Fourier analysis of a space of Hilbert-Schmidt operators-new Ha-plitz type operators

Publicacions Matemátiques, , Vol 34 (1990), 181-197 . Abstract FOURIER ANALYSIS OF A SPACE OF HILBERT-SCHMIDT OPERATORSNEW HA-PLITZ TYPE OPERATORS JAAK PEETRE If a group acts via unitary operators on a Hilbert space of functions then this group action extends in an obvious way to the space of HilbertSchmidt operators over the given Hilbert space . Even if the action on functions is irreducible, the action on H .S . operators need not be irreducible . It is often of considerable interest to find out what the irreducible constituents are . Such an attitude has recently been advocated in the theory of "Ha-plitz" (Hankel + Toeplitz) operators . In this paper we solve this problem the space of H .S . operators over the Hilbert space L Z (A, p .) of square integrable functions over the unit disk A equipped with the Dzhrbashyan measure dpa (z) = (ce+1)(1-jzj')°dA(z)(a > -1) . This complements the earlier results . In particular we discover many new Ha-plitz type operators . The question of their smoothness properties (S pestimates etc .) is however only touched upon . Introduction If a group acts via unitary operators on a Hilbert space of functions then this group action extends in an obvious way to the space of Hilbert-Schmidt (H .S .) operators over the given Hilbert space . (More generally, one can consider H .S . operators from one Hilbert space into another with a different group action on each of these spaces .) Even if the action on functions is irreducible, the action on H .S . operators need not be irreducible . It is often of considerable interest to find out what the irreducible constituents are . Such an attitude has recently been advocated in the theory of "Ha-plitz" (Hankel -}- Toeplitz) operators . Example 1 . Consider the spaces A = A''2(A) (definition in Sec . 1), A 1 = the orthogonal complement of A in L 2 (A,pa) (definition in Sec . 1), A = the space consisting of all conjugates of the functions in A . Operators from A into A have been studied from this point of view in [JP1] ("small" Hankel operators of higher weight) . Similarly, operators from A into A 1 have been studied in [BJP] ("big" dittos) . In both cases the group is the projective group of conformal selfmaps of the unit disk A in C or a double cover . 18 2  J . PEETRE Example 2 . The study of the case when A is replaced by the unit ball in Cn has been initiated in [P1] . Example 3 . H .S . operators on L 2 (Rn) under the action of the "ax+b"-group (dilations + translations) are analyzed in [JP2], [P2] (so-called "paracommutators") . In this paper we want to decompose the space of H .S . operators over L 2 (A, m«) . This complements the results of [JP1], [BJP] . In particular, we discover many riew Ha-plitz type operators . The question of therr smoothness properties (S p - estimates etc .) is however only touched upon . There are good hopes that one could be able to play the same game with other symmetric domains (in higher dimensions), in the first place the ball (cf . The organization of the paper is as follows . Sec . 1 introduces the notatioíi . In Sec . 2 we consider various invariant measures on the space of H .S . operators over L 2 (A,íc a ) . In particular, this allows to write the space in question as a tensor product where one factor is the space L 2 (G), G being the group of all conformal se1fmaps of A . The results of this Section are of independent interest and we intend to retum to them on a subsequent occasion . In Sec . 3 we then apply Plancherel's theorem in the latter space to produce the desired decomposition of L2 (A , fca) . Sec . 4 is devoted to some examples showing how known type of (big) Hankel operators [JP1], [BJP] (Example 1 ultra) fit into our new scheme . In Sec . 5, returning to the general case, we write down the expression for the H .S .-norm of our new Hankel operators . Sec . 6 briefly touches upon the S p -theory, still in its embryo . In Sec . 7 we sketch an alternative infinitesimal approach to our problem . It is not entirely successful, as we have not been able to carry out the spectral analysis of the Casimir operator encountered . In Sec . 8 we specialize this to the case of operators from A"'2(A) into itself (generalized Toeplitz operators) . Finally, there is an appendix to which we have deferred some (dull?) calculations pertaining to Sec . 2 . We intend to apply this result in a different context in a subsequent publication . Acknowledgment . I am greatful to the referee for his many comments, the scholarly ones and also the humoristic ones . He also detected a missing factor 1/(167r 2 ) in formula (6), Sec . 2 . In addition, I am obliged to Richard Askey and André Unterberger for precious information . 1 . Mainly notation Let us thus begin by fixing some notation : FOURIER ANALYSIS OF HILBERT-SCHMIDT OPERATORS  18 3 0 - unit disk in the complex plane C . T - its boundary (the unit circumference) . dxdy idzdz dA(z) =  7r  =  2-  - normalized Euclidean area measure on 0 . dI(z) _ (1dA(I))2 - invariant (Poincaré) measure on A . dp a (z) = (a + 1)(1 - Iz1 2 )°dA(z) (a > -1) - Dzhrbashyan measure . L 2 ( 0 , [t,,) = the space of square integrable functions with respect to dit a . Aa~2(A) = Hol(A) n L 2 (á, /la) - Dzhrbashyan (or weighted Bergman) space, the space of holomorphic functions in .L2(0, pa) .' The corresponding norm and the corresponding inner product will be written 11 - IIa and ( ., .)a respectively . The space of H . S . operators on L 2 (A, p,~) can be identified with L 2 (0, pa ) L 2 (A, /t a ) =L 2 (A x A, ha), that is, a H .S . operator T = TF on L 2 ( 0 , /~ a) iS given by a kernel F(z l , z 2 ) such that IITFII 2 = LA IF(zI,z2)I 2 dha(zl)dUa(z2) < WG= SU(1,1) - MSbius group . G = PSU(1,1) = SU(1,1)/ i 1 - projective group ; the elements x of G are thus fractional linear functions of the type x(z) = e z + d , where ( c  d ) is in G, uniquely determined by x up to sign . G - universal covering group of G or G . It will be necessary to distinguish carefully between these groups . An element of G is an element x of G plus a choice of a determination of the square root P . Similarly, an element of G is an element of G plus a choice of a determination of the logarithm log x' (then we can define arbitrary powers of x') . dH(x) denotes the Haar measure on any of these groups . The group G acts on elements f (z) of L 2 (O,i¿a) or A" ,2 (A) according to the rule f(z) H f(x(z))(x (z))  (= f( áz + d )(cz + d)-(«+2)1 That these group actions are unitary follows from the formula díta(x(z)) = IXI(z)I«+2dMa(z) . (x EG) . 'In [JPR], p . 63 this space was called the Bargmann-Bergman-Besov-Dzhrbashyan-FisherFock-Segal space . As the referee suggests, one could as well have included the narres Petersson and Hecke too . The reason why we have given preference to the narre Dzhrbashyan here is that we think that hitherto one has not in the literature payed sufficient tribute to the achievements of the Armenian school of analysis . 18 4  J . PEETRE Similarly, a kernel F(zl, z2) experiences the change F(zi,zz) i -- > F(x(zl),x(z2»(x ' ( zl» 241 ( 77 (z2» 241  (x E G) . Our main concern is thus to decompose the space S2(L 2 (0, p, « )) of H .S . operators under the last action . Begin by writing 2 F(zl, z2) = K«(zl, zz)Fo(zi ) z2) where K«(zl, z2) _ (1-zlz2)-(«+2) i s the reprodcing kernel in A«, z( A) . Then Fo transforms as Fo(ZJ,z2) - Fo(x(zi),x(z2))  (x E GSlc!), this in view of the well-known formula ( 2 )  K«(x(zl),x(z2)»'(zj)  (! 7 ( - z - 2» 42 =K«(zi~zz)  (~ E G) . + 2  2 We agree to refer to Fo as the "reduced" kernel . Thus we may work with Fo instead of F . In terms of Fo the H .S . norm is given by IITFII 2 =  (1 - I zi l 2 )( 1 - 2Iz21 2 ) / «+2 IFO(zi z2)I 2 dI(z1)dI(z2)- f A (  I1-ziz2l Its invariant character is thus apparent . Clearly 2 . Invariant measures dJ=dI0dI is an invariant measure on A x0 . But, as the group G does not act transitively on this set, it is not unique . There are other interesting invariant measures on A x 0 or rather on A x A \ (diagonal), the manifold of all ordered pairs of elements of A (or oriented hyperbolic segments) . 2 0ne motivation for this construction comes from Berezin's program for quantization (cf . [P3]) . The referee comments that "the idea of factorization of automorphic forms goes back to many, many people, probably much before Berezin (Poincaré, for example)" . Let thus an ordered pair (z 1 , z 2 ), with z 1 E 0, z 2 E 0, be given . Consider the unique (oriented) geodesic through z1 and z2, directed from z2 to z1 . There is a unique transformation 1 in G such that its preimage is the interval (-1,1) and such that the preimages of z 1 and z 2 are two symmetrically situated (about the origin 0) points p1 and P2 on (0,1) and (-1, 0) respectively : (4)  z1 = «PI), z2 = E(p2)  (1 % pl =- p2 % 0) . Let A denote the exponential of the hyperbolic distance between z 1 and z 2 (Le . log A _  Ps  dx  _ 1 log 1 +p1 - 1 log 1 _  +p2 = log 1 +pl or  1 +pl ~  -  a =  ) . n1 1x 2 2 1pl 2 1 - p2  1 - p,  1 - pl Thus the pair (z1, z2) is uniquely determined by the pair (1, A), and the manifold of all ordered pairs gets identified with the product G x R+ . Notice that G acts on the first factor via multiplication from the left (see (4)) and trivially on the second factor (the hyperbolic distance is preserved), Le . t o (x(z1),x(z2)), where x E G, there corresponds the pair (x j, A) . It follows that an invariant measure on the set 0 x 0 \ (diagonal) is given by FOURIER ANALYSIS OF HILBERT-SCHMIDT OPERATORS  18 5 We can further identify the group G itself with the manifold of geodesics equipped with a base point . Let 11 and S2 be the endpoints of the geodesic on 18 6  J . PEETRE T and let m be the exponential of the hyperbolic distance from the base point to the Euclidean midpoint z* of the geodesic (= are of a Euclidean orthogonal circle) . We may regard (S1, (2, m) as coordinates for the corresponding group element 1 . Then the Haar measure on G can be written Let further m i and m2 measure the distance from z 1 and z2 can be rewritten as _ dJ _  1  m 2 - m2 dJ' 16,r2 m1In2 . Id(, ¡ . jd(2 j dm - 1( ;- ' (212  m . dJ' - id(11 - ld(21  drn l dm 2 1S1 - S2 12  m1  m2 A computation, which I defer to an appendix, reveals that the Radon-Nikodym derivative óf the measure (3) with respect to the one in (5) or (5'), is given by (A priori one can say that it must be a homogeneous function of degree 0 in m1 and m 2 .) If we put A = e°, where the quantity u thus has the interpretation of hyperbolic distance, then we have in (6) essentially the hyperbolic sine, sinh Q, which perhaps looks more convincing . 3 . Plancherel theorem For the group G this theorem tells us that under left multiplication by group elements (left regular representation) the Hilbert space L 2 ( G) decomposes into an orthogonal sum 00 00 = 82 (ñ - iX-1) .1 . L 2 (G) = 1 : ® D  , ® j : ® D  , ® C ® C m-1 m-1 to z* . Then (5) (the case of G is in [L] from which the present case G easily follows) . It consists thus of a "discrete" part and a "continuous" part . Here I will focus on the discrete part only, taking each summand separately . As D  , and D  , are each others conjugates it suffices t o consid er D  , . It is known that D  , is isometrically isomorphic te Aa,2(0) ® AQ ,2 (0) where Q = 2m - 2 . Notice that (in contrast to G) for the group G only even degrees occur! Thus D  , is not irreducible but contains A,3,2 (A) with infinite multiplicity . In the following discussion it is useful to bear in ones mind the following purely algebraic fact - essentially a consequence of Schur's lemma (cf . [JPW]) . Lemma . Le¡ V be any simple module (oven any C-algebra) and consider an irreducible submodule W of the direct sum V®V®V ® . . . (with elemns x1 ® x2 ® . . . ) . Then ¡he elements of W are of the form plx ®p2x ® . . . where xE V runs through this module and pl,p2, . . , are fixed complex numbers . The general form of an element of D  , is where '¿ thus is analytic in the first argument and anti-analytic in the second argument . In particular, D  , contains any function of the form with 0,0E A 0,2 (A) (cf . [L], p . 181) . If we fix i :~ 0 and let 0 vary we get a submodule of D  , isomorphic to AQ ,2 (A) . Choosing an orthonormal basis {Oj} in Aa'2(A) (e .g . the standard basis obtained by normalizing the monomials {zl}) we decompose D  , into a direct sum of irreducible submodules . Next pick an orthonormal basis {Bi} also in the space L 2 ((1, oo), A,,), where (Notice that they are of the form FOURIER ANALYSIS OF HILBERT-SCHMIDT OPERATORS  187 f(f) - J  ~(f(t), ~(f~(t))md~a(t), 0 .f(1) = f 0(1(t)ft)(j'(t))mdM#(t) 0 2  =  2ñ  ) 2(a+2)  dñ 1s~  dA~(a)  52 + 1  (a - IX -1 ) 1\ . with Oj(t) E AP 2(A) . We then obtain ( 1 - Iz1I 2 )(1-Iz2I 2 ) _  2 ,  / 2 ) 11 - z1Z21 2 ~ -F 1 Then for every H .S . operator TF the corresponding reduced kernel F o (see Sec . 1) admits the decomposition FO(z1, z2) _  A=(1)Bi(A) . i Let us restrict our attention to the case when all the Ai belong to D  that is, Aj(j(t)) = f ~Y'ij(S(t))Y'j(t)(S~(t))mdhQ(t) 7 FO(z1,z2) _  ~i ;(~(t)) ;(t)( ~(t)) dl~Q(t)Bi(~) . i>j 18 8  J . PEETRE Invoking the lemma (or rather the general philosophy behind it) we see that the most general irreducible submodule is gotten by taking all the O=i proportional to one and the same function 0 in A Q,2 (0), that is, O ;j = pijo for suitable scalar pi7 . Such an irreducible submodule is thus spanned by reduced kernels of the form with a(t, A) = Epii0 .i(t)Bi(A) . i,J By change of variable (see (1)) this formula can be written that is, where with (8) similarly for A o : FO(zl, z2) = L O(e(t))a(t, A)(1'(t))'dpp(t) 0 Fo(zl,z2)=  (t)a( -1 (t),~)(( _1)~(t))mdf~a(t), 0 (7)  Ao(z1, z2, t) = a(17 1 (t), ~)((~-1)~(t))~` . Reintroducing the reproducing kernel we can finally write t he kernel itself as A(z1, z2, t) = K«(z1, z2)Ao(z1, z2, t) . Notice that the kernel A obeys the transformation law (cf . (2)) A(x(zl ), x(z2), x(t)) = A(z1, z2, t)(x'(z1))241(x7(z2))~ (7'(t)) -m l Direct check that Such a kernel defines an invariant submodule : FO(X(zl), x(z2)) = f O(t)Ao(x(zl), x(z2), t)dpp(t) 0 Fo(z1, 21 2) =  O(t)Ao(zl, z2, t)dpa(t) 0 F(z1, z2) = L O(t)A(zl, z2, t)dPP(t) 0 Ao(x(zl), x(z2), x(t)) = Ao(zl, z2, t)(x~(t))-~ O(x(t»Ao(x(z1), x(z2), x(t»ditp(x(t» O(x(t))(x'(t))'Ao(z1, z2, t)I X I (t)I-2'dhp(x(t)) v = f o(t))w(t))'Ao(z1, x2, t)dNtp(t), 0 where we in the last step once again involved (1) . Remark . In hindsight we could perhaps have written down this formula right at the beginning, but we find it more instructive to follow the path of discovery . FOURIER ANALYSIS OF HILBERT-SCHMIDT OPERATORS  189 4 . Examples Example 1 . m = 1 . Conventional big Hankel operators [A], [AFP] correspond to the kernel jz2 F(z1,z2) = K .(z1,z2) J  ~(C)dC _ i (If we introduce the primitive b of 0 (Le . b' = 0) we get the usual expression F(z1, z2) = K .(zl, z2)(b(z1) - b(z2»-) Check of the covariance : (=1) Fo(x(zl),x(z2» = f z(z=2) = O(t)dt form (7) . Indeed, one has to take z1 - z2 A O( z 1, z2) _ (1 - z1~2(1 - z2í)2 . «x(t))x'(t)dt . But this is not of the form (7) . It is however easy to rewrite the formula in the A(z 1 , z2, t)  (1 - (12  ~~ 1 . z,  1  1 2  Zl 1  1  1  z 1 - z2 = t C1-z1t1 -z2t~ _ (1-zl~(1-z2 t) . Example 2 . m= 2 . We have the (big) "Lagrange-Hankel" operator [BJP], [P1] corresponding to Fo(z1, z2) = b ' ( zl) -}- b'(z2) - 2 b(z 1 ) - b(z2) . zl - z2 To get the transition to our present form of notation we invoke the third order derivative b"' = 0 of the given b . (This is in accordance with the requirement of "Ból's lemma" [GP] .) Then we find F+ O ( zl z2 ) _ j  (z1 - O( z 2 - ~) ~(C)dC x(=2)  zl - . z2 As in Example 1 one can write down the corresponding kernel A o , The method in [P1] is however preferable, and applicable also for m > 2 . 19 6  J . PEETRE On the other hand But (as k - k( 2 ) (1 Thus and 2  ( 2 ikm1 2 _ 1 + Ik1 2 m 2 - 2Reikm(1(2 IxI  I1 - ikm1 2 1 + k2 m 2 - 2Reikm 1 - 1x12 = 2Re(ikm((1(2 - 1)) ik 2 ik(L - 1)  ik((2 - (i)( 1 + S2 1 ) - 2ik((2 - (1) . (1  (2  (1  (1 (2  (1 It follows that (1 - IZ12)2 = 41(1 - (212m2 (1 - 1x112)2(1 - 1x212)2  I1 16 1( km11(214mikrn2l4 . i  2  dx l dz l dx 2 dz2  _  1  m 2 - _  _  Id(, 1 - 1d(21 _dml dm2 (27r)  x (1 - 1x 11 2 )2( 1 - 1x212)2  167r 2 m 1 m2  1(1 - (21 2  m 1 m2 as previously claimed (Sec . 2, formula (6)) . References [A] S . AXLER, The Bergman space, the Bloch space, and commutators of multiplication operators, DukeMath . J . 53 (1986), 315-332 . [AFP] J . ARAZY, S . FISHER AND J . PEETRE, Hankeloperators on weighted Bergman spaces, Amer . Math . J . 110 (1988), 989-1053 . [BJP] J . BOMAN, S . JANSON AND J . PEETRE, Big Hankel operators of higher weight, Rend . Circ . Mai . Palermo 37 (1988), 65-78 . [GP] B . GUSTAFSSON AND J . PEETRE, Notes on projective structures on complex manifo1ds, Nagoya Math . J . 116 (1989), 63-88 . [JP1] S . JANSON AND J . PEETRE, A new generalization of Hankel operators (the case of higher weights), Math . Nachr . 132 (1987), 313-328 . [JP2] S . JANSON AND J . PEETRE, Paracommutators - boundedness and Schatten-von Neumann properties, Trans . Am . Math . Soc . 30 5 (1988), 467-504 . FOURIER ANALYSIS OF HILBERT-SCHMIDT OPERATORS  197 [JPR] S . JANSON, J . PEETRE AND R . ROCHERG, Hankel forms and the Fock space, Revista Mat . Ibero .Amer . 3 (1987), 61-138 . [JPW] S . JANSON, J . PEETRE AND R . WALLSTÉN, A new look on Hankel forms over Fock space, Studia Math . 95 (1989), 33-41 . [L] S . LANG, "SL 2 (R)," Addison-Wesley Publishing Co ., Reading, Mass . - London - Amsterdam, 1975 . [P1] J . PEETRE, "Hankel kernels of higher weight for the ball," Technical report, Lund, 1988 . [P2] J . PEETRE, Paracommutators - a brief introduction, open problems, Revista Mat . Univ . Complutense Madrid 2, número suplementario (to appear) . [P3] J . PEETRE, The Berezin transform and Ha-plitz operators, J . Operator Theory (to appear) . [P4] J . PEETRE, Some unsolved problems, In : Colloquie Mathematica Societatis Janos Bolyai, Alfred HaarMemorial Conference, Budapest (Hungary), 711-735, 1985 . North-Holland, Amsterdam, 1986 . [R] G . RIDEAU, Sur la réduction du produit tensoriel des répresentations de la série discréte de SL(2, R), Ann . Inst . H . Poincaré Seci . A . 4 (1966), 67-76 . [UU] A . UNTERBERGERAND J . UNTERBERGER, La série discréte de SL(2, R) et les opérateurs pseudo-différentiels sur une semi-droite, Ann . Se¡ . Ecole Norm . Sup . 17 (1984), 83-116 . [W] J . WILSON, Some hypergeometric orthogonal polynomials, SIAM J . Math . Anal . 11 (1980), 690-701 . Keywords . Hankel operator, reproducing kernel, decomposition of group representations, Hilbert-Schmidt operator 1980 Mathematics subject classifications : 47B35, 30C40, 43A65, 43A80 Matematiska Institutionen Stockholms universitet Box 6701 S-113 85 Stockholm SWEDEN Rebut el 16 de Juny de 1989