous component of a selfadjoint operator (certainly, one should take into account the "non-orthogonalily" caused by the non-self-adjointness of L ). It is also important for the similarity problem to know which factor of the characteristic function ~ corresponds to N ~° . Note also that all difficulties of the problem appear already in the case when the imaginary part of V is of finite rank

(lvl)~(L-~lY'~eH~(E)~

of

(~E ~). Let us present here one more assertion closely related to the problem discussed above and especially to the spectral decomposition of L. PROPOSITION 2o An.operator to a. dissipative operator operator

M

L

on a Hilbert space ~

L~s~

is similar

if and onl~ ~f there exists an

wit.h

~--.- 0

More,over, if such

L~i,s s

exi,sts then this

fyi~, ,,the additional inequality

M

can be, chosen satis-

"~,a~,k M ~ ~,.,k ! m Lass.

151

REFERENCES 1. S z . - N a g y tors on H i l b e r t - Budapest,

B., space~

C. H a r m o n i c

analysis . t N o r t h H o l l a n d - A k a d d m i a i K~ado,

of o p e r a Amsterdam

1970.

2. S z . - N a g y in H i l b e r t

F o i a ~

B. On u n i f o r m l y b o u n d e d

space.-

Acta Sci.Math.,

1947,

linear transformations 11,

152-157.

3, H a d o x o C.H. O0 yo,~oBz.qxno~o~mz c a ~ o c o n p m t e ~ ~ ym~Tapm~ ouepaTo~u.-@/~u.a~a~. R ero np~. (B neqaTz). 4o D a v i s

Ch., P o i a ~

ristic f u n c t i o n s 1971, N I-2, 5. v a n Math.,

and t h e i r

1980,

with bounded

dilation.

characte-

- Acta Sci,Math

127-139.

C a s t e r n

6. H a d 0 E o

C. O p e r a t o r s ~-unitary

42, N I-2,

J. A p r o b l e m

of Sz.-Nagy.

- A c t a Sci.

189-194.

C.H. A 6 c o m ~ x o

Eenpep~

cnezTp Ee~zccznaa~sEoro

onel0a1'opa z ~ym~zoma.~Re~ Ho~e.~. If. - :~an.Ha!r~.cemm.~O~, 19T?, V3, 118-135. 7. H a d o x o C.H. 0 c~ryxap~ou onexTpe HecaMooonpe~Imoro onepaTopa.- 3au.m~m.ce~mn.~0W~, 1981, I13, 149-IV?. S . N. NABOKO

(C.H.HkBOK0)

CCCP, I98904, ~ m r m p a ~ , NeTpo~Bopeu, k s = q e o ~ i @axy~Te~ ~elmH~Ko~ yn~epc,TeTa

152

A PROBT.~ ON OPERATOR VALUED BOUNDED ANALYTIC FUNCTIONS

4.12. old

Let ~

, ~ * be

two

H i l b e r t spaces and J~ (~,, ~)*)

of all bounded linear operators mapping ~

the space

into ~ * . The following

was proved i n ~]. THEOREM. Suppose

O is a bounded ~(~, ~*) -valued function

anal,7%ic i n t h e unit disc D

. The followin~ assertions are equiva-

l.en%: (a) there exists a bounded anal,ytic in ~

~J(~)~ ~))

-valued function

and satisfyin~ (1)

(b) the Kernel function ~8

:

is positive definite, i~e.

(2) K=i

, where ~ e

f o r ~ y finite systems {~4...A.}, {~... ~. } Condition (1 ) obviously implies that

cIAl<

).

The QUESTION is whether (3) implies ~2} with She same

(3} 8

or at

least with some, possibly different, positive constant. In the special case when d i m ~ and d i m ~ * < co the equivalence of (1) and (3), and thus the equivalence of (2) and (3),follows from the Corona Theorem of L.Oarleson, cf.[2]. A proof of the equivalence of (2) and (3) in the general case, and possibly with operator

153

theoretic arguments, would be an important achievement. REFERENCES 1. S z . - N a g y B., F o i a ~ C. On contractions similar to isometries and Toeplitz operators. - Ann.Acad.Scient.Pennicae, Ser.A.I. Nathematica 1976, 2, 553-564. 2. A r v e s o n W. Interpolation problems in nest algebras. J.l~anc.Amal., 1975, 20, 208-233. tl

Bolyai Inst. of Math. 6720 Szeged Aradi V~rtan~k tere I

B. SZOKEFALVI-HAGY

Hungary

CO~ENTARY

This interesting question has been considered in several publicatioms, but the answers are only partial. Using some refinements of T.Wolff's corona argument, V.A.Tolokonnikov [3~ (see also [4], p. IO1) and M.Rosenblum E7] proved (independently) that (3)--7 (I) if ~ ( ~ e ~ ~) ~ ~ mate of the solution O form) looks as follows:

. Moreover, Tolokonnikov obtained an esti. This estimate (in somewhat simplified

T) where

For small values of

8

a better estimate is

due to Uchiyama E6~

154 V. I. Vasyunin has shown that

V.l.Vasyum.in [5] has proved that (3) ~ (I) if ~ < e o @ (with the estimate C~(6)~< ~ O~ (6 %) ). In ~5] it is also shown that 01(~) ~ 4 that C~(%)>i ~ 5 -~*4 ( ~ ~,S, ) and that if the implication (3) > (I) were true for ~ , ~ -----~%W%~----- ~ , then the estimate of ~ wouldn't be better than 6~c~(~6 -~/8 ) (i.e. that ~oo (~)~ ~ ¢ * ~ (@ 6-~/~ ) ). Tolokonnikov has noted also that if ( 3 ) ~ > (I) were always true thence(6) would be finite for every 6 ~ ( 0~ ) (~npublished). If ~ C ~ * and ~ ~ < + ~ then assertions (a) and (b) in the Problem are equivalent to the possibility "to enlarge" to a SQUARE matrix ~ analytic in D and satisfying ~=~I

~

,~

II~(~)II ~ ~

, A ~ D II(~C~))-~II<~

(see

[5]). There exists a connection between the corona theorem ( f o r ~ = ~ ) and the left invertibility of the vector Toeplitz operator ~@~ ( ~I], see also [8]).

REPERENCES 3..T o ~ o K o H H H ~ O B B.A. 0~eHEH B TeopeMe Eap~ecoRa o KOpOHe H KOHe~4onopo~eHHMe ~ e a r ~ aare6pu H~ . - ~yHK~.aHa~. H ero npH~., 1980, 14, ~ 4, 85-86. 4. H H E 0 a ~ C K H ~ H.K. ~ e ~ M H o6 onepaTope CABHra. M., HayKa, 1980. 5. T o a o K o H H H K O B B.A. 0~eHKM B TeopeMe KapaecoHa o ~OpOHe. H~eam~ aare6p~ H ~ , saAaya C~Ke~a~bBH-Ha~2. - BS/I.HayYH.CeMHH. ~0~4, 1981, I13, 178--198. 6. U c h i y a m a A. Corona theorems for countably many functions and estimates for their solutions. Preprint, 1981, University of California at Los Angeles. 7. R o s e n b i u m M. A corona theorem for countably many functions. - I n t e g r a l equat, and operator theory, 1980, 3, N I, 125-137. 8. S c h u b e r t C.P. The corona theorem as an operator theorem. - Proc.Amer.Math.Soc., 1978, 69, N I, 73-76.

155 ON EXISTENCE OF INVARIANT SUBSPACES OF Cm-CONTRACTIONS

4.13.

be a compeletely nonunitary Cl0-contraction ~) with Let T can be s~pposed the characteristic function B ~ H~(E E,) so T acting on the space

Ke = H~CE,) e @H'(E) as follows

H~(E~)

where Z is multiplication by Z (the shift operator) on and W8 is the orthogonal projection from H~(E,)onto K0 , i.e.

where ~ + is the Riesz projection from operator acts as follows

T*f-_

{-f(o) ';z

,

onto

H2 .

The adjoint

(1)

eKe

Recall that T ~ C,Q iff ~ is inner and ~-outer. Any C H -contraction is quasi-similar to a unitary operator and this allows us to prove that the lattice of all T-invariant subspaces (Lat T ) is non-trivial (see [I]). In our case we have only a quasi-affine transform intertwining T and the ~-residual part of its unitary dilation, i.e. multiplication by ~ on c~sAsL2(E~),whereA.=(I-00") Vz. V~iat can we obtain from this for finding a non-trivial invariant subspace? We can suppose that Ker ~ ( ~ ) s { p } and Ker ~(~)~-----{~) for every ~I < ~ , because otherwise T or T has an eigenvector. Hence ~_(E~) ~ K e ~ ~*--- { ~} and therefore we have to investigate only the case ~ b ~ $ E = d ~ w s E , --o° . Indeed, if ~ L ~ E < o o then there exists an antianalytic solution ~ , of the equation ~ * ~ , = ~ , but the fact that ~ is inner implies that ~ # $ ~ ~ ~L~ E Note that Ke~ ~ ~,La(E,) I~(E~) and ~) A l l used terminology can be found i n [ I ]

or [ 2 ] .

156

P.&,L'(E~)cK,.

P+

Her~ intertwines T a n d t h e ~-resldual part of its unitary dilation, namely IA.< (E~). @* Let SC now be an arbitrary vector in K8 and & ~ ~e% Then

{T~,P÷k.)=(z"~,k~)

=

~<~,kw>Eg"~

,

T where ~ is the normalized Lebesgue measure on the c i r c l e T

.

If

then the multiplicity of T is greater than I, i.e. there exists no cyclic vector. Indeed~for every ~ ~0 we can choose a vector such that < ~ ( ~ ) , k~(~)>E =0 a.e. So every nonzero vector 00 generates a non-trivial invariant subspace. Further we shall suppose rank A , ( ~ ) ~ 4 a.e. Putting ~ = = { ~ T ~i~k ~(~) : ~~ , choose a vector ~ , ~ e % ~ *

k~eKe~O*

such that Uk~(~)~ E, :~ ~T\~

for a.e. 5 e &

and k~(~):O

fora.e

. Now we have

Xe~ 8"= A~L'(E.) : { 9k~" 9 eL' } and

T That is p . ~ k . is cyclic for T * i f f there exists a nonzero vector ~ e K 8 such that < Z ~ . > i s a cyclic function for the multiplication by Z on i.e.

U(~),

=0}4:0 Choosing 0G of special type we can obtain various sufficient conditions for the existence of a non-trivial invariant subspace. PROPOSITION. Each of the followin~ conditions implies the existence o~ a non-trivial invariant subsoace of T

1)

B~eL' such t~t

•

~ {~e~" < (P.,k.)(~),k~c~)>ET0}*0;

157 2) ~ L ,

~" such that

~,~ll~L.~;

3) ~[,,,C~ , I'1¢(~,)~'0 4) th,e

~p~.,{h,.G)~,e~,.}÷E.;

and

P_ (11P+k~ll~,)

antianal,ytic function

admits ,# ps,,eudo-

continuation to the unit disc. CONJECTURE. For every inner

~ - o u t e r function

~

there exists

T , (defined by (I)) of the form

a nonzero npnc,yclic vector for

P+ h,~ ,,here k, e Ke'~ g~. If the CONJECTURE is not true a counter example must have a number of very pathological properties and may be a candidate for an operator without invariant subspace at all.

REFERENCES 1. Sz.-N a g y B., F o i a ~ C. Harmonic analysis of operators on Hilbert space, North Holland/Akad~miai Kiad~, Amsterdam Budapest, 1970. 2. H ~ E o ~ ~ c ~ ~ ~

H.K.

~eE~

od onepaTope c~m~ra, M., HayEa,

I980. R.TEODORESCU

Universitatea Bra~ov Facultatea de ~atematic~ B-dul Gh.Gheorghiu - DeJ 29, 2200 Bra~ov, Romania

V.I.VASYUNIN

CCCP, 191011, ~eH~Hrpa~,

(B.H.BAC~mS)

~OHTaHEa 27, ZOMM

158

4.14. old

TITCH~ARSH'S THEOREM FOR VECTOR FUNCTIONS In one version (from which others can be derived) Titchmarsh's

theorem states: i f ~ a n d ~ L~(~+) s u c h t h a t ~ * ~ a n d

i f

t h e n

~

Fix ~

~

v a n i s h e s

m u s t

rnishes on (0,~) .

o n

v a n i s h

, and denote by M ~

a r e f u n c t i o n s o f v a n i s h e s o n (0,~), n o

i n t e r v a 1 (0,6),

o n (0,~)

the set of all

~

. Here is a PROOF. such that ~ * ~

va-

is a closed subspace invariamt umder shifts to

the right. Beurling's theorem states that transforms of functions in ~

M

, the space of Fourier

, is exactly ~ H g

, where

~

is inner

in the upper half-plane and H ~ is the Hardy space on the half-planeC Since ~

contains all functions vanishing on (0,1) ,(~(~))m~p(~%)

is

an inner function too. The known structure of inner functions implies that ~ ( % ) ~ X p ( ~ )

for some

~

, 0<~G~I

. This means that ~

tains all functions that vanish on ( 0 ~ ) , nishes on (0,~-~)

. Hence ~ = ~

Suppose ~ Hilbert space

and ~

@

, so ~

con-

and it follows that ~

va-

must vanish on(034) . @

are functions in ~ ( ~ + )

with values in a

, and suppose the expression

j"

G

CI;

0

vanishes for 0<~c~
What

ne~l~,

is

zation that

th

t h e

of are

right

ere

cl

shift

a

simple

s u b s p a c e s

o s e d , s ,

cha

and

inva

riant

contain

M

ra o f

L~(~+)

under all

h i n g o n (0,~) ? By the vectorial version of Beurling's theorem (see

tions

c t eri-

lunc-

vanis

[I]) the

problem is equivalent to describing the inner functions Q such that Q(~)-I ~ p ~ is also inner. In the vectorial context, an ~nner function

Q

is analytic in the upper half-plane, takes values in the

space of operators on H values ~(X)

Q(~)

, satisfies ~Q(g)J g I

, and has boundary

that are unitary for almost all real ~

. In our case

has spectrum (the support of its Fourier transform) i n [ 0 ~ ]

and so is entire. We obtain inner functions of this kind in the form e~cp $% A where A

is a constant self-adjoint operator satisfying 0 ~ A ~ I

The corresponding subspace M

is easily described. Let ( ~ )

, •

be the

159 spectral resolution of A ; thus H$ = @ for ~ ~ 0 and~=H for $ > I . ~ is the set of vector functions ~ such that ~(~) lies in ~ $ for almost every $ . A straightforward extension of Titchmarsh's theorem would assert that the integral above vanishes for 0 ~ ~ I only if the inner product vanishes identically for such ~ . This is equivalent to saying the inner function of ~ necessarily has the form ~ p $ ~ A . This is not true, as shov~ by an example of Donald Sarason. His example leads to a method for constructing such inner functions. Set ~ ( ~ ) ~ ( ¢ ~ p ( - ~ / ~ ) ) ~(Z) ; then the unitary function ~(~) has s p e c t r u m i n [ - { , % J . Write ~ = ~ ¢ % T with ~ , T setf-adjoint. The fact that ~ is unitary means that ~ and ~ commute at each point, and ~ 4 Suppose ~ is two-dimenqional and ~ = ~ I , 0~ • ~ I . Then on the real axis ~ must be (! ~ where ~ and ~ are entire functi-

T~=I.

ons of exponential type at most -~- , ~

a~d ~ ~q-I~l~=~-'~ ~. -~ -~-

~he choice

is real on the real axis,

~ ' ~ = ~ co~'~,

~=~~

, gives

Can the s t ruc t ur e of Q cribed simply in general, even when H is two-dimensional?

be or

des-

REFERENCE I. H e 1 s o n H. Lectures on invariant subspaces. NY-London, Academic Press, 1964. HENRY HELSON

Department of ~ath. University of California Berkeley, California 94720 USA

16o 4.15.

SO~ FUNCTIONTHEORETIC

PROBLEMS CONNECTED WITH THE THEORY OF SPECTRA~ ~ASURES OF ISO~rRTRIC OPERATORS

Let V be a completely non-unitary isometric operator in a separable Hilbert space H with the defect spaces N and M :

V~HeN -~HeM where it is supposed for definiteness that 0 < ~ N-~M.~®. Let PB denote the orthogonal projection of H onto the subspace L, and let T v - V P ~ . The operator V defines in the unit disc D an operator-valued holomorphic function

%vm=z% (l-~Tv

f IM

which is called the characteristic function of V Consider the class B(M~N~ of all operat or-valued contractive holomorphic functions in ~ taking values in the space of all bounded operators from M to N . Let ~u ( M, N) =

--{%~B(M,~):~(0)=0],

It is known that ; ~ E ' B ° and that for every ~ . there exists an isometric V with given defect spaces N and M such that ~ - - ~ (see [I] -[3]). It is also k11ow~1t ~ t in the case ~ N--~ M all unitary extensions of V not leaving ~ are described by the fo~ula

U8:7 v +8 Pn where & is a unito~ "pa~meter", & : N - ' M

a&*: I IM ). The spectral measure E ~ mined (up to a unitary equivalence) by ~ = ~ of the following formula

of UF_ can be deterand ~v with a help

(i) V where

~Q-P,a IN

The spectral measures of ,the minimal u n i t a r y extensions of V leaving H (now the case ~ N~ ~ . ~ M is also p e = i t t e d ) can be a l so determined by .(I ) Where the parameter ~ i s already an a r b i t r a r y function in B ( N , M ) . ~he ~ e c t r a l measure of U8 is absolutely con-

tinuous if and only if the measure ~

in (I) is absolutely continu-

161

ous with respect to the Lebes~ue measure on I • Consider a subset ~ @ of ~ " (~, consisting of functions ~ whose measure ~ in the Riesz-Herglotz representation

(W,N)

N)

l is absolutely continuous for an arbitrary choice of ~ in B(~,M). The inclusion %V ~ B @ ( M , N) is clearly esuivalent to the condition that all minimal unitary extensions of V have absolutely continuous spectral measures~ PROBLEM. Find criteria for a ~iven %

~o

i~ ~°(~,N)

to belong

B~M,N) • ~oto ~h~ ~or ~ ~ B°~M, N)

~

~o~.s~on

u)4 b'(T) implies ~ ~@(M,~) being thus a sufficient (but not necessary) condition. @ Suppose in the sequel that ~ M < +OO and let ~ ( M , N ) (~S~(M,~)) denote the family of all ~ in B ( M , ~ ) v~ (~0(M,bN)) with

LE~.

G%ven

~ £ ~ (M, N)

the follow in~ are equivalent:

I) there exists an isometric operator correspondin ~ measure in (I~

~ :N

~r M

with the

satisfyin~ the Szeg8 condition

2) co.dit%on (~) holds for all isomet~ies ~; ~ - - ~ M

;

continuous for almost all ~ (with respect to the invariant measure on the symmetric space of all isometries ~" N -'p M ).

162

N) B° MN)

We don't know any example of a function in B ~ (M, not satisfying (2). A more subtle sufficient condition for ~ ~Z ( ~ to belong to i~n@( ~ N ) can be deduced4from results of [4]-[6]. Nszmely, fix % " nd denote by ~ and (~4 ~ (tmique) solutions of the factorization problem

in the classes of outer functio~ non-negative at the origin and belonging to

BiN, N)

IZlP~}

and B (M,M)={I~m):k~d/~)~B(M,M),

respectively.Let

It follows that the values of ~.-~. ~ are contra~tions N---I"M a.eo on T • Consi'd%r%%e~Hankel operator 7~ with the matrix symbol ~0 " The operator F ~aps ~ (N) into ~ (M) and its matrix in the standard basis is ( t (-i-k +4))i,~ , where stands for Fourier coefficients of ~0 . Consider subspaces

No-{e~=t~, o,o,...):'~ ~N}, Mo={~ = oZ,O,o,...): ~, M] of ~ ( N )

and ~ ( M )

and put

where .~...(0,~) , "I~ ~f positive square roots

[%od-f ~" .-- rr*)41 MoT

PNod~-#r'r)-'lNol'~' respsctivel,

and finally

"and

163 It turns out that

~D~[,'[,|I)

provided

(4) 0

then

satisfies (4).

RE~L~RK. ~or % ~ B following formula

0

~M

N)

condition (4)holds iff the

[ P+ establishes a one-to-one correspondence between the set of operator (~---~ I - ' "° a' valued contractive, functions/.~_. __,~^ _ ~with _ the same prin-

and all f~c~ions ~ I. 2. 3. 4.

5.

6.

in

B CN, MI -

REFERENCES ~ M B ,~ M ~ M.C. 05 OAHOM ~ a c c e a M H e ~ x onepaTopoB B rHab6epToBOM npocTpaHcTBe. - MaTeM.c6. I946, 19(6I), 236-260. Z H B m ~ U M.C. MsoMeTp~Mec~e onepaTopM c p a B H ~ ~eSe~TH~H ~Hcaa~H, ~Bas~yHHTapHMe onepaTop~. - MaTeM.c6., I950,26,247-264. Sz.-N a g y B., ~ o i a 9 C. Harmonic analysis of operators in Hilbert space. Budapest, Akad.K~ado, 1970. A ~ a M 2 H B.M., A p o B A.3., E p • ~ ~ M.£. 5ecHoHe~H~e ~aH~eaeB~ MaTp~U~ ~ O605~eHH~e npod~eMM KapaTeoAop~-~e~epa ~ M.~ypa.-~yH~.aHaa.H e~o npHa., 1968,2,B.4, 1-17. A A a M ~ H B.M., A p o B A.3., E p e ~ H M.£. 5ec,oHe~Hue 6aOMHO-~SHEe~eBM MaTpHLIM M CBHSaHH~4e C HMMM npofiaeM~ npoAom~eHHa.14sB.AH ApM.CCP, ¢ep.Ma~eM.,1971,6, I81-206. AAaMmH B.M. HeBMpo~AeHHble yHHTapH~e csen~eH~2 No~yyHHTapH~X onepaTopos. -~yH~U.aHaa.~ ero np~., 1973 ,7, BHH.4, I-I6.

V. M. ADAMYAN

(B.M.~) D. Z. AROV (~.3.APOB) M. G. KR~IN

(M.F.EPE~)

CCCP, 270000, 0Aecca, 0AeccEH~ £ooy~&pCTBeHHM~ yHMBepC~TeT CCCP, 270020, ne~aror~ecK~ CCCP, 270057, y~.ApT~Ma 14,

0~ecca, 0AeccEMB MHCTMTyT 0~ecca, ~tB.6

164 4.16.

THREE PROBLEMS ABOUT I. Let

~= il0 ~

morphic in ~

- 0I~ )

is called

~-INNER MATRIX-?UNCTIONS

. A matrix-function (m.-f.) ~-

i n n e r

~

mero-

if

Let ~ ÷ denote the class of m.-f. with entry functions representable as a ratio of an ~-fu_uction and of an outer ~-ftmction. A ~-inner m.-f. ~ is called: I) s i n g u 1 a r if ~4E ~ and 2) r e g u I a r if there exists no nonconstant singular ~ -inner m.-f. Wo such that W W ~ 4 is ~-inner. THEOREM I. An arbitrary tatiom

W =~

and singula r a constant

W s

~-izL~,er m,-f.

admits a represe n-

WS are respectively re~/lar

, wher e ~% and

J-inner m.-f. ; W ~

~

is.. unique!.y determined by~/uD %o

J-nnitary right fac~Qr

2. The importance of the class of regular I-inner m.-f. is explained in particular by its connection with the generalized inof finding all m.-f. terpolation problem of Schur-Nevanlin such that ,

where ~4 , ~£ , ~o ( £ ~ ) are given m.-f. of order ~ , ~4 and ~ are inner, ~ denotes the set of all m.-f. of order • holomorphic and contractive in D ; ~ (~< p~< +co') is the class of m.-f. of order ~ with entries in ~? . ?ix Ith ( k -- 1,2). Whe ra es over the set of solutions of problem (I), the values S(Ko) fill a matrix ball. If the right and left half-radii of %his ball are nondegenerate then problem (I) is called completely indeterminate; this definition does not depend on K@ • Let W ~~ be an arbitrary ~-iuuer m.-f. It has a meromorphic quasi-continuation to the exterior D e of the disc ~ . We denote ~~(~)=~*(~) . We have [2]

165

where

~

and

~

are inner m.-f.,

~

and

~

are outer m.-f.,

~4 ~ 5 ~ , (~;)-( ~ B m . Singular m.-f. W are characterized by equalities ~4 = ~ = I~ in (2). The following theorem shows that it is important to establish a criterion of regularity of a J -inner m.f. TKEOREM 2. Let and let

W =[W~{

be an arbitrai7/

~ , ~Z , ~, b e mo-fo defi=ed in (2~° Then

~ -inner m.-f. problem (I)

with these data is 90mpletel~ ind,eterminate and the m.-f.

~ ~8

,

where

(3) are its solutions, The famil,y problem (I) iff the

{ 56 }

is the set of all solutions of

J -inner m.-f.

W

is regular. For an~

oompletel,~ i n d e t e ~ n a % e problem (I ~ there exists a regular nor m,-f.

~/= [ W ~k]~

one-to-one

, fo r which

that m.-f, this case

formula (~) establishes a

correspondence between the set of all

set of all solutions of 6~ and ~

tary right

J -in--

R r ~ l e m (1). M.-f.

~

~ ~B~

, and the

may be chosen so

4~ in (2) be the same as in problem (1)~ in

is defined 'bz ...... problem ~S) up to a constant

J-tmmi-

factor. @

3. I-inner m.-f. W = [ W4k]4 being arbitrary, let us consider m.-f. ~4 and ~ corresponding~ to it by (2) and define a m.-f.

--

•

+

It takes unitar~j values a.e. on T and ~ E H~ • If a m.-f. ~ unitary on , up t o t h e l e f t we w r i t e

~

~=0

is representable in the ~ , b e i n g d e t e r m i n e d by

with ~ Z % 0 , then . The following theorem holds (see [I] and ~heorem

constant

factor

166 2). THEOREM m,~f.

~I

3. ~

~-inner ~f.W:[~.K] ~ is regular iff,

defined in (4) we have

COROLLARY. Pot a

~Ii=0

~-inner m.-f.

W=

for the

.

[W~I~

to be regular it

is sufficient, that

The proof of Theorems 2 and 3 is based on the results about the problem of Nehari [3-5] to which problem (S) is reduced by a substitution

~=B~~~ .

PROBLE~,~ I. Pind a criterion for a

~-inner m~-f.

W

to be re-

gular without usin~ th e notion of' the index of a m.-f. 4. It is known [6] that a product of elementary factors of Blaschke-Potapov of the Set, 2hd and 3rd kind with the poles, respectively in D , in ~ and on ~ (see~7]) is a J-inner m.-f. We have [1] THEOREM 4. ~

~-~

m.-f.

~

is a product of elementary fac-

tors of the let and the 2nd kind iff it is regular and the m.-f. $~ and

~

associated with ~

by (2) are products of (definite) elemen-

tar2 factors of Blaschke,P0tap0v. REMARK. Both m.-f. ~ and ~ ducts iff

in (2) are Blaschke-Potapov pro-

T

T

~& corresponding condition also exists for a product of elementary factors of only the let (2nd) kind. In this case ~ = I~ ( ~ - - I~) and instead of (6) we have

T ~,t~ T

T

167

T ~4 T

(8) T

COROLLARY. Suppose condition (5) holds for a

J -inner m.-f.

W W~k]~ . The~ W is a product of elementary factors of the let and 2nd kind (only of the let, only of the 2nd) iff condition (6) (respectively (7), (8)) is valid. PROBLEM2.

Find a criterion for a

l-inne~ m,-f. to be ~ ~T° -

duct of elementary faqtors of the lsto 2nd and 3rd kind. Theorem 4 gives in fact a criterion of completeness of a simple operator in terms of its characteristic m.-f. ~ in case when its eigenvalues are not on T . The solution of ~ o b l e m 2 would give a criterion without this restriction. PROBLEM 3. Find a criterion for a

l-inner m.-f. to be a

product of elementary factors of the ~rd kind. Let us point out that such a product is a singular m.-f. A product of elementary factors of the ]st kind arises in the ~angent ~ problem of Nevanlinna-Pick [8] and products of factors of the let and 2nd kind arise in a"bl-tangent"problem in which '~angent"data for ~(~) and ~*(~) are given in interpolation knots ~ D ) . The author's attention was drawn to such a'~i-tangen~problem by B.L.Kogan. Products of elementary factors of the 3-d kind arise in the'~angent"problem which has the interpolation knots on T . The definition and investigation of such problems is much more complicated

[9, ~o]. REFERENCES I.

2. 3.

A p o B ~.3. 05 O ~ O ~ E H T e p n ~ o H H o ~ s ~ a ~ e ~ EH~e~Ni~2THOM npoEsBe~eHH~ Bx~Ee-HoTanoBa. Tes~cH AOF~S~OB. Hh~oxa no T e o p ~ onepaTopoB B ~yHFI~.npocTpa~cTBaX, ~ C E , 1982, 14--15. A p 0 B ~.3. P e a x ~ s ~ M a T p E I ~ - ~ y H ~ no ~apln~HPTOHy. -- MSB. AH CCCP, cep.MaTeM., 1973, 37, ~ 6, 1299-1331. A~ aMH H B.M., Ap o B ~.3., Kp e ~ H M.r. ~ecEoHere PaHEe~eB~ Ma~p~n~ ~ o6o6~eHHHe ss~aqz KapaTeo~opH-~e~epa

168

H.N~rpa.-~ym~.aEaxz3 ~ e r o np~oz., 1968, 2, BWli.4, 1--17. 4. A ~ a M ~ H B.M., A p o B ~.8., E p e ~ H M.P. BeOEoHe~HHe 6~o~Ho-raHEe~eBNe MaTpHI~ ~i CB~SaH~e C HaME npo6J~eMu n p o ~ a m ~ e ~ . - H s B . A H ApM.CCP, MaTeM. t I 9 7 I , 6 , ~ 2 - 3 , 8 7 - I I 2 . 5. A ~ a M ~ H B.M. HeB~pom~eHHNey~TapHNe c ~ e r ~ e ~ n~Tap-

H~X onepaTopoB. - ~ . a H a ~ H s ~ ero npMo~., 1973, 7, BLIn.4, 1-17. 6. A p o B ~.8., C HM a Eo B a ~.A. 0 rpavmvgHx sHa~eH~XX cxo~m~e~ca noc~e~oBaTex~HOCT~ J-cm~a~mx MaTp~-Qy~. MaTeM.saMeTF~, I976, I9, ~ 4, 491--500. 7. H O T a n o B B.H. Myx~T~mmEaT~BHa~ cTpyETypa J--HepacT~-l~Ba~m~x Ma~p~-~y~En~. - T p y ~ MOCE.MaTeM,o--~a, I955, 4, I25--236.

8. $ e ~ ~ ~ H a M.II. KacaTe~Ha~ npo6xeMa H e B ~ - I I ~ E a C EpaTHMM~ TO~IEaM~. --~oE~.AH ApM.CCP, 1975, 61, ~ 4, 214-218. 9. E p e ~ H M.~. 06m~e ~eope~ o nOS~T~BH~X ~ysEn~oHa~ax. B ~H.: Ax~esep H.H., Kpe~H M. 0 HeEoTopux Bonpocax T e o p ~ MOMeH-TOB. XapBEOB, 12I-I50. (Ahiezer E.I., Krein M. Some Questions in the Theory of Moments. Trans.~th.Mon, 124-153.)

, AMS, 1962, v.2,

I0. M e x a M y ~ E.H. l ~ p a ~ a ~ sa~a~a H e ~ - H ~ E a ~ l-paCT~Ba~X MaTpH~-~. - HsBeCTI~.~ B~IC~I~X y~e6H~x saBe~eHm~, MaTeM., 1984 (B negaTe). D. Z.AROV

CCCP, 270020, 0~ecca, K O M C O M ~ C E a ~ yX., 26, Q~eccEE~ r o c y ~ a p c T B e ~ n e ~ a r o r ~ e c E ~ HHCTETyT

169 4.17.

EXTREMAL MULTIPLICATIVE REPRESENTATIONS

Let ~ be the class of entire functions l.tW of exponential type with values in the space of all bounded operators on a separable Hilbert space and such that

W@ 4,

(lX=o3.

there exists an operator-valued hermitian non-decreasing function E o n [0,~] (E(O)--~, V~J~ E = i ) satisfying Fo= e ery

EO,{]

(see [1, 2~. L~t to

H be

lent

0 the weak derivative of a~~

E

• T h e n (I) is e q u i v a -

0 The function W determines H uniquely iff I - W ( ~ ) ~ C~C) and W , d ~ W have the same exponential type [3]. To single out a canonical function from the family of all functions H satisfying (2) in the general case the following definition is introduced. DEFINITION~ Let n H be a weakly measurable function on [ @ ~ ] ( 0~ ~ , IHI~ ~ [ov,~] ) and suppose that for every S~[~,~] C [ ~ ] the function $

is W&

=E

(~) 1~

the greatest divisor (in ~ ) among all divisors of of type S - ~ • Then H is called an e x t r e -

f u n c t i o n

o n

THEOREM 1. For ever2 W

[~t~]'

of exponential type ~

in ~

there

~

exists a unique H

extremal on

[O,~'] and satisf.ying WO,6.

--

=Wcx . This result is a special case of a theorem proved in [2] (compare with [4] ).

170 PROBLEM. Find an intrinsic description of functions

H

extre-

The following theorem shows that the description is very likely to be of local character. THEOREM 2 ([5~[6~). Suppose ~ is extremal on any [~,~] C [ ~ 6 ]

[(~,~)]

S~

hat

is extremal on [ ~ ]

. Conversel,y~ if for evelV

there exists a segment

S

H

CONJECTURE. Le__~t H

[ ( ~ ] C. [~)~]

is,, xtrem l on

such

H

b e a continuous (with respect to the norm

t0PolO~V ~ operator -valued function on mal iff all values of H

. Then it

[@,~]

. Then H

is extre-

are orthogonal projections.

In the particular case A ~ H(~)=4 , ~ 6. [~,~] the conjecture is true by a theorem of G.E.Kisilevskii [7] (in the form given in [8]). Similar questions in case when W ( ~ ) is the characteristic function of a so called one-block operator have been considered in

REFERENCES I. H o T a n 0 B B.H. My~T~a~NaTRBHa~ 0Tpy~Typa --2epacT~l~BaM a T p ~ - ~ y R E ~ . --Tpy~ MOON.MaTeM. O6-Ba, 1955, 4, 125--236. U

2. F ~ a s 6 y p P TH ovpaaEenH~x

D.H. Myx~T~n~mEaT~2HHe npe~cTaBXeH~ ~ ~ o p a a aHaxzT~ecE~x o n e p a T o p - ~ m ~ . - Sya~.aHax.

ero np~x., I967, I, ~ 3, 9-23. 3. B p o ~ c ~ E ~ M.C. Tpey20~HHe ~ mop~aHoBH n p e ~ c T a B x e ~ x~n e ~ H x onepaTOpOB. -Moc~Ba, "HayEa", 1969. 4. B p o ~ c E n ~ M.C., M c a e B ~.E. Tpeyro~nHe npe~cTasxeH~S ~ C C ~ n a T ~ B ~ X onepaTopoB c pesox~BeHTo~ ~Ncno~e~u~ax~o~o ~ Ha. - ~oE~.AH CCCP, 1969, 188, • 5, 971-973. 5. ~ ~ n s 6 y p ~ D.H. 0 ~ex~exsx ~ Mm~OpaHTaX o n e p a T o p - ~ ol~a~n~eHHoPo B~Ka. - MaTeM. ~oc~e~oBaR~, E~m~eB, 1967, 2, ~ 4~ 47-72. 6. M o r ~ x e m c ~ a a P.~. HeMOaOTOH~Ne M y ~ S T ~ N a T ~ B H M e npe~cTaBxea~ o ~ p a a E e n ~ x a s a m ~ T E e c m ~ onepaTop-~m~. MaTeM. ~ccxe~o~a~s, E ~ m ~ H ~ , 4, ~ 4, 1968, 70-81.

171

7. K m c ~ x e B c E ~ ~ r.3. 142BapEaHTH~e Ho~pocTpa~cTBa BOX~TeppoB~ ~CCHIIST~BHHX onepaTOpOB c ~ A e p H ~ ~ H ~ m NO~nOHeHTa~ . - H s B e 0 T ~ AHCCCP, cep.MaTeM., 1968, 32, ~ I, 3-23. 8, r o x 6 e p r H.~., K p e R H M.r. T e o p ~ BO~TepposI~X oNepaTopoB B I-~B6epTOBOM IrpOCTpSRcTBe H e~ ~ p ~ o z e s ~ . MoczBa, "HayKa", I967. 9. C a x s o B ~ q ~.A. 0 /~CClrIaT~BHRX BO~TeppoBr~X onepaTopax.fdaTeM.cSopH~, 1968, 76 (I18), ~ 3, 323--343. yvJ.p. GINZBURG

r rs r)

CCCP, 270039, 0~ecca, 0~eccmm~ T e x R o x o ~ e c ~ m ~ HHOT~TyT nm~eBo~ n p o ~ e H H o c T ~ ~.M.B.~oMoaocoBa

172 4.18. old

FACTORIZATION 0F OPERATORS ON

I. A bounded operator 5-(, 5+') on L£'(I;;l',~) , is called i o w e r t r i a n g u I a r ( u p p e r a n g u 1 a r ) if for every ~ (&~<~ ~< ~)

s:

:%

tri-

5+P,:% 5+%,

A bounded operator 5 on L~(&,~) is said to a d m i t t h e 1 e f t f a c t o r i z a t i o n if 5 : 5 - 5 + where ~_ and ~+ are lower and upper triangular bounded operators, with bounded inverses. I.C.Gohberg and M.G.Krein [I] have studied the problem of factorization under the assumption

S-I ~ 11"~. The operators ~;)4.,~_

S+=I+X+,

<:1)

have been assumed to be of the form

S_:I+X_;

X+,X_~.

( ~ is the ideal of compact operator) Factorization method had played an essential role in a number of problems of the spectral theory. Giving up condition (1) and considering more general triangular operators would essentially widen the scope of applications of this method. EXAMPLE. Consider [21 the operator

%t

v.p.I 0

The operator ~ (5~0)j_ clearly does not satisfy (1). Nevertheless S~ adraits a factorization S~,,,=W& W&* with ~=~ @%C~ and the~lower triangular operator W& defined by the formula,j

-

(b(~4) 0

~t.

173 The following condition is necessary for an operator admit the left factorization: ~

= ~

~

P

is invertible in

L~(~,~) for an~

~

to

~ ( ~ )

PROBLER 1. For what classes of operators condition

(*)

C*)

is suf-

ficient for the existence of theleftfaqtorization? tor

In the general case ~ defined by

S#: ~CZ) ° + ~.f~,

(~)

is not sufficient. Indeed, the opera-

~,-i, v.p.I k~

~,~,' 0<~<~

(2)

0

satisfies C~) but does not admit the factorization, [2]. Note that (*) follows from (~@) defined below: Operator

~

is bounded positive" a n d h a s a

bounded inverse. ~,)

An important particular case of problem 1 is the following PROBLEM 2. Does

~)

imply th e existence of a factorization?

If $-I g ~ the answer is positive [I]I~ is the Matsaev ideal), 2. It is interesting to study problems I-2 for operators of convolution type

0

$ *

CAoS- Ao) ~-: ~ I ~c~)[ Mc¢~)÷~,(b] ~,¢,,

(4)

0

where A01=* I~{~hl~ . ~f (.~ tions of second order make sense:

holds,the fonowi~ matrix-f~c-

Wc.~,~)=I_~ ICS)'P~CAo -zl)-~M,~),

T I

174

B (~)=

THEOREM 1. Suppose that the operat0r $ factorization. Then the matrix-functions N~B(~)

in (~) admits the left

~*W(~,Z)

and

are absolutely continuous and

~W

(5)

where the elements

~

i (''~)

of the mat rix

H (~)

satisfy

lhl(~)=l~c~)Flic~) ,

(6)

and

~('~) I-1%~)+P,~c'~) FI~.c'~)=4. The functions

~

, ~

(7)

can be expressed in terms of ~_, ~+ :

7,~(~)=5[~M,I~(~):ST~~.

(8)

Every operator ~ satisfying (3) and admitting the factorization defines (via (6)-(8)) a system of differential equations (5). The procedure of this type in the inverse spectral problem have been developed by M.G.Krein ~%] provided S ~0 and I- S ~ . Besides, Theorem I means that the "transfer matrix-function" [5] W(~) admits the multiplicative representation W(~,~)=

e~ g

.

(9)

175

If ~ is positive (4) implies that teristic matrix-function of the operator (5), (9) are known ~6, 7]. The equality =- t ,

(~0, ~) is the charac~-~Iz ~o ~I~. Then formulae

0 < ac < co,

(~o)

is

new even in this case. An immediate consequence of Theorem I is the necessity of the following condition for the operator in (3) to admit the factorization. Operator B(~)

$

in (3) satisfies

~)

, the matrix-function

i S absolutel2 continuous and (I0) holds.

Note that all requirements of in example (2). PROBLEM 3. Does

($~)

~**W)

but (10) are satisfied

impl~ the existence of the factoriza-

tion? THEOREM 2. If the operator

~

satisfies

both ~ )

and ( ~ , )

,

then At admits the factorization. REFERENCES I.

2. 3.

4.

5. 6.

r o x 0 e p r M.~., K p e ~ H M.r. T e o p ~ BOJIBTeppoB~X onepaTopoB B P ~ B 6 e p T O B O M ~pocTpaHCTBe ~ ee HpE~lO~eH~. M., HayEa, 1967. C a x H 0 B H ~ ~.A. ~aETOpESaLG~ onepaTopoB B (~) . -SyHEs.a~ax. ~ ero n p ~ . , 1979, 13, B~n.8, 40-45. C a x H o B H ~ SI.A. 06 HHTePpa~BHOM ypaBHeHl~ C ~ p O M , 8aBE-C ~ OT paSHOCTI~ apryMeHTOB. -- MaTeM.ECCJIe~OBaH~, l~z~HeB, 1973, 8, ~ 2, 188--146. E p e ~ H M.r. KOHTEHya~IBH~e s2a~ol~ n p e ~ o ~ e H E ~ o MHOID~e-Hax, OpTOrOHa~BH~X Ha e~HHH~HO~ oEpy~HOCT~. - ~ O E ~ . A H CCCP, 1955, I06, ~ 4, 687-640. C a x H O B ~ ~ ~.A. 0 ~aETopEsau~H nepe~aTo~o~ onepaTop~yHELIH~° - ~ O E ~ . A H CCCP, 1976, 226, ~ 4. Jl E B • E ~ M.C. 0HepaTop~, Eo~e6aH~, BO~HN. 0TEpMTNe 0~CTEMP. M., Hayza, 1966.

176

.

H o T a n o B B.H. Myx~THmmEaT~BHa~ c~pyETypa ~ -EepacT~l ~ B a ~ x MaTp~11-~yHlg~. -- Tpy~M MOOE.MaTeM.O--Ba, 1955, 4, 125---136. L.A.SAHNOVICH

(~.A. CAXHOBM~)

CCCP, 27002I, 0~ecca 3~e~TpoTex~n~ecE~ ~ C T H T y T C ~ 3 ~ m~.A.C.HonoBa

177

4.19.

EVALUATION OF AN INFINITE PRODUCT OF SPECIAL NATRICES

An important r$1e in studying the integrable models of Field Theory is played by matrix-functions of complex variable of a special form [I]. The simplest example is provided by a rational matrix:

(i)

Lo(,z)- z÷p

where ~ is a matrix of size ~ x~ and ~ is a complex number. It is natural to call it the matrix Weirstrass factor for the complex plane C (i.e. a meromorphic function such that ~(~) = ~ ). The next interesting example is given factor for a strip. This function LI is 0<~e~ ~ ~ } has only one at infinity, i.e.

{~EC:

where of

~+

L~Cz)

on

~

by a matrix Weirstrass meromorphic in the strip pole in it and is regular

are non-degenerate diagonal matrices. satisfy the following relation:

t~(~+~) = A L ~ ( ~

with one pole and

The boundary values

~, ~e~,

(2)

"" : "-~-I " One can represent such a matrix-function as an infinite product of functions (I). For :

this purpose introduce the family of matrices

:A" Lo(z÷.OA

(3)

and their finite product

LN4Cz) L,N(z) LNq (z)--. L"N÷~c~} L-N (z). =

(4)

It is easy to show that the regularized limit

(5)

178

satisfies (2). Por for

~A~

formulae (3)-(5) are nothing but Euler's formulae ~=~ , so that

L~C~)=~ , ~ (~ +~?

in ~2]. In this case

We calculated

A = azo,~ (~,-~)

and S+

so

that

-,$~

"'-~Qce(~)= O .

The limit in (5) is defined as follows

where

DN -The limit matrix

L~ C~)

l

lI s~ o

0 1 N~3

"

has a f o r = .

L~(~) = W -~ L,~C~ W,

W: l k C0~)'l

O /, k,cSD)

I I"(4+1~-z) r'(4-1t-~) ! e_~

~,(~)=

t'~~'tz ~_ It~)

'

I

179

~he~e ~ ~ 5~ ~ 5~ 5_. We pose as a PROBLE~ the explicit calculation o f the limit in (5) for ever~j ~

in terms of known special functions.

REFERENCES I. F a d d e e v L. Integrable models in I + I dimensional quantum field theory. CEN-SACLAY preprint S.Ph.T./82/76. 2. p e m e T ~ X ~ H

H.D.,

~ a~

~ e e B

cTpyETyp~ ~ ~HTezp~pyeM~X Mo~exe~ T e o p ~ I988, 57, ~ I.

~.~.

raM~TOHOBH

nox~. - Teop.MaT.$~8.,

L. D. FADDEEV

CCCP, 191011, ~eHEHrps~,

(~.~.~)

• oHTaHEa 2V, ~0MM

N.Yu. RESHETIHIN (H.D.P~II~TMXEH)

180

4.20. old

~ACTORIZATION OF OPERATOR ~GNCTIONS (CLASSIFICATION OE HOT OMORPHIC HILBERT SPACE BODNDLES OVER THE RIEMANNIAN SPHERE)

Let H be a Hilbert space,~,=~(H) the Banach space of bounded linear operators in H , and G ~ L ( H ) the group of invertible operators in L . We put ~ I ~ : I ~ I ~ I } and~_~I~u~°@} " ~ I~I~<0o} and denote by ~(~,~), ~(~÷, ~ ) , ~(T-, ~ ) the groups of holomorphic GL-va!ued functions in a neighborhood of~,T$,~_ respectively. We shall say that two functions ~,T(~,Ta0(T,~I,)) are e q u i v a 1 e n t if~=A_TA+ for someA+_,A±G~(~±,~). PROBLEM. Classify the functions i n ~(T,@~)

with respec t to

thisu notion of equivalence. REMARK. It is well-known that this problem is equivalent to the classification problem for hclomorphic Hilbert space bundles over the Riemannian sphere. I. What is known about the problem? We shall say that ~ is a diagonal function if ~ ( ~ ) = Z t ~ P~ , where ~ < - - - ~ ~ are integers and ~ .... , ~ are mutually disjoint projections in I,(H) such that p1+...+ ~ '[11 ; the integers ~ are called the p a r t i a 1 i n d i c e s o f ~ a n d the dimensions ~ ~71~ P~ will b~ called the d i m e n s i o n s of the p2artial indices ~ It is easily seen that the collection ~, ....~ , ~I""' ~ determines a diagonal function up to equivalence. Por ~ < + ~ it is well-known (see, for exmmple, KIS, ~2~) that every function in ~ ( T , ~ ) is equivalent to a diagonal function, a result that is essentially due to G.D.Birkhoff [3]. ~orS~??~H = @ ° this is not true. A first counterexample was given in [4~. We present here another oounterexample: Let H ~ - ~ @ H~ be a decomposition of H and V G ~ ( ~ I , H ~ ) . Then the function defined by the block matrix

(~-' V

~O)

(I)

is equivalent to a diagonal function if and only if the operator V has a closed image in H~ , as is easily verified. However there are positive results, too: HEOREM I [ 5 ] .

compact for all ~

T et

, ~V

. if , then A

the

are

is equivalent to a dia~onal

181

function whose non-zemrO' partial indices have finite dimensions ,. For A ~ ( ~ , ~ ) we denote by W A the Toeplitz operator defined by WA~-----~(A~) , where ~+ is the orthogonal projection from h~Qq,~) onto the subspace ht(~,M) generated by the holomorphic functions on ~+ . THEOREM 2 [4]. A function A e ~ ( ~ h )

is equivalent to a dia-

~onal function , whose non-zero 'partial indices ~ mension h~(~,H)

~

, if and onl.y if •

~

have finite di-

is a ~re@olm operator in

If the condition is fulfilled,then i % ~ W a = ~

and

~I£

WA=

~> o further results see

and the re

erences in these

papers. 2. A new point of view. In [6] a new simple proof was given for Theorem I. The idea of this proof can be used to obtain some new results about general functions in ~(T,~,) , too, THEOREM 3 (see the proof of Lemma I in [6] ). Every function from @(~,~b)

is equivalent to a rational function of the form

Let A ~ ( T , ~ [ , ) . A couple qO=(%0_, ~+) will be called a @5 section of A if ~_ , ~ are holomorphic H -valued functions on T - , T÷ , respectively, and ~ _ ( Z ) ~ A(~)~. (~) for~T . Then we put ~(~)~-~+(~) for l~I~ ~ and ~(~3= ~_(~) for ~
~<...<~

b grs 1 4 , . . . , ~ { 4 , ~ , . . . ~ }

, Ae~(T,~)

, there exist

(the partial indices of A

), uniqu e num-

(the dimensions of the partial indices)

and families of (not necessar2 close d~ linear subspaces

182

such that

(~) X~}(~)\ ~ t ( ~ ) 0~1~1~

). I f

for some point ~o

I f ~4,...,8~

if and only if ~(x,~,A)--~ (~=~,...,~ ; ~0 is a

~j -section of A

and (p(z)eM}(zo)\M~_I(Zo)

, then ~0(z)eMj(~)\M~_t(2)

for a l l O ~ l ~ l ~ .

are l i n e a r l 7 idependent vectors i n H

poin t %. , Cp}

are .(~c}.~o,k)

-sections of A

and~ f o r some with (~}(Zo)~---~} ,

then the values ~(~), .... (p~(~) are linearl~ independent for all @

(ii) The function A

is equivalent to a dia~onal functi0n if

and only if the spaces ~ ( ~ )

(0~I~I~;~=~,

---,~)

are close d.

For this it is sufficient that at least for one point ~o ~ ~

the spaces

(~o) are closed, Further, it is sufficient that the dimensions are finite with the exception of one of them. (iii) There are Hilbe~% spaces

tor functions ~;:T~--* ~(H~,H)

and ho!omorph~c opera-

such that ~ ( ~ ) = ~ ; ( ~ )

(iv)

for all

forl~

~ ,

(#=1, ....

The proof of this theorem uses Theorem 3, the method of the proof of Lemma 2 in [6] and the open-mapping-theorem. From Theorem 4 we get a collection of invariants with respect to equivalence, the partial indices and its dimensions. However this collection does not determine the equivalence class uniquely~ because, clearly, for every such collection there is a corresponding diagonal function, whereas not every function in ~ ( T ~ ) is equivalent to a diagonal function. It is easily seen that, for ~ £ ~ V = ~ 0 } the function (I) has the partial indices ~4 ~ 0 and ~ 1 ( ~ ( ~ ) ~ H I@ I ~ V for all ~ )o PROBLEM. Are all functions in

~ (~,~)

with such partial in-

183

dices equivalent to a function of the form (1)? PROBLEM. Can we obtain, in ~enera!, a complete classification, addin~ some special triangular block matrices to the dia~onal functions?

REFERENCES I. P r o s s d o r f S. Einige Klassen singularer Gleichungen. Berlin, S974. 2. F 0 X 6 e p r H.H., ~ e ~ ~ ~ M a R H.A. Y p a B B e R ~ B cBepTEax n p o e ~ o R H M e M e T O ~ H ~ X p e m e a ~ . M., "HayEa', I97I. 3. B i r k h o f f G.D. Math.Ann., 1913, 74, 122-138. 4. r o x 6 e p r H.H., ~ a ~ T e p e p D. 06m~e TeopeM~ o ~a~Top~sau~ onepaTop-~R~ oTHoc~Te~no EO~Typa I. roxoMop~R~e ~yHEIMH. - Acta Sci.Math., 1973, 34, 103-120; II. 06o6meH~. Acta Sci.Math., 1973, 35, 39-59. 5. r o x 6 e p r H.H. ~a~aya ~ a ~ T o p ~ s a ~ onepaTop-~yn~s~. - H s B . AH CCCP, cep.MaTeM., 1964, 28, • 5, 1055-1082. 6. ~ a ~ • e p e p D. 0 ~a~TOpmSa~ Ma?p~ ~ o~epaTop~J~E~.

Coo~J~.AH rpys.CCP, 1977, 88, ~ 3, 541-544.

J. LE ITERER

Akademie der Wissenschaften der DDR Institut f~r Mathematik DDR, 1030, Berlin Mohrenstra#e 39

184

4,21.

WHEN ARE DIFFERENTIABLE FUNCTIONS DIFPERENTIABLE?

If ~ • ~ -~- ~ is continuous and ~ is a C * -algebra then there is defined by the usual functional calculus a mapping A {Alin ~-~ { (~) from the linear space of hermitian elements of to itself. What is a necessary and sufficient condition on ~ that for all the function ~A is differentiable everywhere? Taking

A =-~

(1) If

~A

shows that ~

must be differentiableo In fact:

is differentiable for all A

then

~ C

4 (~).

PROOF, Let A be the algebra of bounded functions on an interval [C~, ~]. The differentiability of ~ A at a function ~ asserts that for every ~ there is a ~ such that for any function with II ~ U <

II {(~+k)-

{(~)-~{~).k

II ~ £ .Ukll

#

This shows immediately that

~ a (JC)

must be the mapping ~--~ satisfy I So- tol ~ and k({} the consand take SC ($) to be the identity function taut function So - ½o Then

--({'o ~)~.

~et

S.,t e E~, 6 ]

II kll = ISo-t.I and ~ (OC+~)- f(OC)-- ~ ( ~ ) ~

< ~

is

equal at

½= ~.

to

~ (So) - ~ ({.) - ~'(t.) (s,- to) ~hus

I { (~o)- {(t.)-

changing $@ and

to

~'(t.)(s.-t)l ~< ~ I s.-t.{ , ad~, ~nd ~vidi~ by IS,-~.l

• Intergive

It is even easier to show that if ~ ~ C 4 and A is commutative than ~ A is differentiable. For general A all I know is this: (2) If in a nei~hbcurhood of each point of ~

the function ~

i_~s

equal to a function whose derivative has Fourier transform belonging to

~.~ ( ~ )

then

~A

is differentiable for all A .

PROOF. Of course "each point" in the assumption on

~

can be rep-

185 laced by "each compact set" and since the differentiability of ~A at ~ depends on the values of ~ in an arbitrary neighbourhood of the interval [-II~II, II~II] we may assume ~ itself has derivative whose Fourier transform belongs to L 4( ~ ) . Let 06 and ~ be hermitian. From the identity

d, e~S(oc~-h,) e4s=c = i.e~SC~c~-h,)~e4SOc ol,s we obtain upon integrating with respect to multiplying by e ~%~

eb{(~b)

$

over [ 0,~]

__ e ~{* + L Ire ~s(~+b) ke~(~-s)~&s

and right

.

o

Applying the Fourier inversion formula gives

--~

0

-~

=

0

I÷H,

let us say. The inner integral in i has norm at most Itl" ~Ii and so (since ~ [ ( % ) C ~ - ~ ) ) the double integral makes sense and represents a continuous linear function of ~ o In fact it will define ~A ( ~ ) ~ . To show this it suffices to show that the double i n t e g r a l ~ has norm O ( U ~ U ) as II~ ~ 0 , But the norm of the inner integral in ~ is o ( ~ ) for each ~ and is at most 2 ~I IIWI~ for all ~ and so the conclusion follows from the dominated convergence theorem, @ PROBLEM 1. Fill the 6ap between (I) an d (2). In particular, is ~ C I a sufficient condition for the differ anti ability of for all

A

~A

?

Here is a concrete example. Let A be the algebra of bounded operators on ~2 (~, ~) . If gC is M~ , multiplication by the identity function, ~ ~ ig the integral operator with kernel

186

{', (,)k

(S,~) , then formally tor with kernel

K (s,b

is the integral opera-

f(4;?

(*)

S-'i;

(This is easily checked by a direct computation if ~ mial). Hence we have a concrete analogue of Problem 1:

is a polyno-

PROBLEM 2. Find a necessary and sufficient condition on ~ ~henever

~ ($,~)

that

iA the kernel o~ a bounded operator on ~i(a,~)

then so also is the kernel (*).

HAROLD WIDOM

Natural Sciences D i w University of California at Santa Cruz, Santa Cruz, California, 95064 USA EDITORS' NOTE

Both problems 1 and 2 were extensively investigated by M.~.Birman and M.Z.Solomyak within the very general scope of their theory of double operator integralsC[1], [21 and references therein~ see also previous papers [3], [4]). They obtained a series of sharp sufficient conditions mentioned in Problem 2 and also sharp sufficient conditions for ~ to be differentiable on the set of all selfadjoint operators (Birman and Solomyak considered the Ggteaux differentiation but their techniques actually gives the existence of the Frgchet differential). Let us cite some results, Suppose that as a

[~,~]C

(0,T)

and

~

can be extended from [~,~]

~--~eriodiq function with Fourier series ~

Pu__~t R~(t) = ~ .

~

K'~

(K) e -~-"

(K)

K=-~ if there exists a sequence

IKI~

f

~,~ ~

of ~ositive numbers w,i,th

~

~ < * oO

such that

187 then the kernel (*) defines a bounded operator on ~ 2 (@,~) whenever

K (S~#)

does, In particular this is the case if

< i-oo

(I)

11,=t

Condition (1) is satisfied e.g. if belongs to the H~lder class Aoc with a positive (arbitrary small) O~ or if ~' has absolutely convergent Fourier series. If

~

is defined on the whole real line and

fies the above conditions for any

G~,~ ~

el [~,~]

then ~

satis-

is differentiab-

l e o n the set of all selfad,~oint operators. The Birman - Solomyak theory encompasses many other related problems (e.g. for unbounded selfadjoint operators and for the differentiation with respect to an operator ideal), In particular they considered Problem 2 in a more general setting, namely replacing the quotient ~(S)6-~ -. ~l%) by a function ~ ($, ~) They reformulated this general problem as follows: for which ~ ($,~) is ~(8) ~(~) ~ (8,~) the kernel of a nuclear operator ~ , ~ for any q),tp C L.,~ with ll"[-~,Lp II~d COH~S~II~UL~ ll~llt~ "~ This equivalence leads (via V,V.Peller's criterion [5] of nuclearity of Hankel operators) to a NECESSARY CONDITION for ~ to satisfy the requirements of Problems 2 and I. Indeed, putting ~-= ~----~ we see that ~(~)--~(~) should be the kernel of a nuclear operator. It follows from [5], [6] that this is the case iff Besov class ~

~44 [ ~,

C~

~]

for

any

~, ~ e ~

belongs t,o ithe

~

.

So the condition

is not sufficient in both ..Problems I and 2 •

Let us mention also an earlier paper by Yu.B.Farforovskaya [7] where explicit examples of selfad,joint operators A~j~m wit h sDectr~ i~ [0,~] and of functions

Im

are constructed such that IIA~-S~II-~0,

and

II F cAD-{ (B )II

that the existence of such sequences { AA ' { B~), {~.~

Note follows also

from the above mentioned Peller's results. RE~ERENCES I. ~ z p M a ~

M.m., C o a o M ~ X M.3. ~ M e ~ a ~

o ~0y~

cne~-

188

paa~Horo O~B~Ta. - 3 a m ~ c ~ ~ a y ~ H . c e ~ . ~0MM, 1972, 27, 33-46 M.N. f~Bo~m~e onelm~opm~e m ~ e r l m ~ CT~T~eca m. rlpe~ea~m~ nepexo~ no~ sHa~o~ z ~ , r e r p a ~ . - HpodaeJ~ MaT. ~SH~H, ~S~. ~J, I973, 6, 27-53. E p e ~ ~ M.F. 0 ~e~oTopHx ROBr~X HCC.~0BaH~O:~X no TeopE~ BOSr~~e~ csMocoup.~z~e~x onepaTopoB. B cd.: "IIelBas . ~ e T ~ maTe~T~~ec~. mNoaa" I, l{~eB, 1964, 103-187. ~ a ~ e n ~ H ~ D . & , ~ p e ~ ~ C.F. M~Terp~poBaeze ~ ~ e p e ~ n~poBa~e ~p~TOB~X onepaTopoB ~ npz~o~ea~e ~ Teop~z Bos~yme~m~. T p y ~ ceM~H, nO ~ y ~ . ~ s a J ~ s y , Bopoge~, I956, T.I, 8I-I05. H e ~ ~ e p B.B. 0nepa~op~ P a ~ e x ~ ~ a c c a ~ ~ ~x np~omee~s ( p a n ~ o ~ a ~ a ~ annpo~c~aUz~, ~ayccoBc~ze nponecc~, npo6~e~ ~a~op~T~z oNepaTOpOB).-~aTeM. C6OpH~E, I980,I~3, ~ 4, 539-88I. P e 1 1 e r V.V. Vectorial Hankel operators, commutators and related operators of the Schatten-von Neumann class ~ - Integr. Equat, and 0per~Theory, ~982, 5, N 2, 244-272.

2. B ~ p M a H

3.

4.

5.

6.

7. ® ~ ~ , o ~ o ~ c ~ ~ ~ ~.~. 0~e~a ~ o ~ ca~ocon~s2Nx onepaTopoB 1976, 56, I43-162.

~

~ ~

I ~ ( ~ - ~ ~A)I

. -- 8 a n z c ~ I ~ i a y ~ H . c e ~ . ~ I 0 ~ 4 ,

189 4,22, old

ARE MULTIPLICATION AND SHIPT UNIPOR~KLY ALGEBRAICALLY APPROXIMABLE ?

O. NEW DEFINITION. A family ~ ~ I A00 : 0 0 6 ~ I } of bounded operators on Hilbert space H is called u n i f o r m i y a i g e b r a i c a i i y a p p r o x i m a b I e or (briefly) a pp r o x i m a b i e if for every positive E and for every ~0~/I there exists an operator A~, 8 such that

cae ..0.. b) the ~-algebra (i.e. algebra containing ]~* together with ) spanned by ~ A 0 0 , 8 ~ is finite-dimensional *) In particular, an operator A is called a p p r o x i m a b 1 e if the family I A } is approximable. In this case ~ ~ A, ~ A is approximable also. Given an approximable family ~ and ~ ~ 0 let O ~S denote the algebra of the least dimension ~ %j among algebras satisfying a) and b). The function 6 H(~.,~) ~s'~,~l~,-~ 8 is called t h e e n t r o p y g r o w t h of ~. I. THE MAIN PROBLEM is to obtain convenient criteria for a family of operators (in particular, for a single non-selfadjoint operator) to be approximable, and to develop functional calculus for approximable families. See concrete analytic problems in section 5. 2. KNOWN APPROXIMABLE FAMILIES. The first is I~ } with A~--~*. Indeed, let ~ -----~ ~ (PA~ -- P ~ _ ~ ) where ~ } ~ = I forms an B

-net for the spectrum of A and ~ A } is the spectral measure of A . In this case ~ (8, A ) coincides with the usual 8-entropy of ~ p ~ ~ considered as a compact subset in ~ . Let ~ I ~,f~..., A~, ~ be a family of commuting selfadjoint operators. It is clearly approximable with ~$, ~ defined analogously. The entropy H ( ~ % ) is again the 8-entropy of the joint spectrum in ~ . The same holds for a finite family of commuting normal operators. Let now % be a finite or compact family of compact operators. *) We do not require the Ldentity in A 6 to be the identical operator on ~ in order to include compact operators into consideration. If the identity of A ° is the identical operator I on then ~ a ~oes not contain co~pact operators and defines a decomposition ~ - - ~I ® ~ with ~ ~ <0o and A ~ , ~ ~-~ 1 1 ~ @ ~ , ~ - ~ , ~ ~(H~), /i } . In general it is convenient to consider all algebras in the Calkin algebra. (See D~ for definition~of the theory of C*-algebras).

190

Then the operators A~ can be chosen to refore ~ is approximable. Given an approximable family ~! and of compact operators consider the family Then ~ is approximable. In particular, imaginary part is approximable. Let ~ =

I@ ~

and let

~

have finite rank and

the-

a finite collection~1,...,~~ ~ ~ / any operator with compact

~=~

.

Then

~

~ ~---~ ~[@ ~ ~X~ : ~ ~--- ~,..., ~,~ is approximable . Consider ~ ~---{ ~ , P~ ~ ~ being an orthogonal projection, ~ , ~ • This is a partial case of the previous example because there exists a decomposition ~ ~ S ~ ~ ~ such that ~

C

~ %~ (see ~] for example). X The unilateral shift U is approximable. If ~ denotes the ~-algebra generated by U then it contains the ideal ~ G ( ~ ) of all compact operators and ~ / ~ C ( H ) : C ( ~ ) (cf ,e.g., [3])It follows t h a t ~ is approximable in the Calkin algebra It was actually proved in [4] that any finite family of commuting quasi-nilpotent operators is approximable (see [~). 3. KNOWN NON-APPROXIMABLE FAMILIES THEOREM. If a family ~ = ~ U~, ~=~,..., ~ ~ of .unitar~ operators is approximable then the

C*-algebra generated by I U ~

is amenab-

le~ in particulart i f ~ is a group algebra then the ~roup is amenable See for example ~6S for the proof. ~=4 ' ~ ~ ~ thogonal projections in ~eneral position then ~

is a famil 2 of oris not approximable.

Indeed, set ~ Z ~ -- I . Then ~ = ] J ~ = ~ and is a free product of ~ copies of ~ which cannot be amenable for Therefore a family of two (or more) unitary or selfadjoint operators picked up at random cannot be approximable in general. This implies that the single haphazardly choosen non-selfadjoint operator is not approximable either. The property of approximability imposes some restrictions o n the structure of invariant subspaces (see the footnote t o S e c t i o n 1). Consider a family ~ [~,...,~ of partial isometries bound by the relation .~ U £ U ~ = I , ~ • Then ~ is not approximable [7] although the algebra generated by 4 is amenable

[8]

191

Any algebra generated by an approximable famil~ being a subalgebra of an inductive limit of C~-algebras of type I, is amenable as a C~-algebra K7~. However, the class of such algebras is narrower than the class of all amenable algebras. If an approximable family generates a factor in ~ then it is clearly hyperfinite K6S. All that gives necessary conditions of approximability. 4.JUSTIFICATION OF THE PROBLEM. Many families of operators arising in the scope of a single analytic problem turn out to be approximable, apparently because the operators simultaneously considered in applications ca~uct be "too much non-commutative" (see ~9~, ~0~, problems of the perturbation theory, of representations of some non-commutative groups, etc). Besides, approximable families are the simplest noncommutative families after the finite-dimensional ones. On the other hand an approximable family admits a developed functional calculus based on the usual routine of standard matrix theory. Indeed, functions of non-commutative elements belonging to an approximable family can be defined as the uniform limits of corresponding functions of matrices. Therefore it looks plausible that a well-defined functional calculus as well as symbols, various models and canonical forms can be defined for such a family. This in turn can be applied to the study of lattices of invariant subspaces etc. In particular, if ~ is an approximable non-selfadjoint operator whose spectrum contains at least two points then, apparently, it can be proved that has a non-trivial invariant subspace. It is known that the weak approximation, which holds for any finite family, is not sufficient to develop a substantial functional calculus for non-commuting operators. However, it is possible to consider other intermediate (between the uniform and weak) notions of approximation (see, for instance, the definition of pseudo-finite family in ~ ) . 5. MORE CONCRETE PROBLEMS~ Our topic can be very clearly expressed by the following questions. ^ a) Let ~ be a locally compact abelian group with the dual group ° Given ~ ~ and ~ e ~ consider operators ~ ( ~ ) and V~ ~-~ ~ on ~ (Q) ~ For example, for G=~ let

and for G

T

192

and finally for C v = ~,

I,.S the Pair I~.[,~}

approximable?

The answer to this question requires a detailed, and useful for its own sake, investigation of the Hilbert space geometry of spectral subspaces of these operators. One of the approaches reduces the problem to the following. Consider a partition o f T = .0. ~ by a finite number of arcs

6~

. Then ~(T) ~-- ~--4 ~ ~{~ "

Let ~-----H~,~,~>0 be the subspace of ~ ( ~ ) consisting of functions whose Fourier coefficients may differ from zero only for integers satisfying ~ ~ j l < 5 • Here { ~ stands for the fractional part of ~ and ~ is irrational v~at is the mutual p o s i t i o n of subspaces ~@, 6

. ..... ~ ~ and

in

~

(T)

, i.e. what ar..~........their

stationary an~les, themutual products of the ortho~onal projectipns etc? Since ~ and ~ s a t i s f y V U V - ~ U -I ~ 6~ I (Heisenberg equation)~ the above question can be reformulated as follows. Is imt ' possible to solve this equation approximately in matrices with any prescribed accurac~in the norm topology? The shift U

can be replaced by a more general dynamical system

with invariant measure (X,T, ~ ) ,

. Then U T ~ ( ~ ) =

~(T~)

etherI

and

,V

is

approximable or not depends essentially qn properties (and not only spectral ones) of the dynamical system. The author knows no literature on the subject. Note that numerous approximation procedures existing in Ergodic Theory are useless here because it can be easily shown that the restriction of the uniform operator topology to the group of unitary operators generated by the dynamical system induces the discrete topology on the group. Note also that if the answer is positive~ some singular integral operators as well as the operators of Bishop-Halmos type ~I] would turn out to be approximable which would leadto the direct proofs oftbe

193 existence of invariant subspaces (see sec.4). b) Let ~ be a contraction on ~ . Are there convenient qriteria for

A

to be approximable

expressed in terms of its unitary di-

lation or characteristicfanction? c) Let

X Find a p p r o x i m a b i l i t E c r i t e r i , a,,,,,in terms of Non-negative k e r n e l s

~

0

are e s p e c i a l l y i n t e r e s t i n g .

d) For what countable solvable groups unitary representation of ~

K.

~

in ~ ( ~ )

of rank 2 the regular

generates approximable fa-

milies? Por what general locally compact groups does this hold?

REFERENCES I

D i x m i e r

J~ Les ~ - a l g ~ b r e s

Gauthier-Villard,

et leurs representations

P. Two subspaces.

381-389. 3- C o b u r n

L.

- Trans.Amer.Math. Soc., 1969, 144,

G~-algebras,

ties° - Trans.Amer,~ath.Soc.,

generated by semigroups of isomet-

1969, 137, 211-217.

C. On the norm-closure

of nilpotents.

III. - Rev.

Roum.Math. Pures Appl., 1976, 21, N 2, 143-153. 5. A p o s t o 1 C., F o i a s C., V o i c u 1 e s c u strongly reductive algebras, 6. B e p m z R

rpz~}. I973

Paris,

1969

2. H a 1 m o s

4. A p o s t o 1

*)

- ibid.,

A.M. C q e T ~ e r p y ~ ,

D, On

1976, 21, N 6, 611-633,

d~s~e

E NoHe~.

- B RH. :

N~map~aAT~oe c p e ~ e e ~a ~ono~oz~ecFHx rpynnax. M., ~ p , (Revised English version will be published in "Selecta ~athe-

matica Sovietica",

1983. "Amenability and approximation of infinite

groups"). 7. R o s e n b e r g

J, Amenability of cross products of

ras. - Commun.Math.Phys.~1977,

8. A p

s yua

~

~

~ -a±geb-

57, N 2, 187-191.

B.A., B e p m ~

~

A.M. ,a~Top-npe~cTaB~e~

c~pe~e~soro n p o ~ s B e ; ~ e ~ l~OlmlyTaTl~B~lO~ C*- am~edpu ~ nozyrpynI~ ee Sa~OMOI~HSm0B. - ~ 0 ~ . AH CCCP, I978, 238, ~ 3, 5II-516.

[7].

*) M.I.Zaharevich turned my attention to ~4] and A.A~Lodkin to

194

9~ S z - N a g y B , tors on Hilbert space I0. r o x d e p r

P o i a @ C. Harmonic analysis of operaAmsterdam - Budapest, 1970

M.~., ~ p

e ~ R

M.F. T e o p ~ B o ~ T e p p o B ~ X

o~epa-

TOpOB B rE~I~6epTOBO~npocTI~CTBe ~ ee u p ~ o ~ e ~ . ~., HayEa,I967. A. Invariant subspaces for Bishop's operator - Bull London Math Soc , 1974, N 6, 343-348

11~ D a v i •

CCCP, 198904, ~eHzHrpa~, UeTpo~Bope~, B~6a~oTe~Hs~ina., 2, MaTeMaT~KO-MexaHM~eOK~ ~ K y l B T e T SeHMHrpa~oKOrO y ~ B e p c ~ T e T a

A. M oVERSHIK

(A.M.BEP~)

COmmENTARY BY THE AUTHOR During several recent years a considerable progress in the field discussed in this paper has been made, as well as new problems have arisen. We list the most important facts. A C*-algebra will be called an ~ - a l g e b r a if it is generated by an inductive limit of finite-dimensional 0~-algebras. C~-sub algebras of A ~ -algebras will be called API -algebras, A family of operators generating an ~I -algebra is called approximable. THE PROBLEM was to find conditions of approximability for a family of operators or one (non-self-adjoint) operator and to give quantitative characteristics of the corresponding ~ ~-algebras, etc. I, in ~ a positive answer to QUESTION a), Sec.5 was actually given. Namely, the approximability problem is solved for the pair of unitaries ~ / ~ V .

(.V£)(1)-~-

;l(r,~),

(V÷)(I~ ~- 16(I) ,

~['~(~) , ~, o ~ E ~ This is the simplest of non-trivial cases. In [12S the authors made use of the fact that these operators are the only (up to the equivalence) solutions of the Heisenberg equation:

UVU~V

-I~

~I .

2. In [13],[14] the approximability of an arbitrary dynamical system (i.e. of the pair ( ~ T , ~ ) , where (7JT~)(~)~ ~(T~), ~ ( X , j ~ ) ) , T being an automorphism of ( X ~ ) , ~ ~ -~-
195

The result is based on a new approximation technique developed for the purposes of ergodic theory ("adic realization" of automorphisms, Markov compacta). Later in ~5] conditions on a topological dynamical system (i.e. on a homeomorphism of a compactum)were found under which the skew product C(~) ~ ~ ( ~ ) is an A~ -algebra. 3. These results in turn allowed to describe in some cases an important algebraic invariant of algebras generated by dynamical systems, the ~ -functor, cf.~2,16,1~. This makes possible to apply K -theory in ergodic theory. Nevertheless, we still have neither general approximability criteria, nor complete information on approximable non-selfadjoint operators. Since many important families turned out to be approximable, the questions on construction of functional calculus, estimates of the n o r m s of powers,

resolvents

tion SOME CONCRETE

ere

are of great importance. Let us m e n -

QUESTIONS.

A. Is the ~roup al~ebr a of a discrete amenable algebra approximable? B. Is it. true that an arbitrary aDuroximable operat0r 'has a non-trivial invariant subspace? C. How the

~-functor

(as an ordered ~roup~ of an

A~I

-al__-

~ebra ma~ look like? D. How are the properties of a dynamical s~stem related to the ' entropy ~r~H~h (i.e. th e ~rowth of dimensions of finite-dimensional subal~ebras of an ~ a l ~ e b r a

whi.ch' contains the algebra ~enerated by

the dynamical system)?

REFERENCES 12~ P i m s n e r

~.,

tional r o t a t i o n Theory,

1980,

13. B e p m ~ K

V o i c u I e s c u

C*-algebras

into a n

D. I m b e d d i n g A~-algebra

the irra-

. - J.Oper,

4, 201-210.

A2~. P a B H O M e p H a ~ a~re6pazqec~as a n n p o K c M M a ~ onepaTOpOB C ~ B M r a H y M H O ~ e H H H . - ~OE~l. A M CCCP, 1981, 259, ~ 3, 5 2 6 529. 14. B e p m H K A.M. TeopeMa o MapKOBCKO~ nepHoA~ecKo~ anHpOECHMaL~HM B apro~H~ecEo~ TeopHH. - 3an.Hay~H.ceMHsjl0~, 1982, I15, 72--82.

196

15. P i m s n e r M. Imbedding the compact dymamical system. Preprint N 44, INCREST, 1982. 16~ C o n n e s A. An analogue of the Them isomorphism for crossed products ef a

C*-algebra by an action of

~.-

1981, 39, 31-55. 17. E f f r o s E.G. Dimensions and C~-algebras. Conf.Series, N 46,ANS, Providence, 1981.

Adv. in ~atho, C.B.~.S.Region.

197

A PROBLEM ON EXTREMAL SIMILARITIES

4.23.

be a Hilbert space operator which is similar to an iso-

Let T metry

Two interesting quantities associated with T

k~T,_ where

II II®

are

~.{ ~:IlP~T, II ~- .llP~oo }

is the su~ over ~

AcT) =

~

of the polynomial p

, and

lULl ~,:'H: ~ ~o~o

The quantity ~(T) i s c a l l e d the k -norm of T a n d ~(T) lated to the distortion coefficient; Holbrook [3]. From (I), it follows that, for any polynomial p ,

is re-

n P~T)11~ ~ ~ II li,,111 llPlland therefore

k(T) (~
(2)

Since estimates on ~ ( T ) yield information about functional calculus and spectral sets, this gives one reason why ~(T) is interesting. Another reason is the frequent occurence of the quantity~h~IIL-111

(see, f o r e ~ p l e ,

[2 , p.248J).

One way to try to compute ~(T) ing (1) and

II ~, II l ~,-'11= ~cT~. Suppose L

llU~ II =

is to characterize ~

satisfy-

(3)

satisfies (I) and

e~

% - - . oo

~T~II

(4)

198 for all X

; then is L

a similarity satisfyin~ (3)?

The answer is "yes" in two

(extreme) cases: that in which

II~ II= ~ and that in which II~-~ II= ~ case~4 I I = ~ ( T ) = ~(T) , by (2) and by

if ~

and ~

. In fact, in the latter

are chosen correctly.

If ~ satisfies (I), (4) and Ill U ~ similar and uses the fact that, in this case

, the proof of (3) is

see D] and B ] .

II-

ineq lity Ill, 1 holds, in particular, when T i s a contraction. In that case, strict inequality holds in (2). In D] , I studied ~(T) and ~(T) in connection with some recent results on similarity of Toeplitz operators. One result was that, in most cases, the similarity satisfying (1) and (4) could be computed explicitly. REPERENCES I. C 1 a r k D.N. Toeplitz operators and ~-spectral sets.--Indlana U.Math. J. (to appear). 2. C o w e n M.J. and D o u g 1 a s R.G. Complex geometry and operator theory.--Acta Nath., 1978, 141, 187-261. 3. H e 1 b r o o k J.A.R. Distortion coefficients for cryptocontractions.--Linear Algebra Appl.~ 1977, 18, 229-256. 4. S z . - N a g y B. and P o i a ~ C. On contractiOms similar to isometrics and Toeplitz operators.--Ann.Acad. Sci.Penn. Ser.A.I. 1976, 2, 553-564. D.N.CLARK

University of Georgia Athens, Georgia 30602 USA

199

ESTIMATES OF ~UNCTIONS O~ HILBERT SPACE OPERATORS, SIMILARITY TO A CONTRACTION AND RELATED FUNCTION ALGEBRAS

4.24.

Pot a given class of operators on Banach spaces we can consider the problem to estimate norms of functions of these operators. Sometimes, dealing with operators on Hilbert space, we can obtain sharper estimates than in the case of an arbitrary Banach space. A remarkable example of such a phenomenon is the following J. von Neumann's inequality:

l~l<- 1 ) on Hilbert space and for for any contraction (i.e. the set of all complex poany complex polynomial (denote by lynomials). We consider here some other classes ~f operators. Operators with the ~rowth of powers of orde r consists of operators satisfying ]T~i~c(~+~)~ ~ 0 any such operator on a Banach space we have

~, ~ w 0

. This class

. Clearly, for

It is easy to see (of, [I]) that the fact that inequality (I) cannot be improved for Hilbert space operators is equivalent to the fact that ~m) is a n o p e r a t o r a 1 g e b r a (with respect to the pointwise multiplication), i.e. it is isomorphic to a subalgebra of the algebra of bounded operators on Hilbert space, It was proved by N.Th.Varopoulos E2~ that this is the case if ~ > ~ / ~ and so for ~ 4[~ (I) cannot be improved. It follows from E2], E3~, E4~ that ~61(~) is not an operator algebra for ~ 4/~ and so (I) can be improved in this case, Now the PROBLEM is to find sharp estimates of

I~(T) if~or T

satisfyingiT~[~o¢4+~) ~

Note that some estimates improving (I) are obtained in

This g e n e r a l problem a p p a r e n t l y i s v e r y

difficult.

[1]

o

Let u s c o n s i d e r

the most interesting case o~ ~-~-0.

Power bounded operators, We mean operators on Hilbert space satisfying

[T~

~ O,

~

0

It is well-known (see ref. in

~I] )

200

that such operators are not necessarily p o 1 y n o m i a 1 1 y b o u n d e d (i.e~ I ~ ( T ) I ~ I I ~ ~ A ). In ~ the following estimates of polynomials of power bounded operators are obtained

(2)

Here

II ll

:~ +

where ~ @

~

k~-~

is the injective tensor product;

where V I ~ O a ~ N

^

A

A

~(1~)~ : ~ ( ~ ) ~ - ~ ( l ~ ) , ~ O

,

for some ~6C(T)}

t

is the norm of • in the Besov space B ~ . The fact that the inequalities in (2) are precise is equivalent to the fact that the sets ~, ~ 0 A ~ ~ , B ~ I form operator algebras with respect to the pointwise multiplication. Por ~ this is not the case

~]d For ~; VMOA @ Ht

open. It is even unknown whether

~

the question is

forms a Bauach al~ebrao

If V ~ O A @ H I is an operator algebra then the norms ll'I[~ and ]] "~VMOA ~ HI are equivalent. The question of whether this is the case can be reformulated in the following may. Let ~HI be the set of all F o u r i e r m u 1 t i p 1 iers of ~I , i.e.

. Let

V~

be the set of matrices

N>O

H

I ~k}~,k~O

such that

201

where ~ @ ~

~

is the projective tensor product.

QUESTION I. Is it true that

~,~H ~< > F'~,~V ~ Recal'l

that

=

to show (see b]

(

? is

t

Hankel

matrixo

Tt

is

easy

~/~ ~H

Similarly we can defineM the spaces a~d the ~ e l

the

F'~,= { ~ (,,+... +

tenor

QUESTION 2. Is it

V rl "N

of tensors

)}

{~ZW.,..~.,.,l}~.~0.)

o

true that

M

ll

M

<. ~ , ~ - . ~

~ ~H~: ,,, > r~, ~ V

?

,I

If Question 2 has a positive answer then V ~ 0 A H is an operator algebra (see ~]) and sole, ~ a n d the estimates II %°(T)II ~ ~o~U~II~

~(P ' IVI,IO A~ i.11

cannot be improved,

QUESTION 3. Is it true that

M

VM

M

An affirmative answer would imply that ~ is an operator algebra and the estimate II~(T)II ~ C0~II 'f IIZ is the best possible (~]). Moreover in this case the estimates is attained on the Davie's example (see ~] ) of power bounded non polynomially bounded operator~ Similarit2 to a contraction, Here we touch the well-known problem (see e.g. ~]) of whether each,,,pol2nomiall 2 bounded operator T o_~n Hilbert, space%,,s similar to a contraction (i.e. whether there exists an invertible operator V such that I VTV-II ~ ~ In D] we considered operators ~S on ~ @

where

I

). ~

defined by

is the shift operator on . It was proved in [I] that is power bounded iff ~ belongs to the Zygmund class A I , i~e~ 4 I~I < I . It was also shown in ~I] that among

202 ~ there are many power b o u n d e d operators, n o n p o l y n o m i a l l y b o v ~ d e d . it seems reasonable to try to construct a counterexample to the problem stated above on the class of the operators ~ . It is very easy to calculatethe functions of ~. Namely,

THEOREM. If

~/6 B M O A

(Recall that J~M0 A ~

~ ~ =~

t.hen B~

is polynomially bounded

~-Cl'l,)~,~': ~(~)~---~,('~,), ~ 0

,

for some

PRO01~. By Nehari's theorem (see [6] "We have

. To f i ~ s h

-'Jr

-

the p r o o f we use the f a c t

that

o

(see FT]). It follows that ~( O~l~f~(~C~)I~(~-~)~)~/~O ~

QUESTION 4. Is it true that if ~ S

~re BMOA

and

is polynomially bounded then

?

This question is related to a question of R.Rochberg KS] concernin6 Hankel operators. QUESTION 5. Does there exist ~&

~

with ~

~B~O A

such that

is similar to no contraction? Operators with th e ~rowth of resolvents of order a

here the oper.tors satisfyi~ I ]~A,T)I ~<(ikl_1)-

. We consider

,

It is not difficult to prove that for amy such operator on a space we have

• Bauach

203

(3) w h e r e t h e B e s o v space

Bt

consists

of the functions

satisfying

being an integer greater than ~ • Inequality (3) is the best possible on the class of all Banach spaces • (It is enough to consider multiplication by ~ on B ~ ). The fact that (3) is the best possible for Hilbert space operators is equivalent to the fact that the algebra B V is an operator algebra. Consider the following operators on a commutative Banach algebra

A~, (~,~)~o~ ~ , ~ ,

~A,

It is proved by A.M. Tonge [9] that if all operators ~ are %-absolutelly summing (see definition in S.1 of this book) then A is an operator algebra and by P. Charpentier [4] that if ~ is an operator algebra then all operators ~ can be factored through Hilbert sp&ce. In the case of ~ the o p e r a t o r s ~ look as follows

The space ~

i s iso=orphic to

Grothendiek's theorem (see through Hilbert space are

:~

~ B~ ~

is

~0]).

~11]) t h a t o p e r a t o r s on ~-absolutely summing.

QUESTION 6. Is it true that for an 2 for ~

(see

~

~

I t f o l l o w s from factored the opera-

~-absolutel2 summing?

The answer is positive if and only if ~ is an operator algebra which is equivalent to the fact that (3) cannot be improved. If the answer i8 negative, find estimates sharp er than (3). Even the QUESTION Of estimatlm ~

I T~I

is not ~et solved. It follows from (3)

204

that

~ T ~ I ~< o(~ + ~)g

r e d in hed),

[12]. The best and another

ted for any

. This q u e s t i o n

example

one is due to J.A.

~ < I/~ ~

for ~ =

~

was

I know is due to S . N . N a b o k o

weighted

van Casteren

shifts

T

[13].

conside-

(unpublis-

They c o n s t r u c -

such that ~ ( ~ , T )

I~

REFERENCES I. P e I i e r tors

V.V.

on Hilbert

Estimates

spaces.

2. V a r o p o u 1 o s Inst,Fourier

N.Th.

(Grenoble),

3. V a r o p o u 1 o s - C.R.Acad.Sci.Paris, tcpologiques.

Th~se,

22,

1978. 7. F e f f e r m a n variables.

Sur lea quotiens

1972,

274, A 1 3 4 4 - 1 3 4 6

P.

Q-alg~bres

Orsay,

theory.

Ohm,

9. T o n g e

A.M.

129, type

Math°,

algebra

space~

- Bull.Amer

and St.Univ°, E.M.

HP

Anal.,

to the theory

1971,

operator 1979,

summing

operators 1968,

spaces

of several

arising

in d e f o r m a t i o n

summing

Banach

' s k i

spaces.

operators. A. Con-

- J.Funct.

A.L.

ones.

J., in

P e ~ c z y n ~p -spaces

' s k i

A. Abso-

and their a p p l i c a t i o n s

29, N 3, 275-326, On~6bius

1978, 40, N 3-4, 371-374~ 13. v a n C a s t e r e n self-adjoint

for

8, 225-249.

- S t u d i a Math~, 12° S h i e 1 d s

Notes

Blacksburg,

35, N I, 457-458.

and a b s o l u t e l y

of classical

11. L i n d e n s t r a u s s lutely

Math

137-193.

Math. Proc.Camb. Phil.Soc., 1976, 80, 465-473 I0. L i n d e n s t r a u s s J., P e ~ c z y n tribution

uniformes.

tensoriels

on the tlnit circle.

Inst.

S t e i n

R. A Hankel Banach

des alg~bres

et produits

in Hilbert

1972,

- Proc.Sympos~Pure

- Ann.

1973.

Polytechnic

- Acts Math~,

8. R o c h b e r g

Q -algebras.

N.Th.

P. Ten problems

at V i r g i n i a

on

opera-

7, N 2, 341-372.

1-11.

Soc., 1970, 76, N 5, 887-933. 6. S a r a s o n D. F u n c t i o n theory lectures

of power b o u n d e d 1982,

Some remarks

1972,

4. C h a r p e n t i e r 5~ H a 1 m o s

of f u n c t i o n s

- J.Oper. Thecry,

J.A,

bounded

operators.

Operators

- Pacif.J°Math.,

1983,

similar I04,N

Acts

Sci.Msth.,

to u n i t a r y I, 241-255

V.V. PELLER

CCCP, I9IOII, ~eHm~rpa~

(B.B.~DD]EP)

~OHTaHEa Z7, ~0~4

and

-

205 ESTINATES O~ OPERATOR POLYNO~IALS ON THE SCHATTEN-VON NEUNANN CLASSES

4.25. old

Precise estimates of functions of operators is an essential part of general spectral theory of operators. In the case of Hilbert space one of the best known and most important inequalities of this type is yon Neumann's inequality (cf. [I~ ) :

-<

:

4}

for any contraction T (i.e. FI ~ 4 ) on a Hilbert space and for any complex polynomial q (~A is the set of such polynomials). For a Banaoh space X an explicit calculation of the norm

jjj jjj×

I

is

:

a contraction on

X~

,~e ~At

would be an analogue of yon Neumann's inequality. In 1966 V.I.Matsaev conjectured that for infinite-dimensional h P -spaces, ~.J~[pcoincides with the P-multiplier norm= This means that for any contraction T on 5P " ,J~(T)Jhp<~J~Jpd¢~ J~(~e P, where % is the shift operator on ~P : ~(~o,~,,...)=(0,~o,~I,...). This inequslity was proved in [2] for absolute contractions ~ (i.e. ITIh4~
IT I- TI I

cO~=CT~

I.

I 1%

. In other words for an[ con-

The conjecture is true for p= 4,Z,~.

206 PROPOSITION 1. Inequalit ~ (1) holds for isometries (not necessarily invert!hie ) on the space ~

.

THE PROOF uses an idea due to A.K.Kitover. Consider an operator W:~(~) ~ ~(~) defined where 0<~'~ . Then W ~ ( X ) for any@~p(~) . P u t ~%~---6*~6 • It is easy to see that ~ ( ~ ) ~ ( ~ T ) , ~ , so

byW~=(a,~%,~T~,...),

P

a,

Therefore

'

M~ing $ ~ t we obtain !~(% i~p --~ I~l~p, cult to show that I~l~p= I~1~,. • Let =.~

us

show that l~lzr >- I ~lp

~(.,~C)%J~

. It is not diffi-

, ~E ~A

~p(3~)

. Let

Cb =

, where ~ = ( ~ , @ , @ , . . . ) ~ O ~ .

°

• )liar

s°l~°l~ ~- I~l~.

•

Perhaps it is possible to prove (I) for certain classes of contractions T

(or even for any contraction ~

) applying one of the

following two methods used by the author for hP-spaces. DEFINITION. 1. A net o f o p e r a t o r s [ ~ } on a Banach space X is said to converge to an operator T in the ~I~ -topology if

~m(Tj~,,~)=(~:,~)

~X

~x

~

~o.

2. Let ~ be an isometric imbedding of ~ into ~ , ~ be the norm one projection from ~ onto ~ P (it does exist ~6]) and T be an operator on ~ . An operator ~ on ~ is called a dila-

tionofT

if

PM"J=~T ~, ~ 0 .

In view of Proposition 1 an operator T on ~ has to satisfy (I) if it can be approximated by isometries in the pig-topology or it has an isometric dilation on T2 . Thus one should describe operators on T~ , having an isometric dilation on ~ , and the closure of the set of isometries on Tp in the ~ig-topology. For this aim

207

perhaps it would be useful to apply a known description of the set of ~ -isometries ~ (cf. K6S). it is known that each contraction on a Hilbert space has a unitary dilation on a Hilbert space ~] and can be approximated by unitary operators in the ~ - t o p o l o g y (cf. [4]). The set of operators on ~P~0,1S , ~=~2 , having a unitary dilation on an ~P-space coincides with the closure in the p~-topology of the set of unitary operators and coincides with the set of operators having a contractive majcrant (cf. [3], ~]). (Earlier for positive contractions the existence Of unitary dilations was established in ETl). QUESTIONS. Is it true that I~ an 2 ~ metric dilation? 2~ any absolute on o n ~

~

-contraction has an iso-

-contraction (i.e~ a contracti-

and on ~o@ ) has an isometric dilation? 3) 2 ~

coincides with the set of all contractions on ~p contains the set of absolute contractions on ~p

- 6~@~

? 4) 2 ~ - ¢ ~ @ ~ ?

The affirmative answer to 1) or 3) would imply the validity of Conjecture I. If Conjecture 2 is also valid then this would imply the validity of V . I . ~ t s a e v ' s conjecture because sP can be isometrically imbedded into ~ in such a way that there exists a contractive projection onto its image. In conclusion let us indicate a class of operators satisfying (1). PROPOSITION 2. Let @,~ Then the operator

T

on

~F

be contractions on ~

,TG~@~ ¢ ~

has an isometric dilation and can be

approximated b E isometries. This follows from the fact that lation on a Hilbert space and can be operators o n ~ .

and ~ have a unitary dipt~-approximated by unitary

REFERENCES I. S z . - N a g y B., ~ o i a ~ C. Harmonic analysis of operators on Hilbert space. North Holland - Akademiai Kiado, Amsterdam-Budapest, 1970.

2. H e x x e p B.B. ARaxor ~epaBe~cTBa ~x.~o~ He~Maaa ~ s paHcTBa ~ . - ~ o ~ . A H CCCP, 1976, 231, ~ 3, 539-542.

npOCT--

3. P e i I e r V.V. L'in~galit~ de von Neumann, la dilation isometrlque et l'approximation par isometrles dans . -C.R.Acad. Sci.Paris, 1978, 287, N 5, A 311-314. •

.

°

#

°

208

4. H e x x e p B.B. ARaxor ~epaBeRcTBa ~x.#oR He,Maria, EBOMeTpE-~ecEas ~EXaTa~HS CZaTE~ ~ annpoEc~Ma[~Es ~ S O M e T p ~ M ~ B npocTpaRcT-Bax ~sMepEM~X Sys~um~.

- Tpy~

M~,

1981, 155, 108-150.

5. c o i f m a n R.R., R o c h b e r g R., W e i s s G. Applications of transference: The LP version of yon Neumann's inequality and the Littlewood-Paley-Stein theory. - Proc.Conf. Math.Res. Inst. Oberwolfach, Intern. ser.Numer.Math., v. 40, 53-63. Birkhauser, Basel, 1978. 6. A r a z y J., F r i e d m a n J. The isometries of --C~ ~'"~ into Cp . - Isr.J.Math., 1977, 26, N 2, 151-165. 7. A k c o g I u M.A., ~ u o h e s t o n L. Dilations of positive contractions o~ ~ spaces. - Canad.Nath.Bull., 1977, 20, N 3, 285-292. V. V. PELL~R (B. B. ~ D ~ E P )

CCCP, IgIOII, ~eR~Rrpa~, ~OSTaREa, 27, ADMH

209

4.26.

A QUESTION IN CONNECTION WITH NATSAEV'S CONJECTURE

This problem is closely related to the preceding one [I], where definitions of all notions used here can be found. I propose to verify Natsaev's conjecture for the operator ]~$, |'= p < 2

on the 2-dimensional space

defined by the matrix

I the standard basis of

2

The contraction

~- is of interest

because it has some resemblance with unitary operators on

2-dimensi-

onal Hilbert space and,as is well-known~the abundance of unitary operators plays a decisive role in the proof of von Neumann's inequality~ Moreover T

has no contractive majorant and so the results of Akcog-

lu and Peller cannot be applied to ito Thus it seems plausible that the validity of Matsaev's conjecture in general case depends essentially on the answer to the following QUESTION. Is Matsaev's conjecture true for

7-- ?

REFERENCES I. P e i I e r V.V. Estimates of operator polynomials on the Schatten - yon Neumann classes~ - This "Collection", Problem 4 2 5

A.K.KITOVER

(A.E.EMTOBEP)

CCCP, 191119, ~eHNHl"pa~, ya.EOHCTaHTMHa SaCaOHOBa 14, KB.2.

210

4.27.

TO WHAT EXTENT CAN THE SPECTRUM OF AN OPERATOR BE D!MTNISHED UNDERAN EXTENSION?

Let X be a Banach space, T (i.e. T ~ J~ (~) ).

be a bounded linear operator on X

QUESTION I. Are there a Banach space Y c o n t a i n ~ qperat0r ~ ( ~ ) rum of T

such t h a t T ~

~I~

is exactl 2 the spectrum.*) of

X

and an

an a th e essential spect~ ?

A stronger version of Question I is QUESTION 2. Given ~,T~ ~ (X), ~ ~(Y)

~

can one find a Banach space

X and an isometrical algebra homomorphism ~:JB(~)--~ such that

exactl~the

~(~)IX=~ ~ ( X )

and the s p e Q t ~

essential spectrum of T

B.BOL OB S

of ~(T)

i ss

?

Department of Pure Mathematics and Hathematical Statistics University of Cambridge 16N~ill Lane, Cambridge CB2 ISB, England

EDITORS' NOTE. Related questions are discussed in S.3.

~C

II

*) Apparently, here the essential spectrum of such thatI[(~-~I)~ll ~ 0 for a sequence { ~ }

is

the set

in

~

with

of

211 4.28. old

THE DECOMPOSITION OF HIESZ OPERATORS

If X is a Banach space)let B(X) , ~(~) and ~(X) denote the sets of bounded, compact and quasi-nilpotent operators on X (respectively). T , T a B ( X ) , is a R i e s z o p e r a t o r if it has a Riesz spectral theory a s s o c i a t e d w i t h the compact operators, i.e. the spectrum of T is an at most countable set whose only possible accumulation point is the origin and all of whose nonzero points are poles of the resolvent of finite rank. The set of Riesz operators is denoted by ~(X) • Ruston D] ~haracterised the Riesz operators as T ~ ~ ( X ) < = > the coset T + K(X) is a quasi-nilpotent element of the Calkin algebra E(X)/R(X) . Clearly ~C X) :~ ~(X)* Q(X) . West [2] proved that if X is a Hilbert space then ~(X)----~cX). Q(X) . This decomposition has been generallsed to a C*-algebra setting by Smyth ~3]. The proof is analogous to the superdlagonalisation of a matrix which is then written as the sum of a diagonal matrix and a super• i a g o n a l m a t r i x w i t h a zero diagonal. Nothln~ is knowa about the decomposition problem in ~eneral Banach spaogs ~ !t m ~

be that the decomposab%lit~ of all Riesz

op@-

raters charap,~ri~,~es,,,Hilbert spaces up to equivalence a m o n ~ B a n a c h spaces. REFERENCES I. R u s t o n A.F. Operators with a Fredholm theory, - J.London Nath.Soc.~1954, 29, 318-326. 2. W • s t T.T. The decomposition of Riesz operators. - Proc.LondonMath.Soc.,III Series, 1966, 16, 737-752. 3. S m y t h M.R.F. Riesz theory in Banach algebras. - Nath.Zeit.~ 1975, 145, 145-155. M.R.P.SITTH T.T.WEST

39 Trinity college Dublin 2 Ireland

212

4.29.

OPERATOR ~GE~RAS IN ~ I ~ ALL FREDHO~OPERATORS ARE INVERTIBLE

Let ~(X) be the algebra of all bounded lin@ar operators in the Banaeh space X . An operator A ~ X ) is called a Fredholm operator if ~ ~ A
where ~ is a homeomorphism of the circle T onto itself. This property is valid for the elements of uniformly closed algebras ~) generated by the above operators as well (see ~] and the literature cited therein). The usual scheme of the proof consists of two stages. First we prove the invertibility of Fredholm operators in the non-closed algebra ~ generated by the initial operators (using the linear expansion [2] it is reduced to the same operators but with matrix coefficients or kernels [3]). Then we have to extend this statemaut in some way to the uniform closure C ~ of the algebra ~ . QUESTION I. Let e v e ~ Predholm operator from the algebra (c ~ (X~ algeb~

C~

be invertibl e. ~

Is

ever~ Fredholm operator in

invertible?

In the examples the passage from ~ to ~0S ~ easier if A - I ~ c ~ ~ for each invertible operator QUESTION 2. Let ever~ Fredholm operator and let ~ - ~ C ~ " ~

the

A ~

I @ ever~ Fre~olm operator

becomes A ~ . be invertible,

in the al~ebra

invertible~

We point out two eases, when the answer to Question I is positive ~]. ~)It is supposed that all algebras under consideration contai~ the identity operator.

213

I ° . Algebra

~

is commutative (or

sists of operator matrices with

(~c~(XT

elements

in

and

~

con-

some commutative

&X)

algebra (~o c ~ )2°. X is a Hilbert space and ~ is a symmetric algebra. The answer te Question 2 is positive if one of the following conditions is satisfied (see [1] ). 3 °. The algebra ~ ~ is semi-simple. 40. The system of minimal invariant subspaces of the algebra is complete in ~ . 5 °. The algebra ~ ~ does not contain nil-ideals consisting of finite-dimensional operators. We call a non-zero invariant subspace minimal if it doesn't contain other such subspaces. A two-sided ideal is called a nil-ideal if all its elements are nilpotent. Either of the conditions 3 °, 4 ° implies condition 5°° ~or 3 ° this is obvious and for 4 ° this follows from [4] (comp

with [5]),

RE~ERENCE S I.

2.

8.

4.

5.

K p y n R ~ E H.~., $ e • B ~ M a H III.A. 06 06paTl~OOTH HeEOTOpBE( ~pe~IN~XBMOB~X oHepaTOpOB. - ~13B.AH ~ C P , cep.~HS.-TeXH. H MaT.HayE, I882, ~ 2, 8-I4. r o x 6 e p r H.IL, K p y n H E E H.H. C E R z y ~ p R ~ e ~HTerpax~HHe onepaTopH c EyCO~HO--Henpep~BHMM~ EO3~X~EL~eHTaMH E HX CIgMBOJI~. --T.~SB.AH CCCP, cep.MaTeM., 1871, 35, ~ 4, Co940--964. Kp y n H E E H.H., ¢ e a ~ ~ M a H H.A. 0 HeBO~MO~U~OCTH BBe~eHE~ MaTpE~HOr0 C~LMBO~a Ha HeEoTop~x sxPe(Jpax onepaTopOB. B F~.: ~ e ~ e onepaTop~ ~ E H T e r p a ~ H ~ e y p a B H e m ~ . K~m~H~B~ ~ T E m ~ a , 1881, 75-85. Jl 0 M O H 0 C 0 B B.H. 06 E~Bap~aHTH~X no~npoc~pa~cTBax c e M ~ CTBa onepaTopoB, EO~TERY}0H~ C BHO~He Henpep~BHm~. - CyHZ~.aHa-~m8 E ero n p ~ o a , . 1973. 7. BRII.3, 55-56. Map E y c A.C., ¢ e ~ ~ ~ M a H H.A. 06 anre6pax, nopo~~eHHRX O~HOCTOpOHHe o 6 p a T ~ H o n e p a T o p ~ . - B EH. : ~iccxe~oBano ~ e p e H n ~ s ~ H m ~ ypaBReH~&K~B. ~ T ~ H ~ a . 1983. 42--46.

N. Ya. KRUPNIK A. SoMARKUS I.A.FEL' DMAN

(H.~.m'YSK) (A.C.m~P~C}

CCCP, 277003, ~ZHS~B, ~zma~'4Bc~z~ r o c y ~ a p c ~ e ~ y~BepcXTeT

CCCP, 277G~8, K ~ B . H ~ C ~ X ~ maTe~aTm~ AH MCCP

214 ON THE CONNECTION BETWEEN THE INDICES OF AN OPERATOR MATRIX AND ITS DETERMINANT

4.30.

be the algebra of all Let ~ be a Hilbert space, and ~(~) A ~ ~(~) is called linear bounded operators in H . An operator H/I A < o o . a ?redholm operator if and is called the index The number ~ A = ~ ~A~ ~/I~% of • is the orthogonal sum of ~ copies of the space ~ , If ~ can be represented in the form then any operator % 6 ~(~)

A

an ope

tor m t ix { A

Ajk}

, Let ~ = { and suppose all to be compact. Define the deter~ mi~aDt ~ in the usual way. The order of the factors in eanh term is of no importance in this connection, since various possible results differ from each other by compact summand.N.Ja.Krupnik showed ([I], see also [2], p.195) that ~ is a Fredholm operator if and only if det ~ is. On the other hand it is known that under these conditions the equality

commutators~LA~L,~. Aik]

Aik

(1) does not hold in general(see an example below ). In [3] it was stated that the equality (1) holds if the condition of compactness of commutators is replaced by the condition of their nuclearity. The question of preciseness of conditions -*[~ik~ ~irkt]~~ ~ arises naturally. CONJECTURE I. Let [4]) of the algebra a Predholm operator

~ ~(H)

be any symmetrically-normed ideal (see , different from

jarv={Aik ~

, such that

~

° There exists

[Aik,Ai1~,]~

but (I) does not hold. The weaker conjecture given below is also of some interest. CONJECTURE 2. For an,Y =[Aik ] not hold.

such that

p>~

there exists a Fredholm opera to r

[-Aik , Aj/k, ] e~'p

but (I) does

215

Conjecture 2 is confirmed Note that in the example below only for ~ • ~ . S ~ is the two-dimensional EXAMPLE. Let ~= ~(~) where sphere. There exist singular integral operators such that

Aik

is a Eredholm operator and ~ = ~ ([5], Ch.XIV, ~4). As ~ = 0 (~5], Ch.XIII, theorem 3.2) equality (1) does not ^ hold. It can be assumed that the symbols of the operators ~k are infinitely smooth (for ~ ~ 0 ) and therefore the commutators [~ik ~ A i ~ ] Hence

map

~ ( S ~)

S~([Aik,Ai~k~] ) = 0 ( ~

into ~1~)

W ~ (~ ~)

([6], theorem 3).

(see e.g.[7]).

We note in conclusion that some conditions sufficient for the validity of (I) have been found in ~8-10]. In these papers as well as in[S],[3] the operators in Banach spaces are considered .

RE~ERENCES I.

2. 3. 4.

5. 6. 7.

K p y n H E E H.H. K BOllpocy 0 HOpMSflIBHO~ paspemzMOCTH E EH-~e~ce C~ry~spH~X ~HTeI?pS~HRX ypaBHeH~. -- Y~. sa~.K~E~HeBCEOrO yBL~BepcHTeTa, I965, 82, 3--7. r O X 6 e p r E.~., $ e x ~ ~ M a H M.A. Y p a B H e H ~ B cBepTEax E npoeEn~oHR~e M e T o ~ ~X p e m e H ~ . M., HayEa, I97I. Map E y c A.C., $ex ~ M a H M.A. 06 EH~eEce onepaTopHO~ MaTpHn~. --~#H~u.a~a~. E ero n p ~ . , I977, II, ~ 2, 83-84. r o x 6 e p r M.H., Kp e ~ H M.r. BBe~eHEe B Teop~0xE-He~HNX HecaMoconp~eHHRX onepaTopoB B I~B6epTOBOM npooTpaHCTBe. M., HayEa, I965. M i c h 1 i n S.G. , P r o s s d o r f S. Singulare Integra~operatoren. Berlin: A k a d e m i e - Verlag, 1980. S e e 1 e y R.T. Singular integrals on compact manifolds. Amer.J.Math., 1959, 81, 658-690. H a p a c E a B.M. 0O ac~MnTo~Ee CO60TBeHH~X ~ CEHryJ~HNX ~ o e ~ Jn~He~z~u~x oHepaTOpOB, H O B ~ P~a~EOCTB. -- MaTeM.cOopH~, I965, 68 (II0), 623-63I.

216

8, E p y n H ~ E H.H. HeEoTopNe o S ~ e Bonpoca Teopm~ O~HOMepHI~X C~I~yJZapH~( onepaTopoB c M a T p E ~ EOS~dl~eHTa~m. -- B EH. : HecaMoconp~eHH~e onepaTopm. K~m~eB, ~TY~Bum, I976, 9I-II2. 9. E p y n H ~ E H.H. YCJIOBH~ cy~eCTBOBaHN21 ~L-C~BO/~a E ~OCTaTOqHOrO ~adopa •--MepHHX npe~cTs3xeH~ 0aHSXOBO~ a~re6pH. B E~.: ~ R e ~ e onepaTop~. I~m~HeB, m T ~ , 1980, 84--97. I0. B a c H x e B C E ~ ~ H.~., Tp y xMX ~ o P, K T e o p ~ ~-onepaTopoB B M a T p ~ a~re6pax onepaTopoB. B m~.: ~ e ~ e ouepaTop~. ~ e B , ~IT~m~a, 1980, 3-15.

I.A. l~EL 'DMAN A. S. ~L~RKUS

(A.C.MAPEYC)

CCCP, 277028, I(~,,~eB, ~HCT~TyT MaTeMaT~EH AH MCCP

217 4.31.

SOME PROBLEMS ON COMPACT OPERATORS WITH POWER-LIKE BEHAVIOUR OP SINGULAR NI~BERS

Classes of compact operators with power-like behaviour of eigenvalues and singular numbers arise quite naturally in studying spect~ ral asymptotics for differential and pseudodifferential operators. Presented are three problems related to the theory of such classes. Let ~(~) be the algebra of all bounded operators on a Hilbert space H . Given A in the ideal C of all compact operators in ~ define 5~(A), ~=~,~,..., the singular numbers of ~ . ~or 0 < p < o o let

0

p:{Ae 7.p:s(A)= See [1-4] for details concerning ~-Tp While studying spectral asymptotics the main interest is focused not on the spaces ~, 7.o themselves, but on the quotient spaces

P'

p

The spaces ~p , O
~

, @~6~ --@~*

. Is there a normal ope-

8J ?

It is known (see [6]), that the answer is negative if p=oo . it is due to the fact that in the ~oo -space there is the Index, i.e. nontrivial homomorphism of the group of invertible elements of the algebra ~co onto the group ~ , as well as to the fact that

218

the

spectrum of an element & ~ can separate the complex plane . These two circumstances do not occur if p < 0 0 . An analogous question on self-adjoint classes has the affirmative answer (and it is trivial): if ~ , @*= ~ , then for an arbitra~j A ~ the operator ~ is self-adjoint and belongs to the class ~ . Closely related to Problem I are the following two problems.

(A+A

PROBLEM 2. Let

operat.or.s

A~

PROBLEM

3.

~

, Be~ Let

, ~

, @~=~

. Are there c o m m u t i ~

?

~=~*e dp

,

self~d~oint commut..i.n~..operators

,

0~ = ~

. Are there

?

= p are evidently Problem I and Problem 3 in the case equivalent. To the contrary a positive answer to Problem 2 does not yield automatically the positive answer to Problem ~. REFERENCES I. F o x 6 e p r H.IL, E p e ~ a M.F. BBe~ea~e B T e O p ~ x~ae~B~x aecaMoconp~zea~x oHepaTopoB. M., Hay~a, 1968. 2. B E p M a H M.m., C o x o M ~ E M.3. 0 ~ e H ~ C ~ r y x S p B ~ x ~ c e x ~aTe~0axBa~x oHepaTopoB. Ycnex~ MaTeM.Hay~, 1977, XXX~, ~ 1(193) --

17-84. 3. s i m o n B. Trace ideals and their applications. - London Math.Soc.Lect.Note Series, 35, Cambridge Univ.Press, 1979. 4. T r i e b e I H. Interpolation theory. Function spaces. Differential operators. Berlin, 1978. 5. B ~ p M a ~ M.m., C o ~ o M a E M.3. K o ~ a a K T ~ e onepaTopH co cTe~eHHo~ aO~TOTN_WOfi C ~ H I ~ p R B ~ X ~ c e x . - Ban. aayqn, ce1~H. ~ 0 ~ , I983, I26, 21-80. 6

B r o w n L., D o u g 1 a s R , F i i I m o r e P Unitary equivalence modulo the compact operators and extensions of C~-algebras, -Lect.Notes in Math , 1973, 345, 58-128 M.S.BIRMA~

(M.[U.BHPMAH) M. Z. S O L O ~ A K (~. 3. C O J I 0 ~ O

CCCP, 198904, ~eHHHrpa~,HeTpo~sope~ ~XsMqecKM~ ~ a x y ~ T e T ~eHm4rpa~cEoro yHHBepCMTeTa CCCP, 198904, ~eHm~rpa~,He~po~sope~ MaTe~aT~o-~exaH~ec~ Saxyx~TeT JieH~rpa~c~oro y ~ B e p C ~ T e T a

219

4.32,

PERTURBATION OP SPECTRU~ OF NORNAL OPERATORS AND OF COMMUTING TUPLES

Recent progress has somewhat clarified the subject of perturbation of spectrum of normal operators and of K-tuples of commuting self-adjoints. This note is a summary. Only the finite-dimensional case is treated here. (The infinite-dimensional case is attacked in [I] .) will be a Hilbert space of Kk dimensions. Th~ spectral resolution of a normal operator A will be written A ~ c ~.|=4 4 %41t~ @ o here the ~i~ are orthonormal eigenvectors, with eig~nvalues ~ corresponding; and the notation ~Ce , for any 3CC~ , denotes the linear function~ corresponding to ~ , Similarly for normal B let us write 8 = ~=I ~ V~ V ~ . As the distance ~ between o(A) and O let us use

(S)

j

the minimum being over all permutations of{ PROBLEM 1. Find the best constant

AandB

C

|,2,...,~ ~. such that, for all normal

?

g~ cRA-BII.

(2)

It has long been conjectured that C=| (e.g,, [5] ). And (2) is known with C--4 in some special cases: if A and B are self-adjoint [7] , if A is self-adjoint and B skew-selfadjo~nt [ 6 ] , i~ A and [3 are u n i t a r y [ 3 ] , or i f A - B is normal along with A and B [2] . Yet only recently has i% even been proved that there exists a universal C for which (2) holds in general It has been known for many years that if ~ were replaced by the Hausdorff distance then (2) would h01d with C= | . The following stronger assertion is also familiar (e.g., [3] ):

[4]

PROPOSITION. If ~Bis a set of

KA

~-k + ~

is a set of

k

ei~envalues of

(i) the convex hulls of

~A

and

~8

ei~envalues of B

some X

C

~d

, and if either

are at distance

or

cii)

A

p c R÷ we have gag

>~

220 while ~B ~ { ~[ l~-~l>~p t ~ }

, then

~A-BU.

~

PROOF. The spectral subspace for ~ belonging to ~ A and the spectral subspace for B belonging to ~S have dimensionalities whose sum is > ~ ; hence there is a unit vector ~ in their inter~ section. In case (i), ~ - ~ I ~ * A ~ - ~ * ~ I ~< IIA-~II . In case (ii),

II(A-X)ocll~ p while

II(B-X)~II

p+

SO

a+ B(A-D~c-(B-X)~cll ~ II A-BII '&. COROLLARY (R. Bhatia).

~b~ ~ x l l~il -I ~[I ~ IIA-BII.

j

(Of course, here too we can translate by any ~ C ~ the l e f t - h ~ d expression is c l e a r l y = ~ll~il-l~ll

). Indeed, i f both

the O~ i and the ~d are labelled in order of increasing modulus. Let the maximum be attained for ~--~ , and assume without loss of generality that Ic6Kl < I~KI ~ Then ~A={ o~4,...,c~k } and ~B={~k,...~) satisfy hypothesis (ii) of the Proposition. • Let ~' denote the greatest ~ obtainable for any .~A and B~B satisfying (i) or (ii), or the analogue of (ii) with A and interchanged. Clearly Hausdorff distance ~ ~" ~ ~. EXAMPLE. I. Set o6~----3,oC~= ~ , 063=4 ; and ~ = - - c ~ for all L i The Hausdorff distance is 2 , ~'----8 (attained for ~A {o~ ,og,} and ~B--- { ~ ' , ~ ' I ), but ~ - - £ ~ . Thus the idea of the Proposition can not be used directly to prove that the constant C above is The problem of ~-tuples of commuting self-adjoints may be more important, but so far seems less tractable, I will use the following notation. If A (q , A~'), , A (K) are self-adjoint and commute, then for orthonormal t~4, • .. , ~ and corresponding real (~) (D (n ~ .'" C~j we have A --~ ~, ~; ~= , ~ will let A denote the . j.~ ~ # o m ,w~ A (1) operator-ma~ix of one "column whose J-th entry is ~'K , so that A ----2-- O 6 ~ t ~ ; * , and we may speak of O C : ~ ~ as the i-th eigenvalue of ~ . (As an operator from 0~ ~o 3~ , it doe~ not have eigenvalues in the usual sense° ) Similarly = ~" ,~iV~ V~W. As above, the distance will be

= ~

~ x II~ - ~ w l l ,

the minimum being over all permutations, and the norm being that of

~K.

PROBLEM 2. ~ind the best constant

Ok

such that~ for all ~

-

221

tuples ~A

and ~B

~

oKIIA-BII

The Proposition and Corollary above have exact analogues for this situation (with ~ replaced by ~ ~ ~ K and so forth); their proofs are almost as brief, and may be left to the reader To make precise the relationship between Problem S and Problem 2 (for the case K = ~ ), I recall these elementary facts; FACT 1- For any self-ad~0int saril~ commuting), ~

"t

an__~d L

on ~

(not neces-

II[~]ll ~ IIH + bLil.

FACT 2. For a~7 s e l f - a d j o i n t

(H+~ff(H+ZL) ~d

H

~

an__~d L

,

I1[ 111

IIH+J.

•

I t f o l l o w s a t once t h a t the c o n s t a n t s C i n Problem 1 and Cl i n Problem 2 are r e l a t e d by C ~ 02 ~ ~ C . In particular, C~ is finite, by v i r t u e of the r e s u l t of [ 4 ] a l r e a d y c i t e d . The e x i s tence of a finite CK for K •2 has not been proved CO = [ -3] 0 EXAMPLE 2. A d a p t i n g Example 1, take ~vt 0 , ~y~vvi=[~] = [ ~ ] ; and ~ = - w ~ v ~ for all i " The remarks concerning Example I apply to this modification, in particular ~ = 2 ~ . Now choose the eigensystems

It is not hard to compute that • For a neard/t2. by example I have found that Ii-"-~-~wl12/~ --" = The example shows that C~ > ~ However, it remains reasonable to conjecture that C= | , and (as this would imply) that Ca ~ I suggest, with little evidence, that in general

CK

,I'R" ' REFERENCES

1. A z o f f E., D a v i s Ch. Perturbation of spectrum of self-adjoint operators. To appear. 2, B h a t i a Ro Analysis of spectral variation and some inequalities. - Trans.Amer.~athoSoc. 1982, 272, 323-331.

222

3- B h a t i a

R~,

D a v i s

tion of a unitary operator 4. B h a t i a

R.,

bation of spectral tions. norms.

subspaces

Math.Soc. 7. W e y I

V.S.

and solution

linearer partieller

of linear operator

gauge functions

and unitarily

2, 1960,

between normal

1982, 84, 483-484. H. Das asymptotische

To appear. A. Pertur-

11, 50-59.

operators.

Verteilungsgesetz

Differentialgleichungeno

invariant

- ProcoAmer.

der Eigenwerte

- Math. Ann.,

71, 441-479, CHANDLER DAVIS

equa-

To apppear.

J° Math. Oxford Ser. Distance

Alg.

Ch., M c I n t o s h

and Appl.

L. Symmetric

-Quarterly

6. S u n d e r

- Linear ~ultilinear

D a v i s

- Linear Alg.

5. M i r s k y

Ch. A bound for the spectral varia-

Department

of Mathematics

University

of Toronto

Toronto M5S IAI Canada

1912,

223 PERTURBATION 0P CONTINUOUS SPECTRUM AND NORMAL OPERATORS

4.33.

Let T be an invertible bounded operator on Hilbert space g ° The cont~muous spectrum (~c (T) of -F is defined as ~ (7-)\ Oo Q~-), where (~o (T) stands for the set of all isolated points of the spectrum O ( T ) whose spectral subspaces are finite-dimensional. If the origin lies in the unbounded component of ~ \ • ( T ) then 0 ¢ ~c( T + K)for any compact operator K ° on the other hand, if ~(T)separates 0 and c o , then for any symmetrically-normed ideal~ ~ ~ ~ ( ~ p denotes throughout the Schatten - v o n Neumann class~ 0
an d

0

does not 'belon ~ to the unbounded compo-

0 ¢ O(T). Then a rank one operator K

0co (T+K)

exi, s,ts

if and

with

L tT- ¢ LefT.

See [2], for the proof. Given an operator T denote by g ( T ) the weakly closed algebra of operators on ~ generated by T and the identity I Suppose a is a normal .operator on ~ • Then

M- c

N

iff

, see [3].

herefore

Theorem I together with a theorem of Sarason [ 4] imply the following criterion for the stability of the continuous spectrum under finite rank one p e r t u r b a t i o n s . D e n o t e ~ M the Sarason hull of the spectral measure of a normal operator ~ , defined in [41. THEOREM 2. L e t M

be an invertible ~normal operator

The followin~

are equivalent :

1) 2)

0 ¢ (~c (N + K) 0¢ C - . - , .

for every K, ~@tck (K) - ] ,

In particular, the continuous spectrum of a unitary operator is stable iff Lebesg~e measure on ~ is absolutely continuous with respect to its spectral ~easure, It is sh~wa in [5] that the continuous spectrum of a unitary operator is stable under perturbations of ran~ one iff it is stable under nuclear perturbations. This result can be extended to normal operators with essential spectra on s m o o t h curves. At the same time it does not hold for arbitrary normal operators ~I~ o

224 QUESTION 1. Given an invertible 0perator are the followin~ equivalent?

O~(T+ K)

1)

O~

2)

0 ¢ dc ( T ÷ K)

f o r every K o_~frank one.

~or everz K of f i n i t e rank,

QUESTION 2. Are I) and 2) equiv.alen$..f.0ra~ arbitrar~ normal opera-

N =T

tor

? Is it true that they are equivalent to

3) o ¢ Oc ( N + K)

for every

K ~

U! ~

?

No%e that for ~ = ~+~ , where ~ is unitary and K C ~ the answer to Question I is affirmative [2] . QUESTION 3- Is either of the followin ~ implications 0 ¢ O c ( T + K)~ V ~ , ~@~k (K) < + oo ~ The inclusion T ~ inclusions

L~t

~-'e

~ = 4,2,...

~ ( T -~)

~ ( T -~)true?

being equivalent to the series of

c

•" eT-'

Tc

L~:

T • ... e T

m;

j

(see [3]), it is natural to ask

QUESTION 4. Are the followin~ statements equivalent for any integer ~

?

1)

0¢

"~¢(T'I-K) , V K ,

2)

Lst T-' e

eT-'

• --

%,@tsk(K) ~ KI,, . c

L~t

T e -.. • T. v

By t h e w a y ,

we do n o t

know t h e

answer

even to

the

following

ques-

tion,

QUESTION 5. Is it true that

Tc R (T-')

i~f .........

Lat T"

c

L~tT

.

Many interesting problems arise when considering special perturbations of normal operators. Recall that the problem of stability of continuous spectra in case of normal operators is reduced to the calculation of Sarason hulls. QUESTION 6. Let

N

and

N + K be normal operators. Is it tru.e

225 that

',,,Q,~,k K < +oo K~:~,

-----~ C. N = C

N÷K '~•

p<4 "-~" GN == GN +K 9

For what normal operators

N

K ~ F~ ~

GN

=

GN+K ?

(1)

It was noted in [1] that there are normal operators not satisfying

(1). If ~ and ~ + K are unitary then (I) holds with W = N This is so because the absolutely continuous parts of ~ and ~ + are unitarily equivalent and the Saracen hull of a unitary operator depends on its absolutely continuous part only~ Therefore Question 6 may be considered as a question of "scattering theory of normal operators". Consider a narrower class of perturbations, namely we assume henceforth that N commutes with N + K Then the symmetric difference G N ~ ~ N + K consists of the points of the point spectrum of N or of N + K This reduces the question to the investigation of metric properties of the harmonic measure, L.Carleson has proved in [6] that the harmonic measure of any simply connected domain is absolutely conti~uous with respect to Hausdorff m e a s u r e ~ , where >I/~ is an absolute constant. Using this result, it can be proved that ~N = G ~ + ~ if ~ and N commutes with N + ~. QUESTION 7. Let

and

N +K

be commutin~ normal operators,

Is it true that

REFERENCES I. M a E a p o B H.r. - ~ o E ~ . A H C C C P (to appear). 2~ M a k a r o v N,G,, ¥ a s j u n i n VoI. A model for noncontractions and stability of the continuous spectrum. Complex Analysis and Spectral Theory, Lecture Notes in Math,, 1981, 864, 365-

412,

226

3. S a r a s o n

D. Invariant subspaces and unstarred operator al-

gebras. - Pacific J.Math., 1966, 17, 511-517. 4- S a r a s o n D. Weak-star density of polynomials.

- J.reine

und angew.Math., 1972, 252, 1-15. 5. H z E o x ~ c E ~ ~ H.K. 0 BosMYmeHz~x cHeETpa yH~TapHNx onepaTopoB, - MaTeM.saMeTEZ, 6. C a r i e s o n

1969, 5, 341--349.

L. O n the distortion of sets on a Jordan curve

under conformal mapping. - Duke Math.J.,

N. G. N~uKAROV

(H.r.~POB)

1973, 40, 547-559.

CCCP, I98904, ZeH~Hrpa~, C T a p ~ HeTepro@, ~ e H ~ H r p a ~ c E ~ yH~Bepc~TeT, Ms T e M a T Z E O - - ~ e x a H ~ e c K ~

N. K. N I K O L 'S K I I

CCCP, I91011, ZeRz~rpa~, ~o~TaHEa 27, ZON~

~aEy~IRTeT

227 AI~OST-NOP~LL OPERATORS NOD~O ~'p

4.34.

O. Notations. H- separable complex Hilbert space of infinite dimension; ~(~), 11' II)~ (bounded operators on H , uniform norm); (~p, I' Ip ) = (Schatten- von Neumann p-class, p-norm); ~

={T~(~)

AN(H) =

~ finite rank, 0.
~

, ~= ~ ,

(almost normal operators on ~)=ITem(H):[T*,7] 6 ~ 4 ~

PT

For T ~ ~ N [ ~ ) we shall denote by its Heiton-Howe measure and by(~. its Pincus G-function (see [6], [10], [5~) so that ~T = 4 GT ~X , ~here ~X is ~ebesgue ~easure on ~ _

I. Basic Analogy. It is known that for V 6 ~ N ( ~ ) we have: index (7-~I)=~T(£ ~ for ~ e C such that V - ~ I is Fredholm. In [13] we noticed several instances which suggest that this relation is part of a far reaching analogy, in which the ~ -function plays the same role for ~ -perturbations of almost normal operat6rs as the index for compact perturbations of essentially normal operators. II. Invariance of PT under ~-per%urbations. This should correspond %o the invariance of the index under compact perturbations. In [13] we proved that if T , $ £ ~ ( ~ ) ~ - ~ and T or $ has finite multicyclicity then % = p$ . Our proof in [13] depended on the use of the quasidiagonality relative to ~ . In fact one needs less. Consider

where the liminf

+ is with respect to the natural order on ~4

"

(see [12]), ~oPosnION.

~-~6

~

Ae,~,,T,S~AN(H)

• Then we have

PROOF. As in

k~(T)=O

and

%=%.

[13] (Prop.3) the proof reduces to showing that

%[T*,T]:%[S*,B] such that I[A~,T:II~-O T

and SU!ODOSe

.

k~(T):o there areAB6:~;, A.~I as ~ - . and the s a m e holds also for

Since

replaced by S.Denoting

~ =T- $

we have:

228

i% (IT,T* ]- [ ~ ,S* ]1 1: =I%([X,T*] + [,~*,S])l-:~,~ I%(A~,([X,T*] +[ X*,S])lg I,I,..~oo ~ &~, ~,~p 1%( [ A,,X ,q'~'] + [ A,, X*, S])l ÷ il,.-~oo

+ ~{,~,~u~pCl[ T*, A~]I~ IXI~, +1 [ S, A,,]b.I X* t~): 0. We proved in [13] that multiplicity.

k~(T)=0

for T e

AN(~)

@

with finite

llI. ~uasitrian~ularity° A refinement of Halmo~ notion of quasitriangularity [7] ~vas considered in [11]. The corresponding generalization of Apostol's modulus of quasitriangularity is:

~,p(T)= ~444,~ I(I P)~PIp ~here the liminf is with respect to the natural order on ~ . We proved in [13] that for T ~ ~N(~)~ ~ ( ~ ) = 0 ~PT'<0 . We conjecture an analogue of the Apostol-Foia~-Voiculescu theorem on quasitriangular operators [1]. CONJECTURE 2. i~or T ~

AN(H)

, we have

1# /

~-0w This would imply in particular that PT ( 0 ~ ( | ) Some results for subnormal and cosubnormal operators supporting the conjecture have been obtained in [13]o IV. Analogue of the BDP theorem. The following conjecture concerns an analogue of the Brown-Douglas-Fillmore theorem [3] on essentially normal operators.

229

CONJECTURE 3- Let

t4,~ ~ ~ A ~ ( ~ )

Then there is a normal operator U£~(~s~)

be such that

~£~(~)

~T4 =~T~

and a unitary

such that

This conjecture implies the following CONJECTURE 4. If that ~ 8 ~

T~(~)then

there is

~

AN(H)

such

= normal + Hilbert-Schmidt.

Note that this last statement corresponds to an important part in the proof of the BDF theorem,

the existence of inverses in Ext or

equivalently the completely positive lifting part in the "Ext is a group" theorem (see [2]). Even for almost

normal weighted shifts,

there is only a quite restricted class for which Conjecture 4 has been established

[8].

Note also that Conjecture 4 implies Conjecture

I and that by

analogy with the proof of the Choi-Effros completely positive lifting theorem one should expect the vanishing of

k~

to be an essen-

tial ingredient in establishing Conjecture 4.

RE~ERENCES I. A p o s t o i

C.,

F o i a q

C., V o i c u 1 e s c u

Some results on non-quasitriangular ~th.Pures

Appl.

2. A r v e s o n

D.

operators VI. - Rev.Roum.

1973, 18, 1473-1494. W.B.

A note on essentially normal operators.

Proc.Royal Irish Acad. 1974, 74, 143-146. 3. B r o w n L.G., D o u g I a s R.G.,

F i I I m o r e

P.A.

Unitary equivalence modulo the compact operators and extensions .%L

of C~-algebras. Lect.Notes in Math., 1973, 345, 58-128. 4. C a r e y R.W., P i n c u s J.D. Commutators, symbols and determining functions. 5. C 1 a n c e y 1979, 742. 6. H e 1 t o n

K. J.W.,

- J.Funct.Anal.

1975, 19, 50-80.

Seminormal operators. Lect.Notes H o w e

R.

in iv~th.,

Integral operators,

commuta-

tor traces, index and homology. Lect.Notes in I~ath. 1973, 345, 141-209.

230

7. H a I m o s (Szeged),

P.R. Quasitriangular

- Acta Sci.~ath.

1968, 29, 283-293.

8. P a s n i c u

C.

Weighted

of normal operators. 9. P e a r c y CBMS, Regional Amer.~ath.Soc., 10. P i n c u s

C.

Some r e e ~ t

Conference

Commutators

in operator theory.

in Nathematics

no.36, Prodidence,

12. V o i c u 1 e s c u

Some extensions

Appl., D.

1973,

D.

tions of almost-normal Birkhauser

of integral

equations,

of quasitriangularity.

18, 1303-1320.

Some results

space operators.

3-37. 13. V o i c u 1 e s c u

and systems

121, 219-249. D.

- Rev.Roum.~ath.Pures

D.VOICULESCU

developments

Series

~ 0 ~

1982.

1978. J.D.

11. V o i c u 1 e s c u

ons of Hilbert

shifts as direct summands

INCREST preprint

I. - A c t a T~ath., 1968,

Theory,

operators.

on norm-ideal

- J.Operator

Remarks

operators.

Theory,

on Hilbert-Schmidt

perturbati1979,

2,

perturba-

- In: Topics in ~odern Operator

1981. Department

of ~ t h e m a t i c s

INCREST Bd.P~cii

220, 79622 Buchareat RONJLNIA

231 4.35.

HYPONORNAL OPERATORS AND SPECTRAL ABSOLUTE CONTINUITY

In the sequel only bounded operators on an infinite dimensional, separable Hilbert space H will be considered. An operator T on H is said to be hyponormal if Such an operator is said to be completely hyponormal if, in addition, T has no normal rt, that is, if there is no subspace ~ { 0 1 reducing T on which is normal. If A is selfadjoint with the spectral family ~E~.~ then the ~ "'2, v. ~ j set of vectors ~ in H for which II II is an absolutely continuous function of ~ is a subspace, H @ ( ~ ) , reducing (see, e.g., Kato [I], p.516). If H @ ( A ) ~ { }0 , then A I ~ @ ( A ) is

TV-TT

called the absolutely continuous part of A

, and if H @ ( A ) = H

then A is said to be absolutely continuous. Similar concepts can be defined for a unitary operator. If T is completely hyponormal then its real and imaginary parts are absolutely continuous. In addition, if T has a polar factorization

T-Uffl,

U

ita and if l- [T "T

(i)

then U is also ~bsolutely continuous. (See [23, p.42 and [3], p. 193. Incidentally, such a polar factorization (I) exists, ,and is unique, if and only if O is not in the point spectrum of Y ; see

[4], p.277.) Ingenera~, i f 7 is completely hyponormal, then its absolute value ~Y ~= (T* T )i/zneed not be absolutely continuous or even have an abmolutely continuous part. Probably the simplest example is the simple unilateral shift V for which V * V is the identity. Of course, V does not have a polar factorization (I), but, nevertheless, there are simple_examples of completely hyponormal T satisfying (I) for which IV1 has no absolutely continuous part. Moreover, it has recently been shown by K.F.Clancey and the author [5] that if ~ is selfadjoint on H , then there exists a completely hyponormal T satisfying IT1_= P and having the polar factorination (I) if and only if (i) ~ 0 and g(P) contains at least two points, (ii) 0 is not in the point spectrum of P , and (iii~ neither ~ ~(~) nor min ~ ( ~ ) is in the point spectrum of r with a finite multiplicity. Let a nonempty compact set of the complex plane be called radially symmetric if whenever

~4

is in the set then so is the entire

232 circ:le IZI=Z4 . All examples known to the author of completely hyponormal operators T for which IT I does not have an absolutely continuous part, and whether or not (1) obtains, seem to have radially symmteric spectra. Por instance, if ~(IT~) has Lebesgue linear measure zero, then ~ ( T ) is surely radiallE symmetric; see [6], p.426, also [7]. At the other extreme, if T is completely hyponormal and if there exists some open wedge W m { E : E ~ O and O<~Z<~<~} (or rotated set e & g W ) which does not intersect Q ~(T) then ITI is absolutely continuous. In certain other instances also one can show at least that H (ffl)÷{0} see [7] and the references cited there. The following conjecture was made in

D], CONJECTURE I. Let T

be completel Z hyponormal with a polar fac-

torization (I)~ Suppose that that some circle

IZ~=~

in nonempt E sets. Then

~(T)

is not radiall E s,ymmetric j so

intersects both H@(IT~)#

~ 0}

6(T)

and its complement

•

The following stronger statement was also indicated in [7] and is set forth here, but with somewhat less conviction than the preceding conjecture, as CONJECTURE 2. Conjecture 1 remains true without the hypothesis

(i). REFERENCES I. K a t o T. Perturbation theory for linear operators, SpringerVerlag, New York Inc., 1967. 2. P u t n a m C.R. Commutation properties of Hilbert space operators and related topics, Ergebnisse der Math., 36, SpringerVerlag, New York Inc., 1967. 3. P u t n a m C.R. A polar area inequality for hyponormal specra. - J.Operator Theory, 1980, 4, 191-200. 4. P u t n a m C.R. Absolute continuity of polar factors of hyponormal operators. - Amer.J.Math., Suppl. 1981, 277-283. 5. C 1 a n c e y K.F., P u t n a m C.R. Nonnegative perturbations of selfadjoint operators. - J.Funct.Anal., 1983, 51 (to appear). 6. P u t n a m C.R. Spectra of polar factors of hyponormal operators. - Trans.Amer.Math. Soc., 1974, 188, 419-428.

233

7. P u t n a m

C.R.

Absolute values of hyponormal operators with

asymmetric spectra. - M i c h . ~ t h . J o u r . , C.R.PUTNA~

1983, 30

(to appear).

PURDUE UNIVERSITY Department of Mathematics West Lafayette, USA

Indiana 47907

234 4.36. old

OPERATORS, ANALYTIC NEGLIGIBILITY, AND CAPACITIES

Let ~(T) and ~ ( ~ ) denote the spectrum and point spectrum of a bounded operator T on a Hilbert space H . Such an operator is said to be s u b n o r m a 1 if it has a normal extension on a Hilbert space K , ~ - ~ H • For the basic properties of subnormal operaters see [I]. A subnormal T on H is said to be c o m p 1 e t e 1 y s u b n o r m a 1 if there is no nontrivial subspace of H reducing T on which T is normal. If T is completely subnormal then ~p (T) is empty. A necessary and sufficient condition in order that a compact subset of C be the spectrum of a completely subnormal operator was given in [2]. If X is a compact subset of ~ , let ~(X) denote the functions on X uniformly approximable on X by rational functions with poles off X . A compact subset Q of X is called a p e a k s e t of ~(X) if there exists a function ~ i n ~ ( X ) such that

Q andl

l<4

result was proved in THEOREM. Let ~ extension N = I ~ E

onX\Q

be subnormal on H

~ o_~n K

, KD

vial proper peak set of ~(~(T))

an,d, H , the spaceE(Q) H the minimal

; see

p.56. The follow g

[41. with the minimal norm~ 1

H . Suppose that

Q

is a non-tri-

and that ~(~)=l=0. Then ~(~)H =t= I@}

reduces

T

normal extemsien E(Q)N

,TIE(Q)H

is subnormaZ,

o_~nE (~)K, and

~mrt,h,~,,r,, ,if it, iS,, also assumed that ~ ( ~ ) = O ( Q )

(TI EcOH) G.

, the___nnTlE(e)H

is norm~l. Thus, in dealing with reducing subspaces of subnormal operators , it is of interest to have conditions assuring that a subset of a compact X ( = ~ ( T ) ) be a peak set of ~ ( X ) . PROBLEM I. L~%

X

be a compact subset of C

rectifiable simpl 9 closed curve for which X

) is not erupt,7 and C ~ X

Does it follow that Q

Q =clos

~nd let

C

be a

((exterior of~)n

has Lebes~ae arc length measure O.

mus% be a peak set of ~(X)

?

In caseCis of class C (or piecewise C ), the answer is affirmative and was first demonstrated by Lautzenheiser [5]. A modified version of his proof can be found in [6], pp. 194-195. A crucial step in the argument is an applicatiom of a result of Davie and ~ksendal [7] which requires that the set C ~ X

be analytically negligible.

235

(A compact set ~ is said to be analytically negligible if every continuous function on ~ which is analytic on an open set V can be approximated uniformly on V U E by functions continuous on and analytic on ~ U E ; see [3], p.234.) The ~ hypothesis is then used to ensure the analytic negligibility of ~ ~ X as a consequence of a result of Vitushkin [8]. It may be noted that the collection of analytically negligible sets has been extended by Vitushkin to include Liapunov curves (see[9], p.115) and by D a v i e ( DO], section 4) to include "hypo-Liapunov" curves. Thus, for such curves C , the answer to PROBLEM I is again yeS. The question as to whether a general rectifiable curve, or even one of class C ~ , for instance, is necessarily analytically negligible, as well as the corresponding question in PROBLEM 1, apparently remains open however. As already noted, PROBLEM I is related to questions concerning subnormal operators. The problem also arose in connection with a possible generalization of the notion of an "areally disconnected set" as defined in ~] and with a related rational approximation question. Problem 2 below deals with some estimates for the norms of certain operators associated with a bounded operator T on a Hilbert space and with two capacities of the set @(T) • Let ~(~) and ~ (E) denote the analytic capacity and the continuous analytic capacity (or A C capacity) of a set ~ in ~ . (For definitions and properties see, for example, ~,11,9]. A brief history of both capacities is contained in K9], pp. 142-143, where it is also noted that the concept of continuous analytic capacity was first defined by Dolzhenko ~2].) It is known that for any Berel set

E, ( ~ ~e~E) ~ ~ ~(E) ~ ~(E) see 5 q, pp.9, 79. THEOREM. Let ~

following

proved in # 3].

be,'a bo~_ud,ed op,erator on, a, Hil,be~t space and

suppose that

(i)

( T ~)(T- ~)*~ ~] ~ ® holds for some nonnegative operator ~

and for all Z

in the un-

bounded component O f the complement of ~(T)

. Then ~ ] 4 / ~ ( ~ ( T ) ) .

if, in addition,

and if, for instance,

~p(T)

(I~ holds for all ~

i_~n ~

is contained in the interior of ~ ( T )

(in particular~ if

236

~T)

is empty), then also PROBLEM 2. Does condition (I~, if v a l i d f o r all 7

without an~

rectriction

o r possibl,

even

on

, alwa[s impl[

~p(T) meas

It m a y be noted that if T * then (1) holds for all ~

meas ( (T)

in

,but

1~14/~(~(q5),

?

is hyponormal,

~

that

i..nn ~

so

with~-qT*-T~T

thatTT~-T*T~ @ and, moreover,

see D4] REFERENCES

I. H a i m o s

P.R.

A Hilbert space problem book, van Nostrand

Co., 1967. 2, C 1 a n c e y

K.F.,

P u t n a m

C.R.

The local spectral be-

havior of completely subnormal operators. - Trans.AmertMath.Soc., 1972, 163, 239-244. 3. G a m e 1 i n T.W, 4. P u t n a m Jour.Math.,

C.R.

Uniform algebras, Prentice-Hall,

Inc.,1969.

Peak sets and subnormal operators. - Ill.

1977, 21, 388-394.

5. L a u t z e n h e i s e r

R.G.

Spectral sets, reducing sub-

spaces, and function algebras, Thesis, Indiana Univ., 1973. 6. P u t n a m C.R. Rational approximation and Swiss cheeses. - Mich Math.Jour., 7. D a v i e A.M.,

1977, 24, 193-196. ~ k s e n d a i B.K.

Rational approximation

on the union of sets. - P r c c . A m e r . ~ t h . S o c . , 8.

1971, 29, 581-584.

B E T y m E H H A.r. AH821~TH~ecE6uq eMEOCT~ ~ o ~ e C T B B ss~a~ax TeopzE n p E S J m ~ e H ~ , - YcnexE MaTeM. HayE, 1967, 22, }~ 6,

I4I-I99. 9. Z a i c m a n

L.

Analytic capacity and rational approximation.

Lecture notes in mathematics, 10. D a v i e

A.M.

50, Springer-Verlag,

1968.

Analytic capacity and appriximation problems.

- Trans.Amer.Math.Soc., 1972, 171, 409-444. 11. G a r n e t t J. Analytic capacity and measure, Lecture notes in mathematics, 12. ~ 0 JI ~ e H E 0

297, Springer-Verlag,

E.H.

1972.

0 I~HdJI~eHI4E Ha 3SmEHyTBLY 06~aoTaX H

0 Hy~B--~o~eCTBaX. - ~OF~.AH CCCP, 1962, 143, 9 4, 77i-774. 13. P u t n a m C.R. Spectra and measure inequalities. - Trans. Amer.~th.Soc.,

1977, 231, 519-529.

237

14. P u t n a m C.R~ An inequality for the area of hyponormal spectra. - ~ath.Zeits., 1970, 116, 323-330. C.R.PUTNA~

PURDUE UNIVERSITY, Department of ~ t h . , West Lafayette, Indiana 47907, USA

EDITORS' NOTE. Two works of Valskii

(P.8.Ban~c~, ~o~. AH

CCCP, 1967, 173, N I, 12-14; C~d~pCE.MaTeM.~., I96V, 8, ~ 6, 1222-1235) contain some results concerning the themes discussed in this section.

238

GENERALIZED DERIVATIONS AND SEMIDIAGONALITY

4.37.

Let

A,, A z

be bounded l i n e a r operators on a H i l b e r t

space

A = A A ^ be an operator on the space ~ ( ~ ) of all bounded operators o n ~ ' # ~ defined by ~ ( X ) = A,X-XA , X e If A4 = A = ~ A A,,Az is a derivation of ~ ( ~ ) • That is why the operators /kA4,A~ are called sometimes g e n e r a i i z • d d e r i v a t i o n s • Put ~ == A A * A ~ . A question of whether

was raised by various people (see [1], [2], [ 3 ] ) . Equality (1) i s true for normal operators A4, Az , whence Fuglede - Putnam theorem follows (see [4]). This equality means that We~ ~A = Kez& and so Ke~ ~ = Ke~ a ~ = K e n Y A = Ke% A . zn [3] it is proved that (1)holds when ~4 = A s is a cyclic subnormal operator or a weighted shift with non-vanishing weights.

Let

pe [{,~]

. An operator B

s e m i d i a g o n a 1

if its modulus of

~p(A) = Z ~ {

in

B(X)

is called p -

p -quasidiagonality

HPA-APII

( ~ being the set of all finite rank projections) is ~inite. Denote by ~ p the class of T P -perturbations of direct sums of p -semidiagonal operators. In [ 5] it is proved that (1) is true if one of ~j belongs to ~ 4 • Though that result covers rather extensive class of generalized derivations ( ~4 contains all normal operators with one-dlmensional spectrum and their nuclear perturbations, weighted shifts of an arbitrary multiplicity and polynomials of such shifts, Bishop's operators), it is not applicable to many generalized derivations with normal coefficients. Namely, a normal operator belongs in general to ~ , but not to ~ 4 . It seems reasonable to try to replace the hypothesis of 1-semidiagonality of one of Ai's by 2-semidiagonality of both. &

QUESTION 1. Does (1) hold i f

A4, A ~ 6 ~ 2

?

QUESTION 2, Does there exist an operator not in

~

?

An affirmative answer to the following question would solve QUESTION 2 (see [5]).

239

QUESTION 3- Do there exist AX

-- X A

Ae~(~)

is a, non-zero

, X~

~2

such that

pro,jection?

REFERENCES

I. J o h n s o n rivations°

B.,

W i i i i a m s

- Pacif.J.Math.,

2, W i 1 1 i a m s

J. The range of normal de-

1975, 58, 105-122,

J. Derivation ranges: open problems

Modern OperoTheory,

5 Int. Conf.0per. Theory, Timi§oara

1981, 319-328. 3. Y a n g H o . Commutants and derivation ranges,

- In: Top Birkhauser

- Tohoku Math J.,

1975, 27, 5 0 9 - 5 1 4 4. P u t n a m C, Commutation properties of Hilbert space operators and related topics.

5. m y a ~ M a H -

Springer-Verlag,

Ergebnisse

36, 1967.

B.C. 06 onel)STO~SX ~MHOXeHBS ~ c~e~sx KO~MI~TSTO~OB,

S81~CER H S y q H . C e M ~ H . ~ 0 ~ ,

(to appear).

V. S. SHUL'NAN

CCCP, 160600, B o g o t a ,

(B.C.~E~H)

~JI.MS~KOBOK0r0 6,

Ks%e~m .s~e~mT~m

240 4.38.

WHAT IS A FINITE OPERATOR?

An operator acting on a complex separable Hilbert space is called f i n i t e if the identity operator 1 is perpendicular to the range of the inner derivation ~A induced by , that is,

~AC~) = A ~ - ~ A and ~ runs over the algebra ~ ( ~ ) o f a l l (bounded linear) operators acting on ~ . The notion of finite operator was introduced by J.P. Williams in ~7~, where he proved the following result.

where

THEOREM 1. These are equivalent conditions on an operato r ~ :

(ii) 0 belongs to the closure of the numerical range of ~A~)

for each X

in ~C~)"

(iii) There exists I ~ ( ~ ) * s u c h

that I(~)=~--ll~ll

an__~d

6-A .

The origin of the term f i n i t e is this: if A C - ~ ( ~ ) has a finite dimensional reducing subspace ~ ~ { 0~ , {ei}i=4 ( ~ = ~

~)

is an orthonormal basis of ~ ,

~(B)=(~/~)~. ( ~ 8 i 8 ~

, then ~

Theorem 1 (iii) and therefore Thus,

if

dimension ~ of the family

~

• )

has a reducing subspace of

, then

~= U ~

i s a subset

(Fin) of all finite operators; furthermore, since

{~in} is closed ~ ~ (~4) [7], ~-C COnJeCTURE

satisfies the conditions of

is a finite operator.

~=~T~ZC~):T

(,=4,~,..

and

I

(J.P.winiams

(~in}.

[7] 1. (Fin) = ~ -

As we have observed above, if A q ~ , then it is possible to construct ~ a~ in Theorem 1 (iii) such that ~ does not vanish identically on ~ ) , the ideal of all compact operators. J.H.Anderson proved in [1] (Theorem 10.10 and its proof) that this fact can actually be used to characterize ~ : A ~ if and only if there exists ~ ( ~ ) * such that ~(~)=~ = II~ , and ker I D

241 D~, but ker $ ~ ~ ( ~ ) ; furthermore, if A ~ (Fin)\~ ,then every ~ as in Theoremn~-~1(iii) is necessarily a s i n g u 1 a r functional (i.e., ker ~ O ~ This is true, in particular, if Since (Fin) D ~ - , it is plain that (Fin) contains all q u a s i d i a g o n a 1 o p e r a t o r s (in the sense of Halmos; see [4] ). Moreover, (Fin) contains every operator of the form

A ='T' ®B *K,

(*)

where T ~ ( ~ ^ ~ (for some finite or infinite dimensional subspace of infinite codimension), ~ = 4 ~ for a suitable uniformly bounded sequence ~ ~=~ o f operators a c t i n g on f i n i t e

Wo

dimensional subspoces (i.e , B is a b i o c k - d i a g o n a i operator [4] ) and ~ is compact Let ~ 0 denote the family of all operators of the form (*). It is apparent that

and

CONJECTURE 2 then there exits

(D'A'Herrer°)'(Pin)1= ~'0 K £~(%)

and

Q

; moreover, if A ~ ( ~

I

quasidia~ona! such that

A-K=A (Q Q®Q® ,,.'), Several remarks are relevant here: (1) J.W.Bunce has obtained several other equivalences, in tion to the three given by Theorem I (see [2] ). (2) ~-

properly includes

U~4(~)

[3, p.262],

addi-

[5 , Example

111. (3) In [I, Corollary 10.8], J.H.Anderson proved that every unilateral weighted shift is a finite operator, by exhibiting a functional ~ satisfying the conditions of Theorem I (iii). Recently, D.A. Herrero proved that all the (unilateral or bilateral) weighted shift operators belong to ~-[5].

242

(4) Suppose that

{P~=~ that

A Q ~-

; then there exists a sequence

of non-zero finite rank orthogonal projections such

IIAP~ - P ~ A II"~0 (~-~oo). Passing, if necessary, to a subse-

quence we can directly assume that ~ - - ~ H (weakly, as n-~oo ) for some hermitian operator H , 0 ~ H % ~ . It is easily seen that commutes with H . If ~ has a non-zero finite rank spectral projection, then A ~ ~ . If either ~=0 or ~ =~ , then it is not difficult to show that ~ 6 ~0 • (5) According to a well-knQwn result of D.Voiculescu, A - K ~ Ao~QoQo

(as in (**~ if a n d only if the

)

C~-algebra gene-

rated by ~) in the Calkin algebra ~(~)/~(~) admits a • -representation ~ such that ~ - ~ ( ~ ) = ~ ~6]. Suppose that

A ~ (F~)'

. t~ it posslble to ~ e the s i ~ n a ~

f~c,t!,,onals

provided b~ Theorem 1 (iii) an d a Gelfand-Naimark-Sega ! t,ype constructio n in order to

construct a

~ -representation

desired properties (i~e., so that

~, ~(~)

~

with the

is quasidia~0nal)?

(6) Is it possible I at l east~ to show that the existence of such a S ~ u l a r ~S,

III~I

functional implies that, for each

finite rank pert~bat,ion

A-F ~A ~ _ _

~

, where

A~

F~

,

~th

nF~< B

~>0~ ,

m

ad-

such t~t

acts on a non-zero finitedimen@io-

nal subspace? (An affirmative answer to this last question implies that (Fin) ~ = ~

).

(7) Any partial answer to the above questions will also shed some light on several interesting problems related to quasidiagonal operators. REFERENCES I. A n d e r s o n

J.H.

Derivations, commutators and essential

numerical range. Dissertation, Indiana University, 1971. 2. B u n c e J.W. Finite operators and amenable C*-algebras. - Proc.Amer. Math.Soc.~1976, 56, 145-151. 3. B u n c e J.W., D e d d e n s J.A. C*-algebras generated by weighted shifts. - Indiana Unlv.Math.J.~1973, 23, 257-271. 4. H a 1 m o s P.R. Ten problems in Hilbert space.-Bull.Amer.

243

Math.Soc.~1970, 5. H e r r e r o

76, 887-933. D.A. On quasidiagonal weighted shifts and appro-

ximation of operators.-Indiana Univ.Math.J. 6. V o i c u 1 e s c u

D.

theorem. - Rev.Roum.Math.Pures 7. W i 1 1 i a m s

J.P.

(To appear).

A non-commutative Weyl-von Neumann et Appl.~1976,

Finite0perators.

21, 97-113.

- Proc.Amer.Math.Soc.~

1970, 26, 129-136. DON[INGO A.HERRERO

Arizona State University Tempe, Arizona 95287 USA

This research has been partially supported by a Grand of the National Science ~oundation

244

4.39.

THE SPECTRUM OF AN ENDOMORPHISM IN A COMMUTATIVE BANACH ALGEBRA

The theorem of H.Kamowitz and S.Scheinberg D] establishes that the spectrum of a ~0n~eriodic automorphism ~ : A - ~ A of a semisimple commutative Banach algebra A (over 6 ) contains the unit circle ~ . Several rather simple proofs of the theorem have been obtained besides the original one ([2],[3]e.g.) and its various generalizations found (see e.g.[4],[53,[6]...). It is easy to show that under the conditions of the theorem the spectrum is connected, At the same t~me all "positive" information is exhausted, apparently, by these two properties of the spectrum. There are examples (see [7~, [8]~[9~) demonstrating the absence of any kind of symmetry structure in the spectrum even if we suppose that the given algebra is regular in the sense of Shilov. Let for instance ~ be a compact set in ~ lying in the annulus { : } and containing { ~ : ~ I ~ l l Zl and equal to the closure of its interior int K. Denote by A the family of all functions continuous on ~ and holomorphic in int K. Clearly, A equipped with the usual sup-norm on ~ and with the pointwise operations is a Banach algebra. The spectrum of multiplication by the "independent variable" On A evidently coincides with ~ . On the other hand the conditions imposed on ~ imply that A is a Banach algebra (without umit) with respect to the convolution

~4

F

~(~) ~ ( ~ )

......

corresponding to multiplication of the Laurent coefficients. This algebra being semi-simple, its maximal ideal space can be identified with the set of all integers. Adjoining a unit to A thus turns it into a regular algebra. Obviously the above mentioned operator on A is an automorphism. 1..A.re t.here other necessary conditions, on th.e.spectrum of.~ nonperiodic automorph%sm of % semi-s~mple commut.a.t~.v.~ 'Banach al~ebra besides the t~0........m.entionedabove? In particular, is it obligatory for the spectrum to have interior points when i% differs from ~ ? It is known in such cases (see ~7],KB]) that the set of interior points

245

may not be dense in the spectrum and may not be connected either. Let M A be the maximal ideal space of a commutative and semis~mple Banach algebra A . An automorphism T of A induces an automorphism of the algebra C ~ C(M A) . The essential meaning of the Kamowitz - Scheinberg theorem is that ~c(T) c ~A (T) . It is natural from this point of view to study the inclusion gc (L) c gA ( ~ ) for a more general class of operators L . The case of weighted automorphisms I,@ ~e~ W. T ~ with ~ an invertible element of A has, for example~been studied in [10]. It turns out that the inclusion does not hold for this class of operators. 2,

Does the spectrum of L = ~ ,

automorphism

T

c g ~ t r u c t e d for a

non-periodic

, contain any circle ' centred at %he origin? If it

does then we obtain an instant generalization of the theorem of Kamowitz and Scheinberg. The spectrum of operators, looking like l. , acting on the algebra of all continuous functions on a compact set has a complete description[11 . I f A is also a u n i f o = algebra then Thus 6"A (~) ~ ~ ( ],) provided that I, is a weighted automorphism of two uniform algebras ~ and ~ having the same maximal ideal spaceo

3. Let A be a closed subalgebra of a semi-simple commutative Banach algebra ~ and let ~A--~ M B . Let ~ be a weighted automorphism of A and ~ simultaneously. IS it true then that

6~A(~) =

~B ( h )

?

We O O N ~ a T ~ R that this question has a negative answer. The spectrum of an endomorphism apparently does not have any particular properties even if we suppose that A is a uniform algebrao Given two oompaota ~4 and ~9 it is easy to obtain an endomorphism with the spectrum either ~4"U ~ or ~I' ~% . The only obvious property of spectra is that Iw belongs to the spectrum, when ~ ranges over its boundary and ~ over the set of non-negative integers. 4. Let ~ be a compact subset of ~ satisfying ~ 6 ~ for ~ = ~,~,.., and for all points ~ in the boundary of ~ . Is there an endomorphism of a uniform algebra whose spectrum is eq~l

to ~

?

Spectra of endomorphisms of uniform algebras (and even those for weighted endomorphisms) can be described pretty well under the

246

additional assumption that the induced mapping of the maximal ideal space keeps the Shilov boundary invariant (see D2], where one can find references to preceding papers of Kamowitz). Roughly speaking, things, in this~case, are going as well as in the case of Banach algebras of functions continuous on a compact set. The situation changes dramatically when the boundary, or only a part of it, penetrates the interior. In such circumstances it is common to begin with the consideration of classical examples. Let D be the unit disc in C , let A(~) be the algebra (disc-algebra) of all functions continuous on the closure of ~ and holomorphic in ~ , and let H~(~) be the algebra of all functions bounded and holomorphic in ~ . Both algebras A(~) and H ~ ( ~ ) are equipped with the supnOl~. 5. Every endomorphism of A(O) induces a natural endomorphism of ~ . Do the spectra of thes e endomorphisms coincide? In this connection it is worth-while to note that the answer to an analogous question concerning the algebra of all continuous functions on a compact set and the algebra of all bounded functions on the same set is in the affirmative ~3~. The proof of this result uses, however, a full (though comparatively simple) analysis of the possible spectral pictures depending on the dynamics generated by the endomorphism. The interesting papers ~4~, ~ , ~6~ of Kamowitz (see also [6S, [9~) deal with spectra of the endomorphisms of A ( D ) whose induced mappings do not preserve the boundary of ~ . In the non-degenerate case the spectrum has a tendency to fill out the disc.Discrete and continuous spirals as well as compacta bounded by such spirals may nevertheless appear as the spectrum of an endomorphism. (But only the spirals can appear in the case of Mobius transformations). 6. Is the spectrum of an endomorphism of the disc-algebra a semi-~rouu (with respect to multiplication in ~ ) ? What kind of semi~rouDs can arise as spectra.? 7. Is it possible to say something concernin~ the spectra of 9ndomorphisms of natural multi-dimensional generalizations of the disc algebra? Note that in the one-dimensional case the theory of DenjoyWolff and the interpolation theorem of Carleson-Newman are often

247

involved in the question. The problem of describing spectra for weighted automorphisms is closely related with an analogous one for the so-called "shift-type" operators which have been studied by A.Lebedev ~7~ and A.Antonerich

Dsl

Let A be a uniform algebra of operators on a Banach space X . An invertible operator U on X is called a "shift-type" operator if U A U A . UB lly X a B nach space of f ctio and A is a subalgebra of the algebra of multipliers for ~ . The transformation ~ ~ U. ~ U ~ determines an automorphism T of ~ which induces the mapping ~: ~ A ~ M A . It is assumed that: I) the set of ~-periodic points is of first category in the Shilov boundary 8A ; 2) the spectrum of U : X ~ ~ is contained in S A ; 3) each invertible operator @ : X - - ~ X , ge~ is invertible as an element of A ; 4) the topological spaces ~A and ~ have the same stock of clopen (closed and open) ~ -invarlant subsets. Then ~ ( @ ~ ) - ~ - ~ ( ~ T ) for all ~ in ~ [19]. We con,~ecture that Condition $ is superfluous. If this were true it would be possible (in view of [1I]) to obtain a complete description of @~ ( @ ~ . It is reasonable to ask the same question for other algebras A besides the uniform ones. REFERENCES I. K a m o w i t z automorphisms

H., S c h e i n b e r g

of Banach algebras.

S.

The spectrum

- J.Punct.An.,

1969,

of

4, N-2,

268-276. 2. J o h n s o n

B.E.

Automorphisms

ras. - P r o c . A m . M a t h . S o c . ,

of commutative

Banach algeb-

1973, 40, N 2, 497-499.

3. ~ e B ~ P.H. HOBOe ~o~asaTe~cTBo TeopeM~ O6 aBTOMOp~zsMax 6aEaXOBHX a~re6p. -BeCTH.MFY, cep.MaTeM., Mex., I972, ~4, VI-72. 4. ~ e B ~ P.H. 0d aBTOMOp~SMaX 6aHaXOBRX a~re6p. - ~ym~.aHax~8 E ero np~., 1972, 6, ~ I, 16-18. 5. ~ e B E P.H. 0 COBMeCTHOM cneETpe HeEoTopb~x EOMMyTI~py~E~X oEepaTOpOB. ~ c c e p T ~ , M., 1978. 6. r o p ~ H E.A. EaE BR~JL~ET cHeETp SH~OMOp~EsMa ~cE-a~re~p~? 3au.HSyqH.ce~m~o~0~, 1983, 126, 55-68.

248

7. S c h e i n b e r g S. The spectrum of an autemorphism. Bull.Amer.Math.Soc,, 1972, 78, N 4, 621-623. 8. S c h e i n b e r g S. Automorphisms of commutative Banach algebras. - Problems in analysis, Princeton Univ.Press., Princeton 1970, 319-323. 9. r o p ~ H E.A. 0 cne~Tpe SH~OMOp~SMOB paBHoMepHHx axre6p. B EH. : Tes~cH ~oEx.~oH~ep"TeopeT~xecz~e ~ n p m ~ a ~ H H e BOnpOC~ Ma-TeMaT~E~" TapTy, I980, I08--II0. I0. E ~ T 0 B e p A.E. 0 cne~Tpe a B T O ~ O p ~ s M O B C BecoM ~ TeopeMe EaMoB~um-~s/m6epra. - ~yHE~.aHa~Hs ~ ero n p ~ . , 1979, 1 3 , ~ I, 70-71. II. E ~ T o B e p A.E. C n ~ T p a x ~ - - e CBOICTBa a B T O M O p ~ S x O B C BeC O M B paBHo~eps~x axre6pax. - 3 a n . H ~ . C e M ~ H . ~ 0 M H , I979, 92, 288-293. I2o E E T O B e p A.K. CneETpax~HHe CBO~CTBa r O M O M O p ~ S M O B C BecoM B a~re6pax Henpep~Bm~X ~ ~ ~x np~o~eHH~. - 3an.HayqH.ce~ . ~ I 0 1 ~ , 1982, 107, 89-103o 13. I{ E T 0 B e p A.E. 06 oiiepaTopax B C (~ , ~H~yI~IpOBaHHIgX l~a~ Emm~ OTo6pa~eH~2m. - ~JE~I~.asax~s E ero n p ~ . , 1982, 16, ~ 3, 61-62. 14. K a m o w i t z H. The spectra of endomorphlsms of the disk algebra. - Pacif.J.~ath., 1973, 46, N 2, 433-440. 15. K a m o w i t z H. The spectra of endemorphisms of algebras of analytic functions. - Pacif. J.Math. 1976, 66, N 2, 433-442. 16. K a m o w i t z H, Compact operators of the form @ C@ . Pacif.J.N~th., 1979, 80, N I, 205-211. I7. ~ e 6 e ~ e B A.B, 06 onepaTopax T~na BSBemeHHOrO C~BEra. ~ccepT~, M~HCE, 1980. 18. A H T O ~ e B ~ ~ A.B. 0nepaTop~ co C~B~rOM, n o p o m ~ e ~ M ~e~cTB~eM EOMIIaETHOI l~y211H H~. - CE61~pCE.MaTeM°EypH°I979, 20, ~ 3, 467-478. 19. E ~ • 0 B e p A.E. 0nepa~op~ IIO~CTaHOBE~ C BecoM B 6aHaXOBHX Mo~y~Ex H8~ paBHOMepH~M~ ax~e6pa~m (B negate). E. A. GORIN CCCP, 117234, ~ocFma ~ e H ~ c K ~ e rop~ (E.A.IDPHH) MexaHzEo-~aTe~aT~ecEz~ ~ r y 2 ~ T e T M O C K O B C E ~ rocy~apcTBeRR~ yHzBepc~TeT

A. K. KITOVER

(A.K.EETOBEP)

CCCP, 191119, ~eH~Hrpax, y~ KOHCTaHTNHa 3aC~OHOBa, ~.14, KB.2.

249 4.40.

COMPOSITION OF INTEGRATiON AND SUBSTITUTION

Consider a continuous function ~ on [0,1] satisfying~(0)=0, 0 ~ ~(~) ~ ~ . The function ~ defines a bounded linear operatorI~ On the space C ~o,1~ of all continuous functions on ~0,I~:

0

Recall that a bounded operator T

£%1IT"lt¢

o.

=

PROBT,~.'~. Describ,~ f u n c t i o n s operators

is called quasinilpotent if

~

correspondin ~ to q u a s i n i l p o t e n t

.

Clearly I ~

is quasinilpctent provided

0~< ~ ~ 1 Does the inverse., conclusion hold?

(2) An analogy with the theory

of matrices provides arguments in favour of the affirmative answer. Let I @ ~ } ~ - 4 be a nilpotent matrix with non-negative elements. It follows from the Perron-Frobenius theorem that it can be transformed to a low-triangular form with zero diagonal by a permutation of the basis. The obtained matrix I ~ I defines an operator On ~

with ~(~) < ~

.

Consider now a natural generalization of (I):

(3) 0

where the kernel K >~ 0 is continuous. Orientation preserving home~nerphisms of EO,1] replace permutations of the basis in the finite-dimensional case and preserve the inequality ~ ( X ) ~ X •

250

Consider a counter-argument to the conjecture. IfI~

s i n i l p o t e n t t h e n by Y e n t s c h ' s theorem t h e r e e x i s t zero continuous function S ~ 0 such that

~~ 0

is not qua-

and a non-

@(x)

o

Suppose q}~C @° ( [ 0 , 1 ~ ) . Then e v i d e n t l y ~ C ~ ( K0,1]~and;~(~'(O)= O, K~O,~.., ~ because ~ ( 0 ) ~ 0 . If it were possible to prove that belongs to a quasianalytic class (under some natural restrictions on ), it would imply clearly that ~ ~ 0 which contradicts Yentsch's theorem. Are there conditions on ~ no t demanding @(~)~X but such that a~y solution of ($~ belonss to la lquasianal,ytic Carleman class?

If yes, then there

exists ~

such that ~(~o) • $o

point to in EO,I] but neve~heless I ~ YU. I. LYUBIC

(D.H.~)

for a

is q~slnilpotent.

CCCP, 310077 Xap~EOB n~.~sepz~HcEozo 4 Xap~EOBCE~ IOCy~apCTBeHmm~ yH~Bepc~TeT

CHAPTER

5

HANKEL AND TOEPLITZ OPERATORS

A quadratic form is called Hankel (resp. Toeplitz) if entries of its matrix depend on the sum (resp. the difference) of indices only. These forms appeared as objects and tools in works of Jacobi, Stleltjes (and then Hilbert, Plemelj, Schur, Szego, Toeplitz ...). They play a decisive role in a very wide circle of problems (various kinds of moment problems, interpolation by analytic functions, inverse spectral problems, orthogonal polynomials, Prediction Theory, Wiener-Hopf equations, boundary problems of Function Theory, the extension theory of symmetric operators, singular integral equations, models of statistical physics etc.etc.). It wBs understood only later that the independent development of this apparatus is a prerequisite for its applications to the above "concrete" fields, and Hankel and Toeplitz operators were singled out as the object of a separate branch of Operator Theory. This branch includes: - techniques of singular integrals ranging from Hilbert, M.Riesz and Privalov to the Helson-Szego theorem discovered as a fact of Prediction Theory, and to localization principles of Simonenko and Douglas; - algebraic schemes originating from the fundamental concept of symbol of a singular integral operator (Mihlin), from the semi-mul-

252

tiplicative dependence of Teeplitz operator cn its symbol, (WienerHopf), and culminating in the operator

N-theory;

- methods and techniques of extension theory (Krein), which have attracted a new interest to metric properties of Hankel operators and to their numerous connections; -

other important principles and ideas which we have either

forgotten or overlooked cr had no possibility to mention here. The inverse influence of Hankel and Toeplitz operators is also considerable. For example, many problems of this chapter fit very well into the context of other chapters: Banach Algebras (Problem 5.6), best approximation (Problem 5.1), singular integrals (Problem 5.14). Problems 2.11, 3.1, 6.6, 10.2 can hardly be

severed from

spectral aspects of Toeplitz operators, and Problems 3.2, 3.3, 4.15~ 4.21, 8.13, S.6, from Hankel operators. ~ n y

problems related to the

Sz.-Nagy-Foia~ model (4.9-4.14)can be translated into the language of Hankel-Toeplitz (~)

(possibly, vectorial) operators, because functions

of the model operator

Hankel operators ~G~~

T@

coincide essentially with the

, and the proximity of model subspaces ~ _

and K@~ can be expressed in terms of the Toeplitz operator TG* etc. Hankel-Toeplitz problems assembled in this book do not exhaust even the most topical problems of this direction *), but contain many interesting questions and suggest some general considerations. Many of the problems are inspired by some other fields and are rooted there so deeply that it is difficult to separate them from the corresponding context. We had to place some Hankel-Toeplitz problems (not without hesitation and disputes) into other chapters. Examples can be found in Chapter 3 (3.1, 3.2, 3.3). Moreover, we believe that

*) To our surprise nobody has asked, for instance, whether every Toeplitz operator has a non-trivial invariant subspace...

253

Problem 3.3 is one of the most characteristic and essential problems of exactly t h i s

Chapter and we hope that the reader looking

through this chapter will turn to Problem 3.3 as well. Problems 5.1-5.3 deal with metric characteristics operators (compactness,

spectra, s-numbers).

of Hankel

In connection with

Problems 5.3 and 5.7 concerning operators acting

n o t

in H~we

should like to mention recent investigations of S.Janson, J.Peetre and S.Semmes and of V.A.Tolokonnikov (X-~Y)

(Spring 1983) who have found

-continuity criteria (in terms of symbols) for Hankel and

Toeplitz operators in many non-Hilbert function spaces

X, ~

.

Problems 5.4 and 5.5 treat similarity invariants and some properties of the calculus for Toeplitz operators. Problems 5.6, 5.13 are related with localization methods, problems 5.8-5.10,

5.15 deal with vectorial and multidimensional

of Toeplitz operators and with related function-theoretic

variants

boundary

problems. Problems 5.11-5.14 treat "limit distributions of spectra" (asymptotics of Szego determinants,

convergence and other properties

of projection invertibility methods etc). The theme of Problem 5.16 may be viewed as a non-commutative analogue of Toeplitz operators arising in the theory of completely integrable systems.

The field of action and the multitude of connections of HankelToeplitz operators are so impressive that it became fashionable nowadays to find them everywhere - from bases theory to models of Quantum Physics and ... even where they really do

not c c c u r - ~

254

5oio old

APmOX~aTIO~ OF B0~DED F~CTIO~S BY E ~ T S OF M ~ + C

Every sequence { ~ } ~ 4 of complex numbers defines a Hankel matrix ~ = { ~+k-~ ~ j,K~4 which is considered as an operator in the Hilbert space ~ . By Nehari's theorem ~ is bounded if and only if there exists a function ~ in the algebra ~c~ ^of all bounded and measurable functions on T such that [~=~ (-~) , ~$= 4, 2,..., ~(~) being the Fourier coefficient [ ~ ' ~ n ~ t ~ of . This function is uniquely determined up to a summand from the Hardy algebra H ~ . The norm ] ~ of ~ coincides with diet ( ~ ~ ) . Given ~ ~ C let ~(~) be the Hankel operator corresponding to the sequence {[~>~4 ~ [ = ~ ( - ~ ) ~ ~ = 4,~,,,. Usual compactness arguments imply that for every exists ~ E H ~ such that

~ ~-~

there

A criterion of u n i q u e n e s s of the best approximation ~ as well as a description of all such ~ s in the non-uniqueness case have been obtained in [2 2 . Hartman has shown [3] that F is compact iff there exists a function ~ in the algebra of all continuous functions on T such that F = P ( ~ ) Moreover, if r is compact then for every 6 > 0 there exists ~ C satisfying r(~6)= r and ~ C %Ir~ + 6 • The results of Nehari and Hartman easily imply the following result discovered independently by Sarason [4] : the algebraic sum H~+C is a closed subalsebra o Z L ~ C s e e ~ s o [~] .here ~ properties of this subalgebra are disussed), Hartmau's result implies also the following characterization of ~ + C : an element ~ in ~ belongs to ~+C if and o n l y i f C(~ ) is a compact o p e r a t o r . Let S~(F), ~=4,~...denote S-numbers of Vcounted with multiplicities and let S ~ ( F ) be the least upper bound of the essential spectrum of ( (cf°[6], §7). Clearly, S ~ ( F ) ~ S ~ ( r ) as ~---~ + ~

r*r)~

co

THEOREM. Let

~L

and

~ = ~(~)

. Then

civet (~, H +C) =S~(r).

(1)

255 Given a function possibilities.

in

CASE I. S ~ ( F ) = I F I

~\(H~+C) , i.e. F

consider

the following

does not have

~-numbers

greater than ~eo ( ~ ) " CASE 2. There exists only a finite number ~ , cf S-numbers greater than Soo ~r) • CASE 3. The set of S-numbers to the right cf Soo(~s infinlte~ Formula (I) is a simple consequence of theorem 3.1 from [2a] but for the purpose of this note it is more convenient to connect it with the investigations of [2c]. co For any positive integer k let H K denote the set of all sums ~ = ~ + ~ , where ~ H c° and ~ is a rational function of degree % k , having all its poles in the open unit disc ~ and vanishing at infinity. The set H ~ is neither convex nor linear. ~lls s e t coincides with the family of symbols ~ corresponding to the Hankel operators ,F'c,~(~) of finite rank ~< k . Clearis dense in ly Ho f,.~H ) OH1 c . . . ~ H +C and ~U~I Fill,

H~+C

. Therefore dist (~,H~C)

On the other hand it has been shown in

Co

~

--

[2C]

~st CJ~,H,~). that diet

(;~ ,H k) = $k (.r)

provided SI<(~) >~K÷I(V ) . Hence (I) holds in case 3- In case 2 we , which implies (1) have (see [2c], ~5) S ~ ( ~ ) = ~ S t (~ , H ~ ) also. At last, in case 1 we obviously h a V ~e l~t(j~,

H )=~st(~, H%C).

aive~ j~eL

denote by M~

the set of ~ll

geH +C

satisfying II~ . ~ h ~ ----Soo(P(~)) " In case I M ~ n H ~ ~ ~ and in case 2 M ~ { ] H m ~ ~ . A necessary and sufficient condition for M~ {l H~ to be a one-point set as well as a description of when it contains more than one element, have been obtained. As for case 3 it is VNENOWN: a) Ca._nnM ~

be emptE for some

cribe such ~ b) Can

and if so how to des-

? M~

a@

~ E Lce

consist of a single element for some ~

i_nn

co

L X(.H +C)? c) I_~f m ~ ~ ~ "selected" part of .

.

.

.

"selected" part)?

then is it possible to describe at #east a

M~ .

just as in case .

,2, ,,(w'ith

M~

NH

as a

256

Clearly ~ ~& ~ for # ~ [°~=\(H~+C) if and only if there exists ~ C such that for #4 # - ~ case ] holds, i.e. Ir(~)l = S~(~(~4)) ° Question b) remains interesting for cases I and 2 also. The matter is that there are situations when card M ~ = o o but card ( M ~ ~ H ~ ) = ~ (in case I) and card ( M ~ ~ H ~ (in case 2). Indeed, let for example ~ be an inner function with singularities on an arc ~ c T , ~ ( A ) < 4 , and let g be any function in C satisfying ~ ( r ) - - 4 on A and l ~ ( r ) l < 4 for ~ T\ A . Then ~6----geC and setting # ----~/6 we have

S~(r(#))--~i~t(#,HtC)=~Li~t(#,H5 =]= fig- 1#~. Ho.ever, I1#-~11>~ Almost

for every & ~ H ~ , t 1 ~ > 0

everything

s a i d a b o v e c a n be g e n e r a l i z e d

matr~x-valuod fu~otions F--(#~) ,

or C

.~t~ e~tries

• In this case the norm

should be def~ne~ as

.

~ ~ T~ o

Hilbert-Schmidt norm of tions we refer to [2d].

A

I F(~)I

~

IIF II

where

of

IAI

to the

case o f

belongi~ to F e

L~,# ~

stands for ~ e

In connection with these generaliza-

REFERENCES I. N e h a r i A. On bounded bilinear forms. - A n n . ~ t h . , 1957, (2), 65, 153-162. 2. A ~ a M ~ B.M., A p o B ~.B., Ep e ~H M.F. a) BecEoHe~Hae rsazeaeBH M a T p m m ~ OdO6~eHH~e s s a a ~ KapaTeo~op~-~e~epa $.PHcca. - ~F~m/.aaaa. ~ ero np~a., I968, 2, I, I-I9; b) BecEoH e , H e rss~eaeBH M a T p m m ~ odo6meHHHe s a ~ a ~ EapaTeo~op~-@e~epa n M.~ypa. - ~/~U.asa~. ~ ero npHa., 1968, 2, 4, 1-17; ~) A~aa~T~~ecE~e CBO~CTBa Hap ~ T a raHEeaeBa onepaTopa n o d o 6 m e ~ a ~ sa~a~a ~ypa-Ta~ar~. - MaTeM.cd., I97I, 85 (I28), ~ I9, 38-78; d) BecEoHe~mae 6ao~Ho-rssEeaeB~ M a T p H ~ ~ c ~ s a a m ~ e c ~ n p o 6 ~ e ~ ~po~oaxem¢~. - MsB.AH ApMCCP, I97I, YI, ~ 2-3, 87-II2. S.H a r t m a n P. On completely continuous Hankel matrices. Proc.Amer.Math.Soc., 1958, 9, 862-866. co 4. S a r a s o n D. Generalized interpolation in H . - Trans. Amer.Math. Soc., 1967, 127, 179-203. 5. S a r a s o n D. Algebras of functions on the unit circle. Bull.Amer.~athoSoc., 1973, 79, N 2, 286-299.

257

.

r o x d e p r Ho~., Kp e ~ H M.r. BBe~eHHe B T e o p ~ n o ~ e ~ HNX HecaMoconlo~eH~x oHepaTopoB. -- M., Hs~Ea, 1965. V. M. ADAB~AN CCCP, 270000, 0~ecca, 0~eccEE~ roc.yHHBepcHTeT; D. Z.AROV

( ~. 8.APOB)

CCCP, 270020, 0~ecca, 0 ~ e c c E ~ ne~.EHCTETyT;

M. G. KREIN

( M.r.KP~

270057, 0~ecca, y~.ApTeMa 14, EB.6

COMmeNTARY BY THE AUTHORS Soon after the Collection "99 unsolved problems in linear and complex analysis" LO~&I, vol.81 (1978) was published S.Axler, I.D.Berg, N.Jewell and A.Schields wrote an important paper on the theory of approximation of continuous operators in Banach space by compact operators, where they obtained, in particular, the answers to questions a) and b) of the Problem.The answers to both questions turned out to be negative. So for every function ~ L~ the set M~ is not void and moreover for any ~ co ~ L \(H +C) the set M~ is infinite. These results were obtained in Eli as consequences of two remarquable propositions, which we formulate for Hilbert space operators only. THEOREM. Let

[~)

be a seque.nqe of linear compact operators

on Hilbert space converging in the strong tooolo~y to a bounded operator

T : S-~

~=T

exisSs a sequence

also.that

" Supp°sIe,

J~ ~

T = s - ~

of non-negative numbe.rs such that

and

I T - K I = S=(T)

where

• Then there

K

=

T %

.

~=~

258

COROLLARY. Let T

{T~}

and

theorem. Suppose also that ~ sequenc.,e,s, [6~}. =

=

{ 8~}

satisf E the conditions of the

is not compact. Then there exist two

oT,,,.,.non-ne~ative~numbers such that, ~- O,n=

4

I T-Ko, I=IT-K I= s LT) where ~ @ = ~ J ~

,

~=[6~T~

,and

•

The indicated propositions permit also to give answers to questions of type a ) a n d b ) f o r matrix-functions AS we got to know from [~ question a) was raised before us by D.Sarason.

~: ~x~(C~×~+H~l

Question c) concerning the description of the set ~ its "selected" part remains open.

or

REFERENCE 7. A x i e r S., B e r g I.D., J e w e I I N., S h i e I d s A. Approximation by compact operators and the space H°°+ C . A n n . ~ t h . , 1979, 109, 601-612.

EDITORS' NOTE Question a) is solved also in a different way by D.Lueckim~ [8]. Let us mention also a recent r e s e t of O.S~udberg [9] asserting that the algebra H ~ + BUC (BUC is the space of bounded ~niformly continuous functions on ~ ) does not have the best approximation property, i.e, there e x i s t s ~ such that there is no ~ in HC~+ BUC saris-

REFERENCES 8. L u e c k i n g D. The compact Hankel operator form an M -ideal in the space of Hankel operators° - Proc.Amer.Math.Soc., 1980, 79, 222-224. 9. S u n d b e r g C. No°+ BUC does not have the best approximation property. Preprint, Inst.Nittag-Leffler, 13, 1 9 8 3

259

5.2.

QUASINILPOTENT HANKEL OPERATORS

Hankel operators possess little algebraic structure. This fact handicaps attempts to elucidate their spectral theory. The following sample problem is of untested depth and has some interesting function theoretic end operator theoretic connections. PROBLEM. Does there exist a non-zero quasinilpotent Hanke 1 operator? A Hankel operator A on H ~ is one whose representing matrix )~ is of the form (~{+I ~'i'-0 with respect to the standard orthonormal basis. A well known theorem of Nehari shows that we may represent A as ~ = $ ~ = ~ M q l H ~ where P is the orthogonal projection of ~£ onto H Z ; ~ is the unitary operator defined by (~)(Z)=~(~), for ~ in ~£ , and ~ denotes multiplication by a function ~ in The symbol function ~ and the defining sequence ~ are conA nected by ~(~) = ~ , ~= 0 , ~ , ~ , . The following observation appears to be new and provides a little evidence against existence. PROPOSITION. There does not exist a non 'zero nilpotent Hankel operator, PROOF. Suppose A~0 and is nilpo±ent. Then ker A is a non zero invariant subspace for the unilateral shift U, since AU--U*~. By Beurling's theorem this subspace is of the form ~ for some non constant inner function ~ . Thus, with the representation above, we have =0 and hence ~ ( ~ ) = 0 for ~ . So the symbol function ~ may be written in the factored form ~ = ~ for some ~ in H~ , and we may assume (by cancellation) that and ~ possess no common inner divisors. The operator ~ is a partial isometry with support space ~= ~ e ~ ~ and final space ~ = ~ ~ ~ al , where ~ ) = ~(~) . By the hypothesised nilpotence of ~ -~ = $ ~ it follows that for some non zero func. .~ 0 ~4 belo ng s to W~ . Hence ~ divides . . tion . ~ . in ~ , and ~=W~4 with ~ in H . Since ~ belongs to ~T we have P(~) =0 . This says that the Toeplitz operator T £ , ~ has non t r i v i a l

kernel.

,

But

k2/£ T *

-- k ~ T

= k~T

=

and we have a contradiction of Coburn's alternative:

Either the kernel or the co-kernel of a non zero Toeplitz operator is trivial. @

260

Function Theory. The evidence for existence is perhaps stronger. There are many compact non self-adjoint Hankel operators, so perhaps a non zero one can be found which has no non zero eigenvalues. A little manipulation reveals that ~ is an eigenvalue for ~ if and only if there is a non zero function ~ in ~ (the eigenvectot) and a function ~ in ~ £ such that +

Since continuous functions induce compact Hankel operators it would be sufficient then to find a continuous function ~ which fails to be representable in this way for every ~ 0 . Whilst the singular numbers of a Hankel operator A (the eigenvalues of ( ~ * A ) ~l~ ) have been successfully characterized (see for example [3 , Chapter 5] ), less seems to be known about eigenvalues. O~erator theory. It is natural to examine (I) when the symbol can be factored as q = ~ (cf. the proof above) with ~ an interpolatingBlaschke product. The corresponding Hankel operators and function theory are tractable in certain senses (see[l] ,[2~Part 2] and[3,Chapter 4] ), partly because the functions (~-~A~l~)41~(~-~)-~, where ~ , A i , are the zeros of Wv , form a Riesz basis for ~ O ~ . It turns out that ~ is compact if ~(~4), ~CA~), . is a null sequence. A quasinilpotent compact Hankel operator of this kind will exist if and only if the following problem for operators on ~ can be solved. I

PROBLEM. Construct an interpolating sequence dia~o~l o~erator ~ proper solutions operator on ~ ting matrix

~

so that the ' equation ~ X x i_~n ~

whe n

~0

~ ~

. Here

associated with ~ A ~

A~ and a compact admits no

is the bounded(!)

determined by represen-

(t. i A,i,l~.)'~1~'(.,I- I~ ~)'~/~'

REFERENCES I. C 1 a r k D.N. On interpolating sequences and the theory of Hankel and Toeplitz matrices. - J.Functicnal Anal. 1970,5,247-258.

261

2. H r u ~ 6 ~ v B.S.

S.V.,

N i ~ o l's k i i

N.K.,

P a v I o v

Unconditional bases of exponentials and of reproducing ker-

nels. - Lect.Notes Math. 3. P o w e r

S.C.

Springer Verlag.

Hankel operators on Hilbert space. - Research

Notes in ~athematics. S.C.POWER

1981,N864,

1982. N 64, Pitman, London. Dept. of Mathematics Michigan State University E.Lansing, MI 48824 USA Usual Address: Dept.of Mathematics University of Lancaster Bailrigg, Lancaster LAi 4YW England

262 HANKEL OPERATORS ON BERGNAN SPACES

5.3. Let

IA

~/~

denote the usual area measure on the open unit disk

17 • ~he B e r ~

~,~ce t . t (1?1

c o n s i s t i n g o f those f u n c t i o n s i n

D

, .Let

i s the s u b s ~ o e ~ ~D. ~ A)

of t}(l?,~A)

which are analytic on

P

denote the orthogonal projection of ~ ( D , ~ A ) onto ~ & ( D ) . For ~ S ~ cD, ~A) , we define the Toeplitz operator

and the Hankel operator

H# :L~ (D) -~ L~(]I), & A)e L~(]D) by

T~I~-P(#I~)

and M#l~=(I-P)(#I~),

For which functions

#E L® (D, &A)

is the HaZel oper.tor

compact? If we were dealing with Hankel operators on the circle T rather than the disk ~ , the answer would be that the symbol must be in the space • on the dlsk " 17 , it is " H ~ tC(T) ea sy to s ee that i f ~ E ~ °O + C ( ~ ) , then H~ i s compact. However, it is not hard

to const=ct

an

open set 5 ~

D

with

S nT ~ ~

such that

if

is the characteristic function of S , then H, is compact. Thus the subset of ~ ( D ~ ~ A) which gives compact ~_~nkelt operators is much ~o bigger ( i n a n o n t r i v i a l way) than H +C (~) , and it is possible that there is no nice answer to the question as asked above. A mere natural question arises by considering only symbols which are complex conjugates of analytic functions: For which

~6

i_ss H ~

compact?

I% is believable that this question has a nice answer. A good candidate is that T must be in The importance of this question stems from the identity

H®*C(D).

~lid

H®

for an ~ ~ . ~hus we are asking which Toeplitz operators on the disk with analytic symbol have compact self-commutator. Readers familiar with a paper of Coifman, Rochberg, and Weiss

263

[I] might think that paper answers the question above. Theorem VIII of [I] seems to determine precisely which conjugate analytic functions give rise to compact Hankel operators. However, the Hankel operators used in ~I] are (unitarily e~uivalent to) multiplication followi s f a r bigger than E , the Hankel operators of [1] are not the s ~ e as the Hankel operators defined here, The Hankel operators as defined i n [ I ] are more natural when dealing wlth singular i n t e g r a l theory, but the close o o ~ e o t i o n l r i t h Toeplitz operators is lost, To determine which analytic Toeplitz operators are essentially normal, the Hankel operators as defined here are the natural objects to study. RE~ERENOE I. C o i f m a n R.R., R o c h b e r g R., W e i s s G. Pactorization theorems for Hardy spaces in several variables. - Annals of Mathematics SHELDON AXLER

1976, 103, 611-635. Michigan State University East Lansing, NI 48824, USA

264 5.4. old

A SIMILARITY PROBLEM FOR TOEPLITZ OPERATORS Co, sider the Toeplitz operator T~

acting on

H%~ H~(~)

,

where ~ is a rational function, with ~(~) contained in a simple closed curve ~ . Let ~ be the ccnformal map from ~ to the interior of ~ , and say that ~ b a c k s u p at 6 ~0 if arg ~-1 ~(¢t~) is decreasing in some closed interval [@I, ~ , where 0,< 01, a n d 0i ~
CONJECTURE. Suppose ~ ( ~ )

has windin~ numbe~ ~ ~ 0

. Then W~

is sizi!a r to

i~ 9>0

, and t o

M4 ~ - - . i_Xf "~---0 , where,

M k

• M~

is t h e o , R e r a t o r o f

(2)

multip!ica,tion b~ ~(6¢$)

e

One c a s e o f t h e a b o v e c o n j e c t u r e g o e s b a c k t o Duren it was proved for ~ ( ~ ) ~ + ~ , I¢I>I;I . In this case never backs up, so that MI,..., M ~ are not present tually, Duren did not obtain s~milarity, but proved that fies

LTr----where ~ =~4~(~)

~, where ~ and in (I). Ac~V satis-

D,

is some conjugate-linear operator( ~(~,X+ A¢~) ----+ ~(~)) and ~ is the mapping function for the

inte-

rior Of C~03~(T) . I n [2~, the conjecture was proved in case is $ - to-one in some annulus ~ i~l ~ ~ . Here again ~ never backs up. In [3~, ~ was assumed to have the form

where ~ and ~ are finite zero. In this case ~(~)-----~

Blaschke products, ~ h a v i n g o n l y one and ~ c a n be taken to be 1.

265

The main tool used in [3] was the Sz.-Nagy-Fola~ characteristic function of ~p , which we computed explicitly and which, as we showed, has a left inverse. A theorem of Sz.-Hagy-Foia~, [4], Theorem 1.4, was then used to infer similarity of TF with an isometry. ~oreover, the ,,-!tary part in the Wold Decomposition of the isometry could be seen to have multiplicity I, and so the proofs of the representations (I) and (2) were reduced to spectral theory. If ~ is of the form (3) where ~ and ~ are finite Blaschke products, ~ having m o r e t h a n o n e zero, the computation of the characteristic function of T~ is no longer easy. However, left invertibility can sometimes be proved without explicit computation. This is the case if ~ and ~ have the same number of zeros; i.e., when T~ is similar to a unitary operator. This and some other results related to the conjecture are given in [5]. The Sz.-Xagy-Fola~ theory may also be helpful in attempts to formulate and prove a version of the conjecture when (3) holds with ~ and arbitrary inner functions. For example, it follows from ~3~ that if is inner and ~ is a Blaschke factor, then ~F is similar to an isumetry. For the case in which ~ is net the unit circle, the only successful %ech-lques so far are those of [2], which do not use model theory. We have not been successful in extending them beyond the case of ~ satisfying the "annulus hypothesis" described above. There is a model theory which applies to domains other than ~ [6], but to our knowledge no results on similarity are a part of this theory. ~ore seriously, to apply the theory, one would NEED TO KNOW that the spectrum of r~W is a spectral set for Tp ; a result which does not seem to be known for rational ~ at this time. Pinally, I% seems hardly necessary to give reasons why the conJecture would be a desirable one to prove. Certainly detailed information on invariant subspaces, commutant, cyclic vectors and functional calculus would follow from this type of result. REFERENCES 1. D u r e n P.L. Extension of a result of Beurling on invariant subspaces. - Trans.Amer.Nath.Soc. 1961, 99, 320-324. 2. C 1 a r k D.N., M o r r e 1 J.H. On Toeplitz operators and similarity. - Amer.J.Math., 1978, 100, N 5, 973-986. 3. C 1 a r k D.N., Sz.-Nagy-Foia~ theory and similarity for a class of Toeplitz operators. - Banach Center Publiaations,v 8,

266

Spectral Theory, 1982, 221-229 4. 5. 6.

S z. - I~ a g y B., F o i a 9 C. On the structure of intertwining operators. - Acta Sci.Math. 1973, 35, 225-254. C 1 a r k D.N. Similarity properties of rational Toeplitz operators. In preparation. S a r a s o n D. On spectral sets having connected complement. - A c t a Sci.Math. 1965, 26, 289-299. DOUGLAS N. CLARK

The University of Georgia, Athens, Georgia 30601 USA

COmmENTARY BY THE AUTHOR Since my first note on the similarity problem, the following results have been obtained.

T~O~ a

1 ([8]),

I__~F

simple closed curve

FCT )

; i_~f V > 0

P

i s a r a t i o n a l function. ~ p ~ T

, which is anal,ytic in a neighborhood o,f

, where

V

is the windin~ number of

about the points interior to F the set (o~ T

) where

T (V)•

, ....... where

V

F

; and if

D

_,~ T (y)

F(~)

•

V

~$

THEOREM 2. ([9], [10] ). If F~T)

F(T)

, where

~-

is similar to copies of the ana-

is a normal operator

is absolutely continuous and the

where F

in th e spectrum of V F

is ,equal

backs up and F(e~)=~ .

is a rational function~ if

divides the plane `into disjoint re6ions~ from which the ones

in which the index of

T F -~I

labeled

); if the closures of any two of these

(~ , ~)

i_~s

mapping function

and where V

spectral multiplicity of a 90int ~ to the number of points

TF

is the sum of ~

to the interior of F

whose spectrum is

F(>-)@F

backs up; then

lytic Toeplitz operator 'as s0ciated with the from

~:o

~

(rasp. ~

is negative (resp,positive) are

inters,act St onl~ finitely many points (called the mul-

tiple points of

F

); if the boundary of each

~

,~

is an ana-

267

l~tic curve except at th e multiple points= where it is piecewise smooth with inner angle under

F

of a point

~0

; if no multiple point is the image I

~o~:T

where

never backs up at a multiple point; then

~(~o) = ~

0

; and if

is similar to

h

where ~

~

(resp. T

) is the maopin~ function of ~ for

restriction t° T

and

~

(resp. ~

),

is as described in Theorem I. THEOREM 3. (Wang, ~I~ ). I_~f F ~ C 4 ~ T )

some

(resp.

), each summand is included with multi p!icit E equal to the

absolute value of the index of T F - ~ I and V

o_~n ~

~<~

of a function

, i_~f F ( e ~$)

F

and if

analytic in

i_~s 1-te-1

, ~t(e~$)

$ ~ F ~ e '~$) is orientation preserving, then ~F , where ~

is the mappin~ function from 0

~ Ce ~)

is the


fo__~r

never vanishes is similar to to the interior

F(T) • Theorem I, of course, PROVES THE CONJECTURE posed in my first note on the similarity problem, EXCEPT WHEN the curve F C T ) has singularities. The case of a singularity with nonzero inner angle can be settled using the methods of ~0], but the case of zero inner angle remains open, owing to our lack of understanding of the behavior of and ~i near such a point. Theorem 2 also excludes zero interior angles in the ~ a n d ~ . The hypothesis that F k ~ o ) ~ 0 at the inverse images of the multiple points can be somewhat weakened ~I0] but not removed. In fact, ~0] contains an example of a TF satisfying Theorem 2 (with such a weakening) and a rational, orientation preserving, homeomorphism of T such that t F and TFo ~ are not similar. The case of intersecting loops of F~T) was discussed in ~ . As indicated there, nothing is known unless F ~ ) is the image o f ~ under a function analytic is ~ . Unfortunately, even the proof of Theorem 5 of ~ ~ is incomplete without f u r ~ e r hypotheses, as pointed out by Stephenson ~I~. Theorem 3 is the first attempt at a systematic similarity theory

268

for non-rational

F

, and one hopes that it may be generalized to

the point of a non-rational version of Theorems I and 2. Other examples of similarity for non-rational Equivalence Theorem

~

TF

may be obtained using Cowen's

, which implies that

TFo ~

is unitarily

equivalent to a direct sum of countably many copies of T F

, when

is a non-rational inner function. An obstruction to similarity between T F

and a "reasonably

nice" operator can occur when T F has a boundary eigenvalue, see Clancey [7, ~ 4 . ~ , In fact, as Clancey has pointed out to me, the spectrum of T F

is not a

k -spectral set, for the example given

in ~ [7]REFERENCES 7. C I a n c e y

K~F.

Toeplitz models for operators with one di-

mensional self-commutators (to appear). 8. C 1 a r k

D.N.

On a similarity theory for rational Toeplitz

o p e r a t o r s . - J.Reine Angew.Math. 9. C 1 a r k

D.N.

1980, 320, 6-31.

On Toeplitz operators with loops. - J.Operator

Theory, 1980, 4, 37-54. 10. C 1 a r k D.N. On Toeplitz operators with loops, II. - J.Operator Theory 1982, 7, 109-123. 11. C 1 a r k D.N. On the structure of rational Toeplitz operators. - In:Contributions to Analysis and Geometry, supplement to Amer. J.Math. 1981, 63-72. 12. C o w e n C.C. On equivalence of Toeplitz operators. - J.Operator Theory 1982, 7, 167-172. 13~ S t e p h • n s o n K. Analytic functions of finite valence, with applications to Toeplitz operators (to appear). 14. W a n g D. Similarity and boundary eigenvalues for a class of smooth Toeplitz operators (to appear).

269

5.5.

ITERATES OP TOEPLITZ OPERATORS WITH UNIMODULAR SYMBOLS

Each invertible Toeplitz operator ~ on H ~ can be represented as T~ ~ T ~ T@ where ~ is an outer function with modulus I~I and ~ = ~ / ~ is a unimodular function. The operator ~ , being invertible analytic Toeplitz operator, had simple spectral behaviour. Therefore the Toeplitz operators with unimodular symbols play an especial role (see [I] ). I would like to propose the following questions concerning these operators. Suppose X is one of the function classes H°°~O~ Q C ~¢---~

= I-I°'% C, n FI°°÷ C, QUESTION I , L e t Ko~T~

~0]

C,, C k, C,~ . ~

be a unimodular function in X

. Is it true that there exists ~

II

i~

such that ~

with

0z

~4>0 If the answer is positive, it is reasonable to ask whether the following stronger conclusion can be done. QUESTION 2. Is it true that under the h~potheses of Question I

IIT s II 0 for every non_zero ~ i_~n~

?

It follows from Clark's results ~2~ that the answer to Question 2 is positive for rational functions ~ (see also Commentary to 5-4). In view mf T.Wolff's factorization theorem ~3~ (asserting that each unimodular function t~ in H°°+ G can be represented as ~ 0 with ~r~C and 0 inner) it seems plausAble that if Question I (or 2) has a positive answer for X= ~ c then so is for X ~ ° ° C . Note that for general unimodular functions ~ with ~c~T~ ~ I@} it may happen that ~ 0 1 1 T ~ I I = 0 for any ~ in ~ . For example, if ~ is a measurable subset of T , 0 < ~4ea~ E < ~ and ~ = ~ on ~ and-~ on~ then it follows from M.Rosenblum's results ~4~ that T ~ is a selfadjoint operator with absolutely continuous spectrum on E-~, ¢~. An affirmative answer to Question I for X~-~ C would imply the existence of a non-trivial invariant subspaces for Toeplitz operators with unimodular symbols in ~ C (see some results on invariant subspaces of Toeplitz operators in [5~ ). Indeed, either one of the kernels ~£~T~, ~ T ~ is non-trivial or T~ and T~ satisfy the hypothesis at Question I and so both subspaces

270

are invariant under

T~ and ~

would be a

0~ contraction

. Therefore either one

==I=H~ ~ { ~ )

of these sub@paces is non-trivial

or ~ = {

~},

~ = H~

i~e, T~

(see [6]). But each elf contraction has a

non-trivial invariant subspace

[6].

REFERENCES

I. S a r a s o n

D. Function theory on the unit circle. - Notes for

Lect.at a conference at Virginia Polytechnic Inst. a~d State Univ., 1978. 2. C 1 a r k

D.N. On a similarity theory for rational Toeplitz ope-

rators. - J.Reine Angew.Math., 3. W o 1 f f

1980, 320, 6-31.

T. Two algebras of bounded functions~

- Duke Math J.,

1982, 49, N 2, 321-328. 4. R o s e n b 1 u m M. The absolute continuity of Toeplitz's matrices. - Pacif.J.Math., 1960, 10, N 3, 987-996. 5o P e 1 1 e r V.V. Invariant subspaces for Toeplitz operators. LOMI Preprints, E-7-82, Leningrad, 6. S z . - N a g y B., P o i a ~

1982. C. Harmonic analysis of operators

on Hilbert space, North Holland, Amsterdam,

V. V. PELLER (B.B.~P)

1970.

COOP, 191011, ~ e H s ~ m ~ , #OH~aHKS 27 , ~ 0 M H

271

5.6. old

LOCALIZATION OF TOEPLITZ OPERATORS

Let H ~ and H °@ denote the Hardy subspaces of ~,~(~) and ~°°(T) respectively, consisting of the functions with zero negative Fourier coefficients and let ~ be the orthogonal projection from h~(~) onto H~ . For ~ in ~,~(T) the Toeplitz operator with symbol is defined on H% b y T 9 ~ = P ( ~ ) . ~[uch of the interest in Toeplitz operators has been directed toward their spectral characteristics either singly or in terms of the algebras of operators which they generate. In particular, one seeks conceptual determinations of why an operator is or is not invertible and more generally Fredholm. One fact which one seeks to explain is the result due to Widom [I] that the spectrum 6~(T~) of an arbitrary Toeplitz operator is a connected subset of C and even [2] the essential spectrum 6~e(Tq) is connected. The latter result implies the former in view of Coburn's Lemma. An important tool introduced in [2], [3] is the algebraic notion of localization. Let ~ denote the closed algebra generated by all Toeplitz operators and ~ C be the subalgebra

(H°% C) n (H°°+ C,) of ~ , where C denotes the algebra of continuous functions o u t . Each ~ in the maximal ideal space ~ Q C of ~ C determines a closed subset ~E of M ~ and one can show that the closed i d e a l % in S generated by

is proper and that the local Toepli%~ o p e r a t o r % + % depends only on ~I X~

° Moreover, since ~. ~.

of compact o p e r a t o r s on ~

, properties ~lch

in~=~/%

equals the ideal~ are t r u e modulo

can be established "locally". For example,T~ is ly if--'J~ ~ is invertible for each ~ in M Q C tion results5 are established [4] by identifying Q ~/~ . One unanswered problem concerning local is:

FTedholm if and on. These localizaC as the center of Toeplitz operators

CONJECTURE I. The spectrum of a local Toeplitz operator is connected. In ~] it was shown that many of the results known for Toeplitz operators have analogues valid for local Toeplitz operators. Unfortu-

272

nately a proof of the connecteduess would seem to require more refined knowledge of the behavior of ~ functions on ~ H @@ than available and the result would imply the connectedness of ~,(T?) • A mere refined localization h~s been obtained by Axler replacing X~ by the subsets of ~ L ~ of maximal antisymmetry for H@@+ using the fact that the local algebras S~ have nontrivial centers and iterating this transfinitely. There is evidence to believe that the ultimate localization should be to the closed support X~ i n ~ L ~ for the representing measure ~ for a point ~ in ~H ~ . In particular, one would like to show that if H~(#V) denotes the closure in ~ ( ~ $ ) of the functions ~I X~ for ~ in M ~ , P~ the orthogonal projection from ~ ( ~ ) onto H~(~N) , then the map

extends to the corresponding algebras, where the local Toeplitz operator is defined by

Zor ~

in

. ~f

N

i s a p o i n t i n M L~

, then H ~ ( ~ ) =

C

and it is a special case of the result ~2] that ~ modulo its commutator ideal is isometrically isomorphic to ~ , that the map extends to a character in this case. A generalized spectral inclusion theorem also provides evidence for the existence of this mapping in all cases. One approach to establishing the existence of this map is to try to exhibit the state on ~ which this "representation" would determine. One property that such a state would have is that it would be multiplicative on the Toeplitz operators with symbols in H ~ . Call such states a n a 1 y t i c a 1 1 y m u 1 t i p 1 i c a t i v e . Two problems connected with such states seem interesting. CONJECTURE 2 (Generalized Gleason-Whitney). I_~f~ analytic all.v multiolicative states on S

and 6~

which a~ree on ~@@

such that the kernels of th e t.wo representations defined b Z ~

ar__~e

and an d

are equa!, then the representations are equi~lent. CONJECTURE 3 (Generalized Corona). In the collection Q f a n a ! y ticallymultiplicative states the ones whichcorrespond to pqints °f

273

One consequence of a localization to XV

when ~

is an analytic

disk ~ ,would be the following. It is possible for ~ in L~ that its harmonic extension ~ I ~ agrees with the harmonic extension of a function continuous on ~ . (Note that this is not the same as ^ saying that ~ is continuous on the boundary of ~ as a subset of H~ which is of course always the case.) In that case the invertibility of the local Toeplitz operator would depend on a "winding number" which should yield a subtle necessary condition for T ~ to be Predholm. Ultimately it may be that there are enough analytic disks A in ~ H ~ on which the harmonic extension ~ is "nice" to determine whether or not ~ is Fredholm but that would require knowing a lot more about ~ H ~ than we do now. REFERENCE S 1. W i d o m

H.

On the spectrum of a Toeplitz operator. - Pacif.

J.Math.~ 1964, 14, 365-375. 2. D o u g 1 a s R.G. Banach algebra techniques in operator theory. New York, Academic Press, 1972. 3. D o u g 1 a s R.G. Banach algebra techniques in the theory of Toeplitz operators. CBMS Regional Confer. no.15, Amer.Math.Soc., Providence, R.I., 1973. 4. D o u g 1 a s R.G. Local Toeplitz operators. - Proc.London Math.Soc., R.G.DOUGLAS

1978, 3, 36. State University of New York Department of Math. Stony Brook, N.Y. 11794, USA

274

5.7.

TOEPLITZ OPERATORS ON THE BERGMAN SPACE

Let ~ (0)

~ denote the Bergman space of analytic functions in , and let P be the orthogonal projection of LI ( 0 )

.9

OntO

~" :

Por~.~ . U--(D)

we define the Toeplitz operator with

- p @

rators may be quite different from that of the Toeplitz operators on the Hardy .space H • However i% is shown in [I] that Toeplitz opera%ors on ~ with h a r m o n i c symbols behave quite similarly to •"" n os e on U% I1 , ana. one can prove analogues for this class of many results about Toeplitz operators on H ~ . An important result about Toeplitz operators on H t is Widom's Theorem, which states that the spectrum of such an operator is connected ([2]). This suggests our problem. @

CONJECTURE. A Toeplitz operator on

A~

with harmonic sxmbol 'has

a 9 o ~ e c t e d spectrum. In support of this conjecture we mention the following cases for harmonic ~ in which the spectra can be explicitly computed. I) If

~

is analytic then ~ ( T ~ ) = ~ ( D ) .

2) I f

~ is real-valued then ~ ) = [ ~ q , ~ ] ,

3) If ~ has piecewise continuous boundary values then ~(T~) consists of the path formed from the boundary values of ~ b y Joining the one-sided limits at discontinuities by straight line segments, together with certain components of the complement of this path. For proofs of these see [1]. In connection with our conjecture it should bementioned that there are easy examples of Toeplitz operators on ~ with disconnected spectra - e.g. ) is disconnected since is positive and compact. The proof of connectedness in the case of H breaks down almost immediately in the ~ case. I would expect a solution to the present problem to shed light on Toeplitz operators in general, and perhaps %o lead %o a different proof and a better understanding of Widom's Theorem.

izl -

T4.1zl:

REFERENCES I. M c D o n a 1 d G., S u n d b e r g C. Toeplitz operators on the disc. - Indiana Univ.Math.J.~ 1979, 28, 595-611.

275

2. W i d o m

H.

On the spectrum of Toeplitz operators. - Pacific

J.Nath. 1964, 14, 365-375. CARL SUNDBERG

University of Tennessee Dept. of ~ath. Knoxville, TN 37916 USA and Institut Mittag-Leffler Auravagen 17 S-182 62 Djursholm, Sweden

276 VECTORIAL TOEPLITZ OPERATORS ON HARDY SPACES

5.8.

Let ~ be a separable Hilbert space (dim ~ may be finite), be the algebra oZ all bonded liner ope~tors on ~ ~ d L~(q) be the Banach space of weakly measurable ~ -valued functions ~: ~ ~~ with the norm

T We denote by ~r(~) the Hardy space of functions in L,~('~) with zero negatively indexed Fourier coefficients and by ~ the Riesz projection onto H P ( ~ ) (4
is called a v • c t o r i a 1 T o e p 1 i t z o p • r a t o r. The following criterion of the invertibility of ~ on ~ ( ~ ) has been obtained by Rabindranathau [I ]°

~o~ ~. ~,et ~ C c~) if, ,a,nd only if ~ = ~ I~, ~. 2)

~

•

T~

is,,invertible ,,~ H~c~)

w,h,,ere

is a unitary-valued f u n c t i o n

in

L~(~0 " )

3) there exists an operator-valued function

~

; with ~+-46

e H~°C~) such t ~ t II '~,- ,.~ I/ See also

< 4.

(1)

L®c~}

[2-3] C ~ , ~ = ~ )

~d [4]

(~,~<~).

A sufficient condition for the invertibility of To$ on H ~ (~) has been given by the authors [5] (the case ~ ~ = had been considered earlier by Simonenko [6] ).

THEOREM 2. ~et

p ~. ({, oo) , ~ = ~ * ~ £ ~

al~ conditions of Theorem 1 for which has to be ~eplaced

b,y

~)~4)~

and suppo,~e hold except for (1)

that

277 Then

~

is inver~,ible ,,On

HP( ) HPcI

HPc) I

(and on

).

PROBLEM 1. Are the conditions of 'Thgorem 2 necessary for the

HPc I?

It is shown in [7] that the answer is affirmative if Let us note that the class of operator-valued functions in Theorem 2 admits an equi~mlent description [8]: & = ~ Z , ~(

, ~4

~ Heo(~ ) and the numerical range

G ~o(.~),

~/C~(~))

lies in

a fixed angle with the vertex at the origin and wlth the size less a.e. on T • It is well-known that the problem of inver%ibility of T ~ in ~P(~) can be reduced by means of faotorization [2-4,73 to the problem of boundedness of ~ in weighted C -spaces. In the case ~ ~ = ~ a criterion for boundedness of ~ was given in [9].

than ~/~vP~C(p, p')

What are the conditions for

~p~'~

to be bounded on

tP(~)

?

We don't know the solution of Problems I and 2 even in the case of matrix-valued Toeplitz operators ( ~ ~ < oo) .

REFERENCE S I. R a b i n d r a n a t h a n M. On the inversion of Toeplitz operators.- J.N~th.Mech., 1969/70, 19, 195-206. 2. H e 1 s o n H., S z e g B G. A problem in prediction theory. -Ann.Mat.Pure Appl., 1960, 51, 107-138. 3. D e v i n a t z A. Toeplitz operators on H Z space. - Trans. Amer.Nath.Soc. 1964, 112, N 2, 304-317. 4. P o u s s o n H°R. Systems of Toeplitz operators on H Z . Trans.Amer.Math.Soc., 1968, 133, N 2, 527-536. 5. B e p d E n ~ H ~ M.3., K p y n H H E H.H. TO~HHe EOHCTaHTH B TeopeMax od O ~ S H H ~ e ~ H O C T H cm~Pyaapm~x onepaTopoB B npocTpaHCTBax c BecoM. --B ~ . : ~ H H e ~ e onepaTopM. Km~ZHeB, m T ~ B , I980, 21-35. 6. C ~ M O H e H ~ o ~I.S. KpaeBa~ s ~ a ~ a P m ~ a H a ~ nap ~ Y H ~ d ~ C EsMepmva~m EOS~J~4eHTa~S~ ~ ee npm~eHeHEe E Ecc~e~oBaH~0 C~HPyJLKpm~x ~HTerpaaoB B npocTpsacTBaX c Becalm. - M3B.AH CCCP, cep.MaT. 1964, 28, 277-306.

278

7. K p y n H E E H.& HeEOTOp~e c~e~CTBH~ ES TeopeM~ Xa~Ta--~EeH-xaynTa-B~eHa. - B EH. : 0nepaTop~ B 6aHaXOBLZ( IIpOCTpaHCTBaX. I~-~ H e B , IHT~H~a, 1978, 64--70. 8. C H ~ T E 0 B C E ~ ~ H.M. 0 ~ a E T o p H s a ~ M a T p ~ - ~ y H F ~ , xayc~op~oBO M~ozecTBO EOTOpHX pacnoaozeHo BHyTp~ yraa. - Coo6m. AH Pp.CCP, I977, 86, c.561-564. 9. H u n t R., M u c k e n h o u p t B., W h e e d e n R. Weighted norm inequalities for conjugate function and Hilbert t r a n s f o r m . - Trans.Amer.Math.Soc., 1973, 176, 227-251.

N.Ya.ER~D~K

. .VERBI Sr

CCCP, 277000, IG~I~HeB, ~eBcE~ l~Cy~apOTBeHH~ yHKBepc~TeT CCCP, 277000, E I ~ H e B , MHCT~TyT r e o ~ s H E ~ H reoaor~AHMCCP

279

5.9.

FACTORIZATION PROBLEM FOR ALMOST PERIODIC MATRIX-E3NCTIONS

AND FREDHOLM THEORY OF TOEPLITZ OPERATORS WITH SEMI-ALMOST PERIODIC MATRIX SYMBOLS I. We consider (~ x ~) -matrices G defined on ~ with elements from the usual algebra ~~ of almost-periodic functions and Toeplitz operators TG = ~ G II ~ ~ generated by these matrices. Here ~ is the Riesz projection onto the Hardy class ~ in the lower half-plane, ~ < p < co . It is well known (this fact holds for an arbitrary G ~ ~ ) that condition G ~ ~ ~ is necessary for T8 to be semi-Predholm. In the case ~=~ the converse is true. Moreover, ~ is leftinvertible if the almost periodic (a.p.) index ~ of the function is non-positive and T G i s right-invertible if $~0 . There exists a certain parallel between Fredholm Toeplitz operators and the factorization problem of their symbols, in accordance with which the formula (1)

is valid in the case

~=~

,

G +-~ ~ AP

.

Here

~(~)= ~ ,

the functions (~ ÷~)-~ ~+~ belong to Hardy class H ~ in the upper half-plane and the operator G+ ~_ G+~ is bounded in all ~, Formula (~) with G ~ ~ ~.possessing~ ~ ~ tabove h properties e (with the natural change of e ~ by ~@~e~V4t, o. ,a~Y~tJ, V ~ ) will be called a P-factorization of G . It is easy to °check that the partial a.p. indices $ ~ , , ~ are umiquely defined by G provided G admits P-factorization; and it is not difficult to describe the freedom of choice of G+ . However in the case ~ > ~ not each matrix ~ invertible in ~~ admits P-factorizations. In this connection the following problems appear. PROBLEM 1. Obtain a criterion (or at least more or less ~eneral sufficient conditions~ of existence of

P-factori~ation.

PROBLEM 2. Find out whether the existence of P-factorization of is a necessar~ condition for

T~

to be (semi-)Fredholm. If

not~ then is it possible to chan~e the definition of P-factorizatipn

280

in such a wa~, as to ~et an equivalent of the semi-Predholm ~roDsrty

Zo~

T~

Note that if ~ admits a P-factorization then 7~ is left(right-) Predholm iff @i ~ 0 ( > 0 ) , i = ~ , ,~. Consequently an affirmative answer to Problem 2 would mean that "Fredholm character" of --T~ is the same in all spaces H~ and its fredholmness implies the invertibility. We do not know whether these weaker statements are true if ~ ~ . 2. The class S A P (of semi-almost-periodic functions) is a natural extension of AP . This class has been introduced by D.Sarason [ I] and may be defined, for example, as {~=(0,5+~)~+

The a°p. components

I' ~

(of

~

) are uniquely determined by

~0 A criterion for TG ( G ~ 5 ~ P , ~=~ ) to be semi-Fredholm in the space ~ was obtained in ~] amd was generalized in ~2] to the case of an arbitrary ~ , ~ ~ (~,oo) . The case ~>~ is considered in [3,4], where the fredholmness and semi-~redholmness criter i a h a ~ b e e n established. These results, however, were obtained under the a priori assumption of existence of P u -factorization of a.p.+ components F , H of G . The latter means that the factors ~; , (~*)±( from the P-factorization F = ~+ A Ff belong to the class AP + of those matrices from ~P whose ~ourier exponents are all non-negative; the same holds for ~ . The following problems arise in connection with the question of removing these a priori assumptions. PROBLEM 3. L e~ and

~

are

hess of ~G

G

be an

(~x~)-matrix from

~

~>~,

F

its aop% components. Is it true that the semi-fredholmimplies the semi-fredholmness of

~F

PROBLEM 4. I s t h e set of matricesadmitting dense in the set of all matrices admittin~

a

and 7 H ~

?

Po-factorization

P-factoriz~tion? What

would the situation be like if we restricted ourselves to matrices wit h a fixed (non-zero~set

of partial a.p~ indices?

The positive answer to Problems 3, 4 would allow to extend the criterion for ~r. to be (semi-)Fredholm [3, 4] to the case of arbitrary matrices~ G ~ 5A~ .

281

3. Let us consider a triangular matrix ~ £ AP of the second order. Under some additional assumptions(e.g, absolute convergence of Pourier series of its elements ) the P-factorization property of ~ is reduced to the corresponding question about "

iv%

e

0

] 0cb=

(2) %({)

e

and the spectrum ~ of %0~ ~ is contained in where V >0 define a.p. polynomials C-~,~) . Assuming that card C~) < co %i (i= ~ ' ~ ' ) by the recurrence formula

(3) It is supposed that the sequence ~ of leading exponents of ~i strictly decreases and ~ ~ AP + ~. Analogously to the case of the usual factorization of continuous triangular matrices E5] there exists an algorithm (say, A ) for P-factorization of Bo which is connected with the continuous fraction expansion of ~i/~0 or equivalently with the relations (3). Algorithm ~ unlike in the continuous case does not necessarily lead to the aim. A sufficient condition to make application possible i ~ K ~ K 4 ~ 0 for some K EZ+ . In this case the factors B± are a.p. polynomials. PROBLEM 5. Give conditions for the convergence rithm

A

to obtain a

~ -

(or

P- ) factorization

of the algoof matrix

(2~. These conditions have,to be formulated in terms of entries of

Algorithm A can be applied to obtain a ~ -factorization of matrix (2) if, for example, ~ C ( - $ , 0] or the distances between the points of ~ are multiples of a fixed quantity (in particular, if card CI~) ~ ~ ). Already in the case card ~ = 3 , i.e.

there exist situations when algorithm ~ tions is: #=0 , ~= [ - & and ~ - & / [ )

fails. One of such situa-

is i r r a t i o n a l ,

282

In this case we found another algorithm based on the successive ±~ ~+ application of the transformation ~'-~ ~ B ~ C ~ (A~, CB g A ) preserving the structure of B~ and on the factorization of elements close to the unit matrix. With the help ~ f this algorithm it was established that under the restriction IC~I # I~÷~I a ~-factorization exists with 9~ = 9~ = 0 ., but B± are no more a.p. polynomials. In the case IC~I =I $4+~I the P-factorization of (2) does not exist. Thus, even in the case card ~= ~ the following problem is non-trivial. PROBLEM 6. Describe the cases when matrix (2) admits Pc-) factorization~

%P-(0 r

calculate its a.p. partial indices and construct ,,

if possible~ corresponding f actorizations. The interest to matrices of the form (2) is motivated by the fact that they naturally arise in connection with convolution equations on a finite interval with kernels ~ for which ~ & ( ~ ) ( ~ ) has an a.p. asymptotics at infinity for an & ~ C [3]. REFERENCES I. s a r a s o n D. Toeplitz operators with semi-almost periodic symbols. - Duke Math.J., 1977, 44, N 2, 357-364. 2. C a r i~ H a m B ~ ~ ~ A.M. Cm~ryaapHae m~Terpaa~H~e ypaBHeH~ C EOS~M!U~eHT82a~, m g e ~ m ~ paspMB~ noay-no~TH-nepHo~eoEoro T~na. Tp.T6mmc.Ma~eM. EH--Ta, 1980, 64, 84--95. 3. E a p a o B ~ D.M., C n~ T E o ~ c E ~ H.M. 0H~Te-pOBOOTE HeEoTop~x C~Hrym~H~X ~HTe~sa~H~X onepaTopoB c ~ m T p m n ~ m EOB~HI~eHTaM~ ~ a c c a ~ A ~ CB~S~x c ~ CECTeM ypaBHeHNl~ cBepTEH Ha EOHe~HOM npomeayTEe. - ~oEa.AH CCCP, 1983, 269, ~ 3. 4. K a p a o B ~ D.H. , C n~ T ZO B C E ~ ~ H.M. 0 H~Te-pOBOCT~, ~- E ~--HOpMa~BHOCTI~ CEHIVJL~pHRX E H T e I ~ B H ~ X onepaTopOB C MaTp~m~M~ EOS~EIU~eHTam~, ~ O n y C E S m ~ m paspNBN noay-no~T~nepEo~H~ecEoro THna. - ~Eoxa no Teop~H onepaTopoB B ~ O H a X B R N X npocTpaHCTBaX (TesHc~ ~OF~S~OB), MHHCE, 1982, 81--82. 5. q e 6 o T a p e B F.H. qacTHae HH~eEc~ EpaeBo~ salaam P~Ma~a c Tpey~oJ~HO~ MaTpm/e~ BTOpO~O nop~Ea. - Ycnex~ MaTeM.HayE, I956, II, h 3, 199-202. Yu. I. KARLOVICH CCCP, 270044, 0~ecca, Hpo~eTapcK~ 6y~BBap 29, MopcKo~ rH~pOSHSMYecKM~ MHCTMTyT I.M. SPITKOVSKII 0T~e~eH~e SKOHOMMKM M (M.M. CHMTKOBCK~) ~K0~OrHM ~p0BOr0 oKe~a.

283

TOEPLITZ OPERATORS IN SEVERAL VARIABLES

5.10.

?or ~ the complex numbers, let ~ be a bounded domain in C ~ with closure ~- and with 9 ~ the Shilov boundary of the uniformly closed algebra A(~-) generated by all polynomials in the complex variables ~=(E~,Z~., ~Z~) on ~- . In general, ~ is a closed subset of the topological boundary of ~ . When ~ is one of the classical domains of Cartan or in other cases of interest, ~ is a compact manifold with a "natural" volume element ~ and the space ~ ( ~ ) of ~-square integrable complex valued functions is the setting for our analysis. The closure of ~(~-)in ~ ( $ ~ ) is denoted by H~(~) and this (Hardy) space, together withthe (unique) orthogonal projection operator ~ from ~(9~) onto H ~ ( ~ ) , is a basic object in complex analysis on ~ . For q essentially bounded on 9~(~) , the ~e~

I ib;TZ@ =~(j~)e.rThet ~*r-a2g~ebraiged~t ed fyral~lT!

~th

continuous I is denoted by ~ (~). l " Even for ~ - - ~ x ~ x xD ( M~ times) where ~ is the open unit disc in C , many interesting questions about ~ ( ~ ) remain open after more than a decade of study. Note that for ( MS times), S ~ = T ~, the M~-torus. The structure of ~(T~)is well-understood for P~= I ,2 ~1,2,3]. necessary and sufficient conditions for ~ in to be ?redholm of index ~ are known [2~. It follows from the analysis of [2] that every ?redholm operator of index % i n ~ ( V ~) can be joined by an arc of such operators to .

(T~n)particular,

cI

Here, ~

is an integer and

~i

.

- ~i

PROBLEM I. Classify the arc-components of Fredholm operators in

This question reduces to: PROBLEM 2. Classif,y the arc-components of invertible elements in

284 REFERENCE S I. C o b u r n

L.A.

The

C*-algebra

II. - Bull.Amer.Nath.Soc.,1967, 1969,

generated

73, 722-726;

by an isometry

I,

Trans.Amer.~ath.Soc.~

137, 2 1 1 - 2 1 7 .

2. C o b u r n

L.A.,

D o u g I a s

A n index theorem for Wiener-Hopf ter-plane.

R.G.,

S i n g e r

operators

!.~.

on the discrete

3. D e u g 1 a s litz operators

R.G., H o w e

R.

on the quarter-plane.

On the

C*-algebra

of Toep-

- Trans.Amer.~Aath.Soc.~1971,

i58, 203-217. L.A.COBURN

quar-

- J.Diff.Geom.91972 , 6, 587-593.

State University Department Buffalo,

N.Y. USA

of N e w York

of Mathematics 14214

285 5.11.

SOME PROBLEMS CONNECTED WITH THE SZEG6 LIMIT THEOREMS

1. Any sequence of ( % x % ) - m a t r i c e s {Ci.~Z~+~ { sequence l,,ofm a t r i x - v a l u e d T o e p l i t z m a t r i c e s

determines a , ° .

If a~ ~I ~ T ~ ~ 0 for sufficiently large values of then the question about the limiting behavior o f A ~+4 /A 6 arises. The analogous question arises about A ~ /~~+ ~ provided the nonzero limit ~ ~..

~

&~÷, /A~

exists.

It was G. Szego who studied both questions for the first time. He dealt with the case %=~ and supposed that { Ci }ig z is a sequence of Fourier coefficients of a positive summable function. See [1] for the precise formulations, for the history of the problem and for its natural generalization A

Ci = H(j),

(1)

M being a f i n i t e non-negative Borel measure on 7 . By the Riesz-Herglotz theorem the class of sequences s a t i s f y i n g (1) i s the class of p o s i t i v e d e f i n i t e sequences. We consider here the case when % ~ and { Ci} is an &-sectorial sequence for some & ~ [ 0,g/~) . The latter means that every T~ is & -sectorial i.e. its numerical range (Hausdorff set) lies in the angle { E : l l ~ l ~ t ~ & ~8~} . It is clear that { C i} is an &-sectorial sequence iff there exists a measure ~ satisfying (1) and taking values in the set of & - sectorial Q~%)-matrices on all arcs of 7 • The real part ~ of this measure ~ permits us to construct the Hilbert space ~ =~C~) consisting of ~-tu~les of functions and equipped with the sesquilinear form A ( ~ , ~ ) = ~ ( ~ ) % M ( ~ ) ~* c~ ) Employing the factorization theorems from ~,3] we have proved in [4] the existence of the limit ~ in the case % ~ , &~0 and have obtained the following formula:

T =~M/~ (see [43 for details and the information about where earlier results by A.Devinatz and B.Gyires). Formula (2) is valid in

286 the case ~ G ~ C too; this is the only case when ~ = 0 . We propose the following as an UNSOLVED PROBLEM: find an extensic n of the Sze~B

second limit theorem to the case of

& -sectori-

al sequences. We CONJECTURE that the limit (finite or not) exists for every the regularity condition

~ ~~°~ ~/~+4 &-sectorial sequence satisfying

We are somewhat encouraged in this conjecture by Theorem 2 of Devimatz [5] related to the case provided ~=~ , M is an absolutely continuous measure and ~ satisfies some additional restrictions (including the requirement G ~ L ~ ). We have proved in the case & =0 , ~ > ~ the existence of and h&ve obtained a formula for its calculation using some geometrical considerations from [4]. Before formulating the corresponding result let us remind that under condition (3) there exist two canonical factorizations of the matrix G : the left one and the right one G = G * G~ ( G~ and G~ are outer matrix functions of the class ~ ~ ). Let us denote the Toeplitz operator with the (unitary valued) symbol ~ = G * G~ -4 by T F .

G=G G

THEOREM I. Let -matrices and let ~ G =~/~

Ci}

be a positive definite sequence of (~×%)-

be a measure connected with it b[ formula (I~,

. Then under con d! tic n %3,),there exists a limit

i (4 oo)

if__ f M

{

of the sequence

~ / ~

is absolutel continuous and

+~ ^

. This limit is finit.e

K F <)II

If these conditions are fulfilled then ~ = ( ~ T ~ T ~

~ 4.

We do not know whether there is any kind of general result in the case ~ ~0 , ~>4 . it is well understood now that the existence of ~ (and formulae for its evaluation) may be proved under some additional restrictions with a help of results obtained in another direction (the rejection of the positive-definiteness with simultaneous amplification of restrictions on the smoothness of G ) that we do not touch upon here, see [6] and references in it. 2. Considering an & -secto~ial matrix measure M concentrated on the line, it is possible to introduce a continuous analogue of the

287

space ~

and to establish the following result.

THEORE~ 2. The two statements given below are equivalent:

,

2) the subspace

~

of constant

tien with the subspace least for one)

~$=V

S>0

~-tuples has

where

~$

zero interHec-

{e~:t~$1

for all (at

.

If these conditions are fulfilled and distance from

where ~ = ~ / ~

~ ~

t_~o ~S

i.~n ~

is a "dis tansematrix"which

& = 0

then the square of th e

metric equals to ~ $ ~ is calculated by the formula

0 Here V ~ is the inverse Fourier transform of the function from the left canonical factorization of a.e. on ~ , -~+ is outer and belongs to the Hardy the upper half-plane). For the case %=~ Theorem 2 was already proved discrete analogue of (4) was established in ~4] in the & ~ [ 0 ~ ~/~) , ~ . We propose a natural

matrixG ( class ~

G=G+G in

in [7]; the general case

PROBLEM: ~eneralize the second part (concerning formula (4)) of Theore m 2 to the case" of distances ~n th e skew In this case obscure points to interprete the right-hand side fact is that the inverse Fourier from canonical factorizations of [2,3] in general are not elements

A

-metric C&>o).

already appear after first attempts of the formula of type (4). The transform of the factors ~ ± & -sectorial matrix-functions of ~ .

The problem to find continuous generalizations for the Szeg~ second limit theorem admits different formulations and even in the definite case corresponding investigations form an "unordered set" (see [8] and the papers cited there). There are still more unsolved questions in the case & ~0 but we shall not go into this matter here.

288

REFERENCES

I.

r o a E H c E ~ ~ B.JI., M O p a r ~ M 0 B ~I.A. 0 npe~ea~Ho~ TeopeMe r.cerS. - MsB.AH CCCP, cepml MaTeM., 1971, 35, BNII.2, 408-427. . Kp e ~ H M.F. , c n ~ T E o B C E ~ ~ H.M. 0 ~aaTOp~Ssam~ MaTpm/-~y~zLU~ Ha e~ZHEqHO~ oEp3~KHOCTI~. --~OIO~I.AH CCCP, 1977, 234, ~ 2, 287-290. . K p e ~ H M.r., c n E T E o B C E E ~ 14.M. 0 ~ a K T o p H s a m ~

.

% --CeETopI~a~IBHNX MaTp]alI-~yHEI/~I~ Ha e~141IIiqHO~ OEpjfaHOCTH. -~aTeM.I~CCJIe~OBalIK~, 1978, 47, 41-62. K p e ~ H M.r., c ii E T E O B C E I~ ~ I~.~. 0 HeEOTOpRX 0606~eHE~X HepBo~ n p e ~ e ~ H o ~ T e o p e ~ Cere. - An~l.N~th., 1983,

9, N 1. 5. D e v i n a t z

A.

The strong Szego limit theorem.

- Illinois

J.Math., 1967, II, 160-175. 6. B a s o r E., H e 1 t o n J.W. A new proof of the Szego limit theorem and new results for Toeplitz operators with discontinuous symbol. - J . 0 p e r . T h e o r y , 1980, 3, N 1, 23-39.

7. E p e ~ H M.F. 06 o ~ o ~ sEcTpanox~xmo~o2 npo6xeMe A.H.Ko~MoropoBa. - ~oF~.AH CCCP, I945, 46, ~ 8, 306-309. 8. M E E a e x ~ H ~.B. MaTpH~H~e EOHT~ETyS~w_RPS~e aHa~oI~ TeopeM r.Cer~ o T ' @ ~ e B ~ X ~eTepM~HaHTaX. -- M3B.AH ApMCCP, I982, I7, 4, 239-263.

M. G. KREIN

(M.F.EP~H) I.M. SPITKOVSKII

(H.M. CnETKOBC~ )

CCCP, 270057, 0~ecca, yJi.ApT~Ma 14, EB.6 CCCP, 270044, 0~ecca, H p o x e T a p c K ~ ~yzbsap 29, MopcKo~ rH~oo~s~qecxM~ HHCTXTyT OT~e~eHMe ~ O H O ~ X H X sKo~orxM M~t-posoro oxeaMa

289 5.12.

THE DIOPHANTINE MOMENT PROBLEM, ORTHOGONAL POLYN0~IAT.S AND

SOME MODELS OF STATISTICAL PHrfSICS I. In [I ], [2] it was shown that in investigations of the Ising model in the presence of a magnetic field the following one-parametric Diophantine trigonometrical moment problem (DTNP) appears, PROBLEM. Describe all non-negative m~asures '

rcle T = { ~ ~arameter ~,

~-e~@~ ~[-~,~[]}

~

even in ~

(@, ~)

on the ci-

, de~en~in~ on a

0 ~ ~ ~ ~ , and such that

0

and the moments

1 0

are pol.vnomials (in ~

) of de~ree

the parity of

coincides with the parity of

Mk(~)

is an inte~e r >I ~

(~

~e~

with integer coefficients; ~k

. Here ~e

is the number of ,the nearest nei~hbours in

the lattice). It is known that the description of such measures can be reduced to the description of the corresponding generating functions

2.

Hk( )-Tk _

re polynomials,

4-~ ~

, where T k are Tchebysh~v polynomials,

290

V~a note that in examples I-3 the generating function is ratiomal, whereas in example 4 it is algebraic (this case corresponds to the orm-dimensional Ising model). QUESTION. Has the ~eneratin~function

correspondin~ to a DTMP,

to be algebraic? Fixing a rational value of the parameter ~ , ~ = ~ , p, ~ integers, 0 ~ p ~ ~ , * ~ ~, we see that our DTMP implies the following "quasi-DT~P":

-- --~

~

k e ~O(~)=

~

, CK

being integers,

0

In particular for PROBLEM:

~ = 0 or

~(~-|)

we obtain the following moment

Describe non-negative even measures whose trigonometrical moments are inte~er~. This problem is solved by the known Helson theorem .-4

d,o(e)= Z $=0

(

2~sj

[31:

.-4

o.,,~', o--'~w + ~ g ~ s sOlO $='0

under some additional conditions on @s,~s [2]. II. It was shown in [4] that the theory of Toeplitz forms and orthogonal polynomials is closely connected with some problems of s~atistical physics and in particular with the Gauss model on the semiaxis. In this connection some mathematical problems appear whose solution would be useful for the further investigation of such models. I, Let ~C~) be an even non-negative summable function on satisfying the Szeg~ condition

1

Cel

0

We define the function

0

-

T

291

--

~K z k ,

N

Izl <

k-O

PROBLEM. Find necessary and sufficient condition s for ~ K - ~ 0 k-~

a_gs

(.physically the last condition means the absence of a long-

@rderDara~eter). as

It was shown in [4] that the condition k--co and

~

q L°

implies ~ - - 0

@O

k:o 2, Let

~ (3 (~)

i+ k

be an even non-negative measure on

T

and

~Ik-il = ]-~ i'~S (K-i)~O(@) ', k, i= 0,4,... 0

i s the c o r r e s p o n d i r ~ T o e p l i t z m a t r i x . We denote by ~ k i T (N) : ( I I k - ~ l )N k,i-O verse m a t r i x f o r JIK-jl

/

the in-

PROBLEM. ~,ind an asymptotics for

N

T(N))_~

i,k-0 This problem appears in the study of the free ener~T in the Gaussian model on the semi-axis with an external field (see 141 )° For example, when ~ O ( ~ ) - ~ t ~@~(~), @ >0, this expression tends to ~/~ The numbers (T(N)) ~ok -~, k =0,~,. ,N, are proportional to the coefficients of the orthogonal polynomials ~N (Z) (see [41, [5]). This leads to a P R O B L ~ of a more detailed investigation of the asymptotics of ~N (e~@) as N - ~ ~ in the presence of non-zero singular part of the measure ~ . As we know only the case of an absolutely continuous measure was considered in detail (see, e.g. [6], [7I)° 3. The multidimensional Gaussian model. Calculate the free energy and correlation functions under less restrictive conditions than in [41, [81~.

292

REPERENCES I. B a r n s 1 e y Diophantine

M.,

moment

B e s s i s

D.,

M o u s s a

p r o b l e m and the a n a l y t i c

t i v i t y of the f e r r o m a g n e t i c

I s i n g model.

structure

P~ The i n the ac-

- J,Math.Phys.,

1979, N 4,

20, 535-5522. B a a ~ a M a p O B

B.C.,

B o a o B ~ ~

H.B. Mo~eJ~ Hsasra c

M a ~ T I m M noaeM ~ ~O#aHToBa nloo6aeMa MOMelITOB. - TeOlO.Ma~eM. ~Hs., 1982, 53, ~ I, 3-15. 3, H e 1 s o n

H. N o t e

on h a r m o n i c

ftulctions~

- Proo.Amer.Math.Soc.,

1953, 4, N 5, 6 8 6 - 6 9 1

4. B m a ~ ~ M ~ p o B B.C., B o a o B ~ ~ H.B. 06 O~HO~ u o ~ e ~ O ~ a T ~ C ~ e c ~ o ~ ~ s ~ F J . - T e o p . MsTeM. ~Hs., 1983, 54, ~ I, 822. 5~ B a a ~ ~ M ~ p o B B.C., B o a o B ~ ~ H.B. YlmB~eH~e BzNe- Xo~a, sa~a~a l ~ a ~ a - ~a~6epwa ~ o p ~ o ~ o H a a ~ e ~ o ~ o ~ e ~ . - ~ o E a . A H CCCP, I982, 266, ~ 4, 788-792. 6. S z e ~ ~ ed.,

G. O r t h o g o n a l

polynomials.

AMS C o l l , P u b l . ,

23, 2

1959,

1~ o a ~ H C ~ ~ ~ B.JI. Acm~n~o~ec~oe i i p e ~ c ~ a B a e H ~ e ol~oz'o]~aa.l~HSX MHOI'O~e~OB. -- Ycnex~ Ma~eM.~sy~, 1980, 35, ~ 2, 148-198. 8. Jl ~ H - ~ ~ H.D. l~oPo~el:~m~ a~maoP ~eol~m~ Ce#~. - HsB.AH CCCP, cep.~m~eu., 1975, 39, ~ 6, 1393-1403. 7.

V,S.VLADIMIROV

(B. C. B~A~I~4POB) I.V.VOLOVICH (~.B.BoaOB~)

CCCP, 117966, M o o m m ya. BaB~aOBa, 42 Ma~esaT~eoF~ ~HC~T AH CCCP

293 5.13.

THE BANACH ALGEBRA APPROACH TO THE REDUCTION METHOD FOR TOEPLITZ OPERATORS

Let H ~ denote the Hardy subspace of LZ = L£ (T) , consisting of the functions { with ~ ( ~ ) = 0 , ~ < 0 , and l e t P be the orthogonal projection from [~ onto H~ . For ~ @_L® = L " (T) the Toeplitz operator with symbol & is defined on H i by T(~)~ = P(@@) • Let ~ { H ~) be the Banach algebra of linear and bounded operators on H ~ . Given a closed subalgebra. B of ~ denote by ~ T ( B ) the smallest closed subalgebra of ~ ( H '~) containing all operators T(¢) with Cb6 ~ . Furthermore, let Q(~) denote the so-called q u a s i c o m m u % a t o r i d e a 1 of ~ T ( B ) , i.e. the smallest closed twosided ideal in ~ T ( B ) containing all operators of the f o = 6~). It is a rather surprising fact that this ideal plays an important role not only in the ~red/aolm theory of Toeplitz operators, but also in the theory of the reduction method for operators A ~ T ( ~ ) .%(with respect to the projections PIT defined by p ~ ( ~ ) ~ k I(~)~k) k=O k=O

T(~)-T(~)T~)

(~,~

For /~e~IH ~) write A~rI{P~I if the reduction method is ~,pplicable to

(see [3] for a precise definition). Finally, put Q~=I-P~ and denote by ~ the group of invertible elements of a Banach algebr~ ~ with identity. For A 6 ~ ( H ~) , the following statements are easily seen to be equivalent:

(iii)

A

P AP,÷Q

(iv)

G (H

(~ 7/I~0),

Q,,A Q~, +P,, ~G'~(H ~)

t'l,~tT o

and

294

(.) A~GB{H~), V_~A'~V~EGB{H ~) (~,)

~I(V_~A V~) i
where

and

V~=T(~), V_~=T(~-~) (~eB).

There is an important estimate closely related to (v) (see [I]*)):

k (~)

which holds for every finite collection of functions Gtq~ E L~ Now, given a closed subalgebra of ~ it follows from (I) tha;

k ~,

k~,

defines a bounded projection

S

~T(B)

on

. One can show that

5(A)-~-~ V_~AV~. zf A e ~ T ( ~

) nG~cH )

since ~ T( ~ ). is a and belo~Vs to # ~ T ( ~ )

C

,

A G~T(~

)

,

-algebra. Thus S ( A ) makes sense ~oreo~er, *~ A~R{P~ t h e n (ii) and

.

(v) imply the invertibility of S (~

i~ /~ ~nd S(A4) ~r~in

then

].

G'~(H~).

The following special cases are of particular interest;

(a) A=T(~)~ ,

r

i

i

u

Ji ,

ll,iiii

~) See also H . K . H ~ o x ~ c ~ , p a x ~ a ~ Teop~. - H p e n p ~ ~ 0 ~

0nepaTop~ r a ~ e ~ P-I-82, 2eH~rps~,

~ T ~ _ ~ a . CneET1982. - Ed.

295

(b)A~-~.~Tq~(~i) of course, for

~

,

=-~

where ~'i e {-4, ~}

the invertibility of

and,

T(~)

is

part of the hypotheses. For ~ ~ PC ( ~ i s the algebra of piecewise continuous functions on with only finitely many jumps) the case (b) is of importance in connection with the asymptotic behavior of Toeplitz determinants generated by singular functions (cf.~S]). In the case (a) the conjectuor (b6~X~PC (see [3],[6]), a n d re 1 is confirmed for A~C + ~ in the case (b) for ~=(T~(@))~

(see [4], [7] ). One possible way to attack these problems concerned with the reduction method is to formulate them in the language of Banach algebras and then to use localization techniques (cf.~6]). Define

W~ : H~ "H%y

and denote by ~ ~

$ ~

the collection of all sequences

~$~

operators ~, ~

having the following property: there exist two ~(H~)

(strong convergence .

such that

e i=t on {A I +[

nach algebra with identity. If operators on

H~

[A ~}~=0 ®

) or even if

A ~~

= {

( ~ is the ideal

A ~ ~T(~

~)

of

compact

, thenIP~AP~

(see [5] ). Notice that this is obvious for ~ = T ( @ ) , ~ since W~T(~)W~ = P~T(~)P~ , where ~(~)=~(~/~) be proved that the set

~

, . It can

296

J={{A,}. AcP,TP~,WGW~+C~,T,?aU. I c,, p,l-,,o} actually forms a closed two-sided ideal in ~ , and that the problem of the applicability of the reduction method to ~ ~ ~ T ( ~ ) admits the following reformulation (see [6]):

A~n{P~I~A,~

~(#)

and the

{P~AP~} i,

co~et

invertible

of

~/j

oo~tai~ing

in~/J

.

Note that now localization techniques can very advantageously be applied to study invertibility in the algebra A / ~ . There is a construction which is perhaps of interest in connection with the case (a). Denote by ~ { ~ ~T(~) the smallest closed subalgebra of ~ containing all ~elements of the form

{P~T(~)P~t and

,where

@EB

Jc~tMT(I~)

{~I~}~G'~{PI,,,}T4~) n. . ~- ~-~

containing

phism

.

If

(of.l:2]). ,~ ~

,R: _~.,{P,,} T{B)

the

C(T) c B

then ~ o e C ~ T ( B )

A , s i ~ t o each sequsnoo coset

[A]F~ ~ T ( B ) / ~ o

o

. In this way a continuous homomor-

~ 4~ T(B)/T®

is produced, and

one has ~ ; D J B= L~ then ke~,~ = ~ . A confirmation of this conjecture would imply that CONJECTURE 2. If

which, On its hand, would verify conjecture I for ~--|(~), ~ L . It is already of interest to find sufficient conditions for the validity of (2) in the case ~ ~ m~° . Note that (2) was proved for

B =C+H ®

or B = d ~ P C

in [2]. REI~ERENCES

I~ B o t t c h e r A., S i I b e r m a n n B. Invertibility and Asymptotics of Toeplitz Matrices. Berlin~ Akademie-Verlag, (to appear). 2. B o t t c h e r A., S i I b e r m a n n B. The finite section method for Teeplitz operators on the quarter-plane with

297

piecewise continuous symbols. - Math.Nachr. 3.

(to appear).

r o x 6 e p r H.~., ~ e ~ ~ ~ M a H H.A. Y p a B H e H ~ B cBepTEaX npoeFa~oHH~e MeTO~N EX p e m e H ~ . MOCEBa, HayEa, 1971. (Transl. Math.Monogr., Vol.41, AMS, Providence, R.i., 1974).

4. R e c h f~r

S.,

S i I b e r m a n n

B.

Das Reduktionsverfahren

Potenzen yon Toeplitzoperatoren mit unstetigem Symbol. -

Wiss. Z. d. TH Karl-Marx-Stadt 5. R o c h

S.,

1982, 24, Heft 3, 289-294.

S i I b e r m a n n

B.

Toeplitz-like Operators,

Quasicommutator Ideals, Numerical Analysis.

- Math.Nachr.

(to

appear). 6. S i 1 b e r m a n n fur

B.

Toeplitzoperatoren.

7. B e p 6 ~ ~ E ~ ~ B~X MaTpHS.

B. SILBERNANN

H.8.

Lokale

Theorie des Reduktionsverfahrens

- Math.Nachr.

1981, 104, 137-146.

0 MeT0~e p e ~ J z ~ H ~

- MaTeMaTH~ecEHe Ecc~e~oBaHE~,

c~eneHe~ T~nn~e1978, BHH.47,

Technische Hochschule Karl-Marx-Stadt SektionMathematik DDR-9010 Karl-Marx-Stadt PSE 964

3--11.

298

5.14.

STARKE ELLIPTIZIT~T S I N G U L ~ INTEGRALOPEP~ETOREN UND SPLINE-APPROXI~T ION

Sei ~ eim Kurvensystem in ~ , das aus endlich vielen einfachen geschlossenen oder offenen Ljapunowkurven besteht, die keine gemeinsamen Punkte haben. Des weiteren seien ~ , , ~ ~ paarT~ weise versehiedene Punkte, - 4 < & k < ~ ( k = 4 , . . . , M"I,) u.nd ~(~) I'11, I~--~ k IAk , ~it ~ . ~ ( ~ ) bezeichnen wir den Hilbertraum k'-~ auf ~ me~b~ren ~unktionen ¢ sit ~ 415~ ~ h~ (F) . Wir aller betrachten die singul~ren Integraloperatoren der Gestalt

F mit stuckweise stetigen Koeffizienten ~,~ 6. PO (F) . Bekanntlich gilt ~r ~ Z ( ~.~( F, fl )). Mit ~ (P~) bezeichnen wir /ie kleinste abgeschlossene Teilalgebra yon ~ (~.~( ~, 2)) , die alle Operatoren A. sowie das Ideal ~ der kompakten Operatoren in [ ( F, ]~) enthalt, und mit ~ A das Symbol eines Operators A £ ~(~C) (vgl. [6 ] oder [IO~). I ~ stetige Koeffizienten P~,~ £ C ( ~ ) gilt

umd % ~ SF ist eine stetige 9hniktion auf einer gewissem Raumkurve "A = ~ (~) [7]. Im Falle des Intervalls ~ = [ ~, ~ ] ist ~ der Rand des Reohtecks [@,~]x[-~,4] . Wenn ~ nur aus geschlos~enen Kurven besteht und ~ ~---4 ist, dazm gilt

Die Abbildung Sym ist ein isometrischer Isomorphismus der symmetrischen Algebra ~ ( C ) / ~ auf G(A) mit ~ A* - ~ A Der Operator ~ ~ ~ ( ~ . ~ ( F , ~ ) heist s t a r k e 1 i i p t i s c h, gilt wobsi D positiv definit und T ~ T ~ ist. Wir nenmen A @ -s t a r k e 1 1 i p t i s c h, wenn eine 9kmktion @ @ C(~) @(%) @ 0 (~ % £ ~ ) , existiert derart, d~ 8Astark elliptisch ist. -

299 ~r

"Re

0peratoren der Algebra ~(6) gelten folgende Kriterien: 1°. A £ ~(C) ist genau dann stark elliptisch, wenu

A;'o. Ar = ~I + ~ ~r

dann

wobei

( ~v,6~ C (I~ ))

ist

ge:z~a1~

~ -stark elliptisch, wenn

K~({)

die konvexe Hulls der }Ienge { ~

~F({,~)}(~,~)£A

bei fsstem ~ 6 ~ bezeiohnet. Im Falls f =-~ ist K4(%)= [-~, 4] Die Hinlanglichket der Bedingung ~£ ~ A F0 folgt leioht aus den obengenannten Eigenschaften der Abbildung ~ A (vgl. [9] ); ihre Notwendigkeit ergibt sich aus der Hinlanglichkeit und der Eigenschaft, da2 der Operator ~ £ ~(C) gen&u dann sin Fredholmoperator ist, wenn ~ ~ ~ 0 [6], [10]. Die Notwendigkeit von( 1 ) ist sine direkte Folgerung der Eigensch~ft 1°; ihre Hinlanglichkeit kann man mit Hilts einer Einheitszerlegung der Kurve beweisen [3]. Wegen 6~£~s ={~ Sp(~,~)} (~,Z)~ (vgl. [7] ) zieht die Bedingung

(.~p)

(2) die Bedingung (I) nach sich; f~r konstante Koeffizienten ~ , ~ sind beide Bedingungen (1) und (2) aquivalent. Die starke Elliptizitat ist sine notwendige und hinreichende Bedingung dafur, d ~ f~r den invertierbaren Operator A die Reduktionsmethode bezuglich einer beliebigen 0rthonormalbasis konvergiert [4]° Wenn U = ~ der Einheitskreis ist, so konvergieren fur den singularen Operator A T in h% (T) gewlsse Projektionsmethoden mit Spline-Basisfum/~tionen genau darm, w e ~ A~ @ -stark elllptisch ist [12], [13]. Wit betrachten auf die stuckweise linearen Splines

{k

-wobei

~k

f

-{)({k+ 0

"4

far

{ £ {k{k,~

SONS%,

(k= 0 , . . . , *-4) '

bezelchnen wir den Orthoprojektor in

L~(T)

mt

P.,

auf die lineare H'ulle

300 ~

~ ... $@~ ~ ) +

projektor, der

. .., ~ (~) ~.~ } und mit O~ den Interpolationsjeder beschr~ten F~nktion ~ den Polygonzug ~'~

c~) (+,1

zuordnet. Wenn die Operatoren H ~ c h£ (T )

(fur alle

A~ = ~ 7 ~o

) i~L~er%ierb~r sind trod

~ t t ~ - ~ l l <~o ist, dann schreiben wir in diesem Falle g i l t ~-~ ~ ¢ ~ ~'; ~ ~lle ren

~

~it

Q+~ ~ ~

, i~eoo~der~

AT~]]~P,,,Q~}'~ ,

e~

.--~oo ,

~ll~ ~iem~te~ie:~-

1~m~tionen [12]. Das soeben besohriebene

2rojektionsverfahren

heist Kollokationsmethode (auch Polygonmethode). Analoge Bede~tung hat ~{P~, P,,} (Reduktionsmethode). Es gilt folgender SATZ ([12],[13]). Se~ A T = @I + ~ mit

~, ~ E PC ( r ) . Dann gilt

(~=~I¢~)

A T ~ rl

[ P~, O~}

gena~

dann~ wenn folgende Bedingung erfS_llt ist:

C3)

+&c~.o)~,(-r.-o)(,t-.;)~o,

V.'r.~r, '~y.,~[o,q

Wir bemerken, da~ aus (3) die Invertierbarkeit von AT in L~(~) folgt [7], [10]. Im Falle ~ ~C(~) bedeutet (3) gerade die ~-starke Elliptizitat yon A THYPOTHESE

I. Bedingung (3) ist ~quivalent tier

Elli~tizit~t ' des Operators

AT

im Raum

L~(~)

~ -starken

(0 ~ ~C (T))°

Aus der G~ltigkeit dieser Hypothese wurde insbesondere A T ~ I PI,I,, PK.t folgen. Die Schwierigkeiten beim ~-berpr~en der Hypothese I bestehen darim, da~ ~ AT eine Matrixfunktien und ~ (PC) eine nichtsymmetrische Algebra ist. Von gro~em theoretischen und praktischenInteresse sind Bedingtmge~, die ~ Konvergenz entsprechender Kollokationsmethoden mit gewichteten Splines auf offenen Kurven garantieren (s.z.B. [2~, [8] ). Sei der Einfachheit halber ~= [0,~] , ~ ) = j / ~ t (i =0~,..,~) ~|~' ) die entsprechenden st~okweise lineare~ Splines und ~-J!~) -

301 -41~

=~ ~

(~) ~, "

. Nit ~ C~)

bezeichnen wir den 0rthoprojektor in (~ ~ und mit

den entsprechenden Interpolationsprojektor.

E'fgOTKESE 2.

Sei

A~=~I+65 r-

(~,%)£0C~))

-stark elliptische r Operator. Eann silt

~

,~in

6

Im ~alle ~==~ ergibt sich die Richtigkeit der Hypothese 2 aus dem obengenannten Satz dutch Abbildung yon ~ auf eine Halfte yon ~ und anschlie~ende Fortsetzung der Koeffizienten auf ganz (vgl. ~0], Seite 86). HYPOTHESE 3. Hypothese 2 ~ilt fur i

be!iebi~e Dol.ynomiale Splines

~erade,n Grades.

~r gesohlossene Kurven ~ wurde die Konvergenz der Kollokations - tmd Reduktionsmethoden mit Splines beliebi~ea Grades in nichtgewichteten Sobolewr~umen in [I], [11] ,[I~ untersucht.

LITERATUR I. A r n o 1 d D.N., W e n d 1 a n d W.L. On the asymptotic convergence of collocation methods. - Math.of Comput., 1983. 2. D a n g D.Q., N o r r i e D.H. A finite element method for the solution of singular integral equations. - Comp~Math.with Appl., 1978, 4, 219-224. 3. E 1 s c h n e r I., P r o s s d o r f S. ~ber die starke Elliptizitat singularer Integraloperatoren. - Nath.Nachr. (im Druck). 4. r o

x 6 e p r npoemmomme 5. r o x 6 e p r

M.~., * e a ~ ~ M a H M.A. Y p a B H e H ~ B cBepT~ax MeTO~H ~X pemeH~a. M., HayEa, I97I. M.~., K p y n H ~ ~ H.H. 06 aareOpe, nopo~t~eHHO~ O~HoMepH~ C~I~JIHRH~]M~ ~HTerpazBmes~ onepaTopa~m c EyCO~HO-Hen p e p ~ m e ~ ~ o s ~ m m e H T a ~ m . - ~HKA.aaaa. ~ ero npza., I970, 4, .~ 3, 2 6 - 3 6 . 6. r O X 6 e p r M . H . , K p y n H ~ E H.H. C~I'y.Tr...Ep~e ~ T e z p a x ~ m~e onepaTop~ c Eycovao-~enpepmmm~ ~ o ~ m m e H T 8 ~ , ~ ~ ~x cm~oza. - E s B . A H CCCP, cep.MaTeM., I97I, 35, ~ 4, 940-96Io 7. r 0 X 6 e p r Mo~., K p y n ~ ~ E H.H. BBe~eH~e B Teopm0 O~HoMepH~X cm~yaapH~x EHTerpa~H~X onepaTopoB. Kmm~eB, ~T~mam, 1973o

302

8. I e n

E., S r i v a s t a v

R.P. Cubic splines and approxi-

mate solution of singular integral equations. - Nath. of Comput., 1981, 37, N 156, 417-423. 9. K o h n I.I., N i r e n b e r g differential operators.

L.I.

A n algebra of pseudo-

- Comm.Pure and Appl.Math.,

1965, 18,

N 112, 269-205. 10. M i c h i i n raloperatoren.

S.G., P r o s s d o r f - Akademie-Verlag,

11. P r o s s d o r f

S.

S.,

Singulare

Integ-

Berlin, 1980.

Zur Splinekollokation fur lineare Opera-

toren in Sobolewraumen.

- Teubner - Texte zur ~ath. "Recent

Trends in Math.", 1983, Bd.50, 251-262. 12. P r o s s d o r f S., S c h m i d t

G., A finite element

collocation method for singular integral equations. - Math.Nachr.~ 1981, 100, 33-60. 13. P r o s s d o r f

S.,

R a t h s f e 1 d

Methoden fur singulare Integralgleichungen stetigen Koeffizienten. 14. S c h m i d t

G.

Pinite-Elemente stuckweise

(im Druck).

On spline collocation f~r singular integral

equations. - P r e p r i n t der DDR, Inst.f.Math., S.PR~SSDORF

- Math.Nachr.

A. mit

P-Math.-13/82, Akademie der Wissenschaften 1982. Institut fur

~thematik

Mohrenstra~e

39,

1086 Berlin, DDR

AdW

303 5.15.

HOW TO CALCULATE THE DEFECT IW3MBERS OF THE GENERALIZED R~EMANN BOUNDARY VALUE PROBLEM?

The question concerns the problem of finding functions ~ satisfying the boundary condition

~P

p Here ~,~ £ L , k £ U ,4<~<~, ~ is a non-singular orientation preserving diffeomorphism of T ("the shift") with the derivative in ~p~,0<~<~ . The case of orientation-changing shift & comes to this'J~by an evident replacement of ~(~) by A~[~ ; & by ~ ; ~ by ~ • The investigation of (I) and of its generalizations is connected with a number of questions of elasticity theory ([1],Ch 7), the rigidity problem for piecewise-regular surfaces [2], etc., and has already a rather long history, starting with A.I.Markushevitch's work of 1946 (see [3] and a detailed bibliography contained therein). Fredholmness conditions and the index of the operator corresponding to the problem (I) are known and don't depend upon "the shift"~ If ~4~ is sufficiently small, then under certain additional conditions on & (e.g. II~ ~ < ~ C p , - ~ ~E~ ~ being the class (introduced in [5]) of multipliers not affecting the factorizability) one of the defect numbers of (I) is equal to zero, and therefore defect numbers don't depend upon "the shift". I.H, Sabitov's example (see [3], p.272) shows that this is not the case in general.

PROBLEM. Calculate the defect numbers of the problem (1). Find the c.onditions on the' coefficients

~,~ , under which the defect num-

bers do not depen d upon "the shift". The defect numbers ~ and ~' of the problem (1) without "shift" (~(~)={, { £T ) are connected [3,4] by formulas ~= ~ ( ~ i ~ ) + e~uX(~,0)-I~ 61= ~4 + ~ ' ~ ' ~ with partial indices ~ 4 , ~ of the matrix

(here the defect numbers are calculated over ~ ). The problem to calculate partial indices of matrix-valued functions even of this special kind, however, is far from final solution.

S04 Under assumptions

(2) (where

~ = @-@I6, ~ . i s

an o u t e r f u n c t i o n w i t h

I~+1 = l m l

a.e.),

as

it is shown in [6], ~I and ~ are expressed in terms of the multiplicity of S-number I of the Hankel operator 0 ~ P ( P is the Riesz projection of ~P onto H P, Q-- ~ - P ). using this fact and the results of V.M.Adamjan, D.Z.Arov and M.G.Krein ~7,8,9] the defect numbers of problem (I) are expressed in [6] in terms of approximation characteristics of its coefficients. The elimination of restrictions (2), and the generalization of the above-mentioned results to the weighted spaces seems to be of interest. In the case ~ ( { ) ~ the calculation of defect numbers of (I) may reduce to the problem of calculation of the dimension of the kernel of the block operator

Pw2

'

composed by "slLifted" H~nl~el and T o e p l i t z o p e r a t o r s (here W , ~ = ~0~ is the so-called "shift" (translation) operator). It is interesting to remark, that these operators also appear while investigating the so-called one-sided boundary-value problems, studied in [I0] .Thus, the investigation of the problem ~(%)=0v~ +~, ~ 6 ~?, ~ with an involutory orientation-changing "shift" ~ under ordinary conditions ~3] ~" ~ = ~ ~@~)a 0 can reduce to the study of the operator ~ W ~ 0 : both have the same defect numbers, their images are closed simultaneously and so on. Thus, the new information about "shifted" Toeplitz and Hankel operators may be employed in the study of the boundary value problems with the shift. In conclusion it should be remarked, that by an analytical continuation of ~ into the domain { ~ £ C :I~I>4] and by a conformal mapping, the problem (I) can be reduced to the problem of finding the pair of functions in Smirnov classes Ev+, satisfying a "nonshifted"boundary condition on a certain contour. We note, by the way, that the related question of the change of partial indices of matrix-valued functions under a conformal mapping (i.e., practically, the question of calculating defect numbers of vectorial "shifted" Riamann boundary value problem) put by B.V.BoJarski~ [11], has received no satisfactory solution so far. The authors are grateful to I.M.Spitkovskil for useful discussions.

305 RE~ERENCES

I. B e K y a H.II. C~0TeM~ C~HI~JIS!0H~X BHTST!0aalSHSX y!08BHeHm~ ~ HeEOTOI~e I"!08H~Hse sa~a~, M., HayEs, 1970. 2. B e z y a N.H. 06o6~essue aHa.Tl~T~l~ecF,~e ~ . r ~ a a , M., (I:~, 1959. 3. ~I a T B ~ H ~ y Z r.c. KpaeBue sa,~aqa a caHr~Slo~e aHTer!oaa~Hue y l m B S e ~ s CO c~Baroa, M., HayEs, 1977. 4. C II a T K O B 0 K ~ ~

H.M. E Teop~H O606~esHo~ KpSeBO~ Sa~SR~

PaUSHa B F~accax L P . - Yl~p.~aTem.mypH., I979, SI, ~ I, 63-78. 5. C n z T E 0 B C E Z H H.M. 0 ~HOZZTe2SX, He B~lZ~0~X Ha SSETOpSSyemOCTB. - ~oF~.AH CCCP, I976, 231, ~ 6, I300-I308. 6. ~ I R T B ~ H N y E r.c., C l l z T E O B C E ~ R 14.M. TOWHee o~eHE~I ,~e~e~T~Ux ~ac8,,'l 060611~eEHo#r KIBeBOf~ sa,~a~a Pa~aHS, ~8KTOp~sa~zs @I~ZTO~UX ~ a T p ~ - Q ~ y ~ t ~ ~eEOTOl~e npo6~e~u n p ~ 6 ~ e H a s s e p o s O l ~ U z ~ySF~ZS~Z.- MaTe~.c6OpH., 1982, 117, 2 2, 196-214. 7. A ~ a ~ s H B.M., A p o B ~.3., K p e ~ H ~.r. 0 ~ecKoHe~HUX ~as~e2eBux aa~pa~ax ~ 0606~eHH~X 3a~sNax EapaTeo~opa-@e~epa *.Pacca. - ¢~HF~.a~a~. z e r o n!oz2., I968, 2, ~ I, I-Ig. 8. A ~ a u s H B.~., A p o B ~.3., K p e ~ H ~.r. Bec~ose~sue ~am~e~eBs MaTp~II~ a o6o~meHm~e s a ~ m ~ KapaTeo~op~-~e~elm z H . ~ I e . - ~ H ~ s ~ . a H a ~ . a ero nlO~., 1968, 2, ~ 4, 1-17. 9. A~

a Ms H

B.M.,

Ap

0 B

~I.3.,

K p e ~ H

M.r. A~m~Ta~ec-

E~e CBO~CTB~ nSp ~ a A T a raH~e~eBs onelmwop8 ~ o6o6=eHHms sa~a~a ~ypa - TaEsr~. - MSTe~.c6opH., IgVI, 86, ~ I, 33-73. I0. 3 B e p o B ~ ~ 3.~., ~I ~ T B ~ H ~ y ~ r.C. 0~HOCTOpOHHae KI~SBHe 38~8~Z T e o p ~ 8 H a ~ T ~ e C E ~ X ~ H . - HsB.AH CCCP, cep. ~STe~., 1964, 28, ~ 5, I003-I036. II. B o , p C E a ~ B.B. AHS~I~S I)SSI~8~ISMOOT~ I~IOSHZEHRX sa~aq TSOID~ ~ H E ~ R ~ . - S EH. : HCCAe~0BSHRS no COB!O.IlpOO~leua~ T e o I ~ ~SyHE~R ~ o ~ n ~ e z c H o r o nepe~esHoro, M., ~ , 1961, 57-79.

(~.~.~~)

CCCP, 220000, 0~ecca, ~ . H e T p a B e ~ o r o 2, 0 ~ e o c E ~ roc?~spc TBeH~u2

G. S. LI TVINCHUK

yHMBepC~ Te T

Yu° D. LATUSHKIN

(r.c.m4TBHRNYK)

306 POINCAPd~-BERTRAND OPERATORS IN BANAOH ALGEBRAS

5.16. Let

A

be an associative algebra over ~ . A linear operator is said to satisfy the Pomnoare-Bertrand identity if .

~E#~ A for all X,Y~A

~CX" EY+ EX.Y) =~X. ~Y+XY . THEOREM. Suppose ~

(~)

satisfies' (1)~Then

(i) the formula

Xx y

=

gX'Y+X'gY

defines an associative product in al~ebra by A~ ); (ii) The mappi~s

A

- ~et

~ ± {

A (We denote the corresponding

are homomorphisms from

A± =J~C~tD , N±=Ke~C~4)

algebra apd

N!~ A±

is

a

• ~hen

A&

A± ~ A

~wo-sided i dgal, Also,

into

is a sub-

A = A+ + A_

N+~N-=O (iii)

The mappi~ of the q,uotient ' a,,lgebras

e: A+/N+--- A_/N_ ~iven by e :(~+0X - - ~ - ~ ) X (iv) each

X~A

i s an al~eb~ isomorphism;

can be tmiquely decomposed as

X=X+-X_ ,X± cA±, e~+)=X_ (we denote by X~

the residue class of X ±

E~JL~I~o Let A =W

modulo

N

e

be the Wiener algebra. Define

by

~hen ~

~×= I X,

if X

is analytic in ~

[ -X,

if X

is antisa~alytic in

,

satisfies (I).

PROBLEm. Por

A = W ® M@t C ~ c)

o~ A = L ® M~t c~ c)

describ~ alllinear operators satisfyiz~ (I~.

,

307

NOTES. I. This problem arises as a byproduct from the studies of completely integrable systems. This connection is fully explained in [1]. For partial results in the classification problem cf. [I],[2]. I n m o s t papers the problem is considered in the Lie algebraic setting In that case equation (I) is replaced by

+

y]+Cx,y]

which is commonly known as the (classical) Yang-Baxter equation. 2. Given a solution of (I) an operator X ~-~-X ×g Y can be regarded as an analogue of the Toeplitz operator with the symbol Y . It seems interesting to study the corresponding operator calculus in detail.

REFERENCES I. 2.

C e M ~ H O B -- T ~ H ~ & H C E E ~ M.A. I~TO ~aEoe ExaccH~ecEa~ t-MaTp~a. -~.a~a~. ~ ero n p ~ . , 1983. 17, ~ 4. B e ~ a B ~ R A.A., ~ p ~ H ~ e ~ ~ ~ B.r. 0 pemeH~i~x ~ a c c ~ e c E o r o y p a B H e ~ H~r~a-Ba~cTepa ~ npOCT~X a~re6p ~ . @yH~.a~a~. ~ ero n p ~ . , I982, I6, ~ 3, 1-29. --

~.A. SE~'NOV-TIAN-SHANSKY (M.A. C E ~ 0 B - T H H ~ C K ~ )

CCCP, I9IOII, ~eH~Hrpa~, ~oHTa~Ea 27, ~ 0 ~

CHAPTER

6

SINGULAR INTEGRALS, BMO, H p

This chapter is a natural

c o n t i n u a t i o n o f t h e p r e c e d i n g one:

e i g h t problems opening the c h a p t e r deal with s i n g u l a r i n t e g r a l s . two f i r s t

a r e " o l d " (and a r e e s s e n t i a l l y

1977 b r e a k t h r o u g h i n

The

i n f l u e n c e d by C a l d e r 6 n ' s

f,~-estimates of Cauchy-type integrals

on

Lipsch~tz curves (see S.5 and Commentrary therein). Others cover various aspects of the theory of singular integrals (continuity,

two-

sided estimates as in 6.8 and even the exact values of their norms as in 6.6). Unlike the preface of the "old" Chapter9 we dispense here with emotions caused at that time by the very appearance of BHO and the real

HP-spaces.

The

HP-BMO ideology has shared the destiny of

all sis~lificant theories (see Introduction to Chapter I) being now together with

~,P's or

C - a necessary prerequisite for analytical

activity, as though they (i.e. BMO and H P

)"existed always" but pas-

sed unnoticed for a period of time.

BMO is ubiquitous as is seen, e.g., from the items of this (and not only of this) Ghapter. In Problem 6.10 BmO is intertwined with famous coefficient problems for univalent functions.

In some problems

its presence or influence is not so explicit (as e.G. in 6.11, 6.12~ 6 14

or

in

Problem

6.13, dealing with a quantitative variant of

the John - Nirenberg inequality) but nevertheless undeniable

The sa-

309

me can be said about VMO-setting of the "old" Problem by Sarason (S°6). Various aspects of the

~P -theory (real or complex) are dis-

cussed in 6°4, 6,9, 6~13, 6,15-6,19. Other interesting connections are represented by items 6.9 aud 6.10. These problems are of importance for Toeplitz operators (see Chapter 5)~ The solution of S,6 found by T.Wolff (see Commentary in S,6) yields a useful factorization of unimodular functions in ~ + operators with

(~$

C

leading to a factorization of Toeplitz

G) -symbols,

The prediction contained in the last

phrase of Section 2 in 6.9 was more than justified: the

n e g a t i v e

solution obtained by T.Wolff (see Commentary to 6~9) also has an important application,

namely,

the existence of a non-invertible Toeplitz

operator whose symbol has a Poisson extension bounded away from zero This disproves the famous conjecture of Douglas. We conclude by the indication of Problem S.11 inspired by the abstract

HP-theory of Coifman - Weiss.

(We first included the problem

into this Chapter, but people became aware of it before the volume was ready and got so interested that we had - at the last moment - to remove it to "Solutions".)

310

ON THE CAUCHY INTEGRAL ANDRELATED INTEGRAL OPERATORS

6.1.

old Let P be a rectifiable curve in 6 . The Cauchy integral of and integrable relative to arc length is a function defined on defined as:

Recently A.P.galderon [I] has proved the existence almost everywhere of nontangential boundary values for the function C ( ~ ) (denoted as C(~) (~) ). This poin%wise existence theorem follows from the following estimate proved in [I] by an ingenious complex variable method. THEOREM I (A.P.Calder6n). There is a constant

mlllllfO= constant

~

II 'll <

~o~ ~o > O, there exists a

for which:

It is not hard(by using singular integral techniques) to reduce the existence a.e. result mentioned above to theorem I. SEVERAL IMPORTANT QUESTIONS remain open. L

Is the restriction

, ,, of theorem 1? timate

II~'I~ < T0

necessary to obtain the ,es-

• ! s method as well as other techniques Calderon

are unable to eliminate this restriction. IX. Since the operator 0(#)(~) exists almost everywhere for all functians in L~(P, I ~ I ) existence of a weight

cop (~)

, it is natural to conjecture the ( > 0 a.e.) for which

311 ( T h e existence of such a weight for a weak ~ estimate is guaranteed by general considerations related to the Nikishin-Stein theorem.) IIL The integral operator appearing in Theorem I is related to a general class of operators like Hilber~-transforms, of which the following are typical examples. a) The so called commutators of order -f,-~

b) ! Ac o+'b-

Ac.-lb

~>0

( -lb

(Here ~

l~

),,

It is easily seen that ~heorem 4 is equivalent to the following estimates on the operators A ~ : A

for some constant C . The boundedness in L~ of the operators in a) b) c) has been proved in E2], E3~ by using Fourier analysis and real variable techniques (which extend to ~ ). Unfortunately the estimate obtained (by these methods) on the growth of the constant in (,) is of the order of ~V (and not 0 h ). It will be h$~hl~ desirable to obtain a ~roof of Calder~n's result which does not de~end on special tricks or complex variables. Any such technique will extend to higher dimensions and is bound to imply various sharp estimates for operators arising in partial differential equations.

312

REFERENCES I. C a I d e r 6 n

A.P.

On the Cauchy integral on Lipschitz cur-

ves and related operators. - Proc.N.Ac.Sc.1977, 4, 1324-27. 2. C o i f m a n R.R., ~ e y e r Y. Com~utateurs d'integrales szmgulieres et op~rateurs multilinealres.--Ann, Fourier

(Grenoble),

3. C o i f m a n

1978, 28, N 3, xi, 177-202.

R.R.,

M • y • r

Y.

Multilinear pseudo-

differential operators and commutators, R.R.COII~N

Inst.

to appear.

Department of ~athematics Washington University Box 1146, St.Louis, M0.63130 USA

YVES ~EYER

Facult~ des Sciences d'Orsay =

°

r

Universlte de Paris-Sud Prance

CO~ENTARY The solution of Problem I is discussed in Commentary to S. 5.

313

6.2. old

SOME

PROBLE~gS CONCER/~ING CLASSES OF DO~IA~TS DETER/gINED BY PROPERTIES OF CAUCHY TYPE INTEG~%LS

Investigation of boundary properties of analytic functions representable by Cauchy-Stieltjes type integrals~ni a given planar domain G (i.e. functions of the form ~b-~ (~-%f~); if ~G ~ = ~ we denote this function by ~ ), as well as some other problems of function theory (approximation by polynomials and rational fractions, boundary value problems, etc.) have led to introduction of some classes of domains. These classes are defined by conditions that the boundary singular integral ~V (D ( ~ = ~ ) should exist and belong to a given class of functions on ~ or (which is in many cases equivalent) that analytic functions representable by Cauchy type integrals should belong to a given class of analytic functions in G . see [I] for a good survey on solutions of boundary value problems. An important role is played by the class of curves (denoted in [I] by ~p ) for which the singular integral operator is continuous on mP(r)~ ~ ~ ~p if and only if

Vco LPCr)

llSrC°)llCpll°°ll P•

(I)

This means that the M.Riesz theorem (well-known for the circle) holds for r . Some sufficient conditions for (1) were given by B.V.Hvedelidze, A.G.D~varsheishvily, G.A.Huskivadze and others. I.I.Danilyuk and V.Yu°Shelepov (a detailed exposition can be found in the monograph [~ ) have shown that (I) is true for all p>~ for simple rectifiable Jordan curves r with bounded rotation and without c u s p s . Some general properties of the class ~p were described by V.PjHavin, V.A.Paatashvily, V.M.Kokilashvily and others. It was shown, e.g., that (I) is equivalent to the following condition:

Ep(,G)

being the well-known V.I.Smirnov class (cf. e.g, [~) of over functions # analytic in 6 and such that integrals of I~IP some system of closed curves (with , ) are bounded. That is~ ~p can be characterized by the property

314

where

~ is a conformal mapping of ~ onto ~ . Another class of domains (denoted by ~ ) has been introduced and investigated earlier by the author (cf.[4] and references to other author's papers therein). We quote a definition of K that is closely connected with definition (2) of ~p : ~ K if for any function ~ in ~ , analytic and representable as a Cauchy type integral, the function C~o~) ~' ( g being a ccnformal mapping of D onto 6 ) is also representable as a Cauchy type integral: K)

Note that by Riesz theerem it is sufficient for F ~ p that~he function ( ~ o ~ ) ~ in (2) be representable in the form with ~ L P C T ) . This allows to consider K as a counterpart of ~p for p= I (it is well-known that to use (2) directly is impossible for p=1 even for F = T ). It is established in [5] (see also ~]),using Cotlar's approach, that the classes ~p coincide for p > 4 . Thus the following problem arises naturally. PROBLEM I. Do the classes

~p