New Hilbert Spaces from Oldt

Transcription

1 New Hilbert Spaces from Oldt Donald Sarason It is a cliche that Hilbert spaces are often useful in connection with problems whose initial formulations do not mention them. A familiar case in point is the classical moment problem [3J. It is also a cliche that, in attacking problems about Hilbert spaces and Hilbert space operators, one is often led to construct new Hilbert spaces. Occasionally, one of these new Hilbert spaces originally lies hidden, lurking in the shadows, and, once revealed, makes possible an unexpectedly elegant solution. Don Sarason, 1985 My aim in this article is to illustrate the preceding remarks through a discussion of unitary dilations, a subject originated by Paul Halmos. The first section contains the basic facts about unitary dilations, including the existence theorem of B. Sz.-Nagy. The second section concerns one of the central results of the subject, the lifting theorem of Sz.-Nagy and C. Foias. An elegant proof of the lifting theorem due to R. Arocena will be presented. In what follows, all Hilbert spaces mentioned are assumed to be complex and separable. All operators are assumed to be linear and bounded. The symbol I will denote the identity operator; the space on which it acts will be clear from the context. Unitary Dilations If a is a complex number of mudulus at most 1, then a can be made the (1,1) entry of a two--by-two unitary matrix, for example, the matrix tto Paul Halmos, for caring about mathematics and about people.

2 196 Donald Sarason New Hilbert Spaces from Old 197 In one of his first papers on operator theory [5], Halmos observed that the preceding construction, if suitably interpreted, applies not only when a is a number but also when it is a Hilbert space contraction. Namely, if T is an operator of norm at most 1 on the Hilbert space H, then (as will be explained shortly) the two-by-two operator matrix U _- [ (I - T*T)1/2 T (I - -T* TT*)1/2J defines a unitary operator on the direct sum If EElH. (The square roots here are the positive square roots of the given positive operators.) If one identifies H with the first coordinate space in H EBII, one sees that one can produce the action of T on a vector in H by first applying U to the vector and then projecting back onto H. In Halmos's terminology, U is a dilation of T, and T is a compression of U. To prove that U is unitary one verifies by direct calculation that U*U and UU* both equal the identity on HEElH. The verification uses the equality T*U -TT*)1/2 = (I -T*T)1/2T*, which Halmos deduced starting from the obvious relation T*U - TT*) = (I - T*T)T*. From the latter relation one infers that T*pU - TT*) = p(i - T*T)T* for every polynomial p. If (Pn)l is a sequence of polynomials converging uniformly in the interval [0,1] to the square-root function, then Pn(I - TT*) tends in norm to (I - TT*)1/2 and Pn(I - T*T) tends in norm to (I - T*T)1/2. In the limit one thus obtains the desired equality. Shortly after Halmos's paper appeared Sz.-Nagy [11]extended his ideas. Suppose T is a contraction operator on the Hilbert space H. According to the theorem of Sz.-Nagy, there is a unitary operator U acting on a Hilbert space H' containing H as a subspace such that, not only is U a dilation of T, but also un is a dilation of Tn for every positive integer n. The Sz.-Nagy dilation is more intimately related to T than the one of Halmos. It has been referred to as a strong dilation, and as a power dilation; usually, though, now that its study has developed into a branch of mathematics in its own right [12], it is called simply a unitary dilation of T. The dilation of Sz. Nagy can be taken to be minimal, that is, not admitting a decomposition into the direct sum of two unitary operators the first of which is also a dilation of T. Such a minimal dilation is essentially unique and so is called the unitary dilation of T. A few simple examples will serve to illustrate Sz.-Nagy's theorem. Let L2 denote the usual Lebesgue space relative to normalized Lebesgue measure on the unit circle. It has a natural orthonormal basis, the Fourier basis, consisting of the functions en(z) = zn (n E Z). Let{,~ (usually called H2) be the closed linear span in L2 of the functions en "vith n ~ 0 and L~ the span of the functions en with n < O. The bilateral shift is the operator 5 on L2 of multiplication by the independent variable I:(51)(z) = zf(z )]. The subspace L~ is invariant under 5; the restriction of 5 to L~ is called the unilateral shift and will be denoted by S+. Let 5_ denote the compression of 5 to L:", or, what amounts to the same thing, the adjoint of the restriction of 5* to L~; it is unitarily equivalent to 5-'t, the so-called backward shift. It is then very easy to verify that 5 is the minimal Sz.-Nagy dilation of 5+ and 5_. It is also the minimal Sz.-Nagy dilation of the zero operator on the one-dimensional subspace spanned by the basis vector eo. (Even when the original contraction T (j,ctson a finite-dimensional space, its Sz.-Nagy dilation acts on an infinite-dimensional one, except in the trivial case where T is a finite-dimensional unitary operator.) If one sanguinely assumes the Sz.-Nagy dilation exists and proceeds to analyze it, one learns from the analysis how to construct it. Thus, suppose T is a contraction on the Hilbert space Hand U is its minimal Sz.-Nagy dilation, acting on the Hilbert space H' containing H as a subspace. The orthogonal projection in H' with range H will be denoted by P. Let H+ be the closed invariant subspace of U generated by H; it is the closed subspace of H' spanned by the vectors Unx with x in Hand n = 0,1,2,... Let Hrj = H+ e H, the orthogonal complement of H in H+. It is asserted that the subspace Hrj is invariant under U. To verify this it is enough to check that the vector U(I -P)Unx is orthogonal to H whenever x is in Hand n = 1,2,... [This is so because the vectors (I - p)unx, with x and n as described, span H[j.J If y is in H then the inner product (U(I - P)Unx, y) is the difference of two terms. Because x = Px and y = Py, those terms can be written as (PUn+1 Px, y) and (PU pun Px, Yl' But each of the two preceding inner products equals (Tn+lx, y), so U(I _p)unx is orthogonal to H, as desired. The operator U thus maps Hrj isometrically into itself. Moreover, the restriction U I Hrj must be a pure isometry, that is, it has no invariant subspace on which it acts as a unitary operator. (This follows because H+ is the smallest U-invariant subspace containing H.) The purity of the isometry U I Hrj means it is a shift. Namely, if we let L+ = Hrj e UHrj, then we can write Hrj as the orthogonal direct sum of the subspaces un L+, n = 0,1,2,... (Halmos's paper [6] contains a lucid discussion of these matters. The subspace L+ is called by him a wandering subspace of the operator U.) The subspace Ho = H' eh+ is invariant under U*, being the orthogonal complement of the U-invariant subspace H+. From the minimality of the dilation U one infers that the restriction U* I Ho is a pure isometry, so that He; is the orthogonal direct sum of the subspaces u*n L-, n = 0,1,2,..., where L - = Ho e U* He;. The U invariance of the subspace Hrj implies that the subspace H- = H' e Hrj is U* invariant; it is the U*-invariant subspace generated by H and is the orthogonal direct sum of H and He;. The whole space H' is the orthogonal direct sum of H, Ht, and He;. We introduce the defect operators DT = (I - T*T)1/2 and DT* = (I TT*)1/2, and the corresponding defect subspaces 1)T and 'DT*, the closures of the ranges of DT and DT*, respectively. If x is a vector in H then, as one easily sees, the vector U x - Tx lies in the subspace L +; the square of

3 If)8 Donald Sarason New Hilbert Spaces from Old 199 its norm equals IIxl12-IITxI12, which is the same as IIDTxI12. We thus have a natural isometry of the subspace DT into the subspace L +, defined on vectors in the range of DT by DTX ~ U X - T x. The range of this isometry is in fact all of L+. Indeed, one can obtain a set of vectors that spans L+ by projecting onto L + all of the vectors Unx with x in Hand n = 1,2,... Eut n Unx = Tnx + L Un-k(UTk-Ix - Tkx). k=1 'The first term on the right is in H, and the kth term in the sum is in Un-k L + and so is orthogonal to L + for k = I,...,n -1. The projection of Unx onto L+ thus equals UTn-Ix-Tnx, showing that the vectors Ux-Tx with x in H already span L +. The isometry defined above thus maps 'DT onto L + and so provides a natural identification between those two spaces. By similar reasoning, there is a natural isometry of the subspace 1)T' onto the subspace L-; it is defined on vectors in the range of Dr- by DT"X ~ U*x - T*x. We now have a fairly good picture of the action of the operator U. On tbe subspace Hit it acts as a shift, sending each of the mutually orthogonal sllbspaces unl + (n = 0,1,2,... ) to the next one. On the subspace He; it acts as a backward shift, sending each of the mutually orthogonal subspaces U *n L - (n = 1,2,... ) to the preceding one, and mapping L - itself into the orthogonal complement of He;. If x is in H, then the difference Ux - Tx lies in L+ and has the same norm as does DTX. To fill out the picture, we need to understand better how U acts on the subspace L -. The vectors U*x - T*x with x in H form a dense subspace of L -. The image of U*x - T*x under U is x - UT*x, which is the difference between x - TT*x and UT*x - TT x. The first of the two preceding terms equals DT" DT" x; it is, in other words, the image under Dr" of the vector in 1)T" that corresponds to U*x - T*x under the natural isometry between L and 'Dr". The second term, UT* x - TT' x, lies in L+; under the natural isometry between L+ and 'Dr it corresponds to the vector DrT*x, which is the same as T* DT'x [by the equality (I - T*T)I/2T* = T*(I - TT*)1/2, mentioned earlier]. To summarize and paraphrase: the image under U of a vector in L- has two components, one in H and one in L+; if the original vector corresponds to the vector Y in 1)r' under the isometry between L- and Dr", then the component of the image in H is Dr' y, and the component of the image in L+ corresponds, under the isometry between L+ and DT, to the vector -T*y. While the discussion above was premised on the existence of the minimal Sz.-Nagy dilation of the contraction T, the information obtained can be interpreted as a set of instructions for the construction of the dilation. Following these instructions, we let Hn be a copy of the defect space 'Dr for n = 1,2,... and a copy of the defect space Dr- for n = -1, -2,... We also let Ho = H, and we let H' be the direct sum of all the spaces Hn. We define the operator U on H' by letting it take the vector E':"oo EBxn to the vector 2:::':"00 EBYn given by Lifting Theorem (i) YUI (iv) IWII = IIXII Yn = Txo + Dr-X-I { Xn-I DTXo - T'X_I = U2Y; (ii) X = P2Y I HI; (iii) YH{ c Hi, and YH{o c Hio;,, for n:;:; -1 or n 2 2 for n = 0 for n = 1 A straightforward verification shows that this U is unitary and that it is the minimal Sz.-Nagy dilation of T. The uniqueness of the dilation is established by the preceding analysis. The analysis shows that a unitary equivalence between two contractions extends to a unitary equivalence between their minimal Sz.-Nagy dilations. (This is the precise statement of the uniqueness part of Sz.-Nagy's theorem.) The construction just given can be found in the book of Sz.-Nagy and Foias [12]. It refines a proof of Sz.-Nagy's theorem due to J.J. Schaffer [10]. The original proof of Sz.-Nagy was less direct. The lifting theorem of Sz.-Nagy and Foias says that any operator commuting with a Hilbert space contraction T is the compression of an operator commuting with the minimal unitary dilation of T. More generally, it describes the operators that intertwine two contractions. Suppose TI and T2 are contractions on the respective Hilbert spaces H 1 and H2. For j = 1,2 let Uj be the minimal unitary dilation of Tj, acting on the Hilbert space Hi containing Hj as a subspace. The orthogonal projection in Hi with range Hj will be denoted by Pj, and the other notations from the analysis in the first section will be carried over, with the appropriate subscripts appended. (Thus, in place of H+ we now have Hi, etc.) The inner product in Hi will be denoted by (-, -)j. Suppose X is an operator from HI to H2 that intertwines TI and T2, in other words, that satisfies XTI = T2X. The lifting theorem then states that there is an operator Y from H~ to H~ satisfying the following conditions: In the case where TI = T2, condition (iii) guarantees along with condition (ii) that Y is a dilation of X in the sense of Sz.-Nagy. The very special case of the lifting theorem where Tl = S+ and T2 = S_ is especially interesting. In this case we can take H~ = H~ = L2 and

4 200 Donald Sarason New Hilbert Spaces from Old 201 UI = U2 = S. Since now Hio is trivial and Hi is all of L2, condition (iii) becomes automatic. ' An operator from L~ to L=- that intertwines S+ and S_ is called a Hankel operator. The matrix of such an operator, relative to the usual orthonormal bases for L~ and L=-, has constant cross diagonals, a property that characterizes Hankel operators. The entry on the nth cross diagonal is the Fourier coefficient with index -n of the image under the operator of the constant function l. If X is a Hankel operator then, according to the lifting theorem, X can be obtained from an operator Y on L 2 that commutes with S and has the same norm as does X; to apply X to a vector in L~, one first applies Y and then projects onto L=-. It is an elementary result that the operators on L2 that commute with the bilateral shift S are just the multiplication operators induced by the functions in LOO of the unit circle, and that the norm of such a multiplication operator equals the essential supremum norm of the inducing function. Thus, a Hankel operator X equals a multiplication operator followed by a projection. The function that induces the multiplication operator must clearly have for its Fourier coefllcient of index -n (n = 1,2,... ) the entry on the nth cross diagonal of the matrix for X. The upshot is the following theorem of Z. Nehari [7]: Let Cl, C2, C3,.. complex numbers. Then the matrix Cz C3. C lcicz C3... j induces a bounded operator on 2 if and only if there is a function 1in Loo of the unit circle such that Cn = j (-n) (the Fourier coefficient of f with index -n) for all positive integers n. In the case of boundedness there is such an f with equal to the norm of the operator. Nehari's theorem solves the following one-sided trigonometric moment problem, now usually called the Nehari interpolation problem: Given complex numbers CI, Cz,..., when does there exist a function 1in the unit ball of Loo such that j( -n) = Cn for all positive integers n? The answer, according to Nehari's theorem, is that such an 1exists if and only if the matrix above has norm at most 1 as an operator on z. The Nehari problem contains as special cases the classical interpolation problems of Nevanlinna-Pick and CaratMorody-Fejer; one will find more details on this in [9]. The general lifting theorem can be applied to a host of interpolation problems, many of which are discussed in the book of M. Rosenblum and J. Rovnyak [8]. V.M. Adamyan, D.Z. Arov, and M.G. Krein [2]discovered an approach to Hankel operators and the Nehari problem that is analogous to the operatortheoretic approach to the classical moment problem. In one respect, though, it is more subtle than the latter approach, for it involves a Hilbert space that initially lies hidden. be We owe to Arocena [4] the realization that the Adamyan-Arov-Krein approach can be adapted to yield a proof of the Sz.-Nagy-Foias theorem. This article ends with a presentation of Arocena's proof. Without loss of generality we assume the operator X that intertwines the contractions TI and T2 is of unit norm. We form the algebraic direct sum of the two subspaces Hi and Hi", which we denote by Ko. To avoid confusion between orthogonal direct sum and algebraic direct sum, we use the symbol -+- to denote the latter: Ko = Hi -+- Hi". On Ko we introduce an inner product, denoted by (-, ')0, by setting (Xl + Xz, YI + Y2)O = (Xl, Ylh + (xz, yz)z + (XPIXI,YZ)z + (XZ,XPIYI)z. The assumption that IIXII = 1 guarantees that this inner product is positive semidefinite: (Xl + Xz, Xl + XZ)o = Ilxllli + Ilxzll~ + 2 Re (X PIXI, xz)z ~ IlxIili + Ilxzll~ - 211x111Iilxzllz ~ o. The space Ko is a so-called pre-hilbert space. We can produce a Hilbert space from it in two steps. First we form the quotient space Ko/N, where N is the subspace of Ko consisting of the vectors having self-inner product O. The quotient space Ko/N is a bonifide inner product space, and its completion, which we denote by K, is thus a Hilbert space. (It is our "new" Hilbert space.) We denote the inner product in K by (.,.). The space Hi is identified in the obvious way with a subspace of Ko, and this identification is isometric, in other words, the inner product of two vectors in Hi is the same when they are regarded as vectors in Ko as when they play their original roles as vectors in Hi. Thus, we can identify Hi with a subspace of the Hilbert space K. The same is true of Hi". The natural injections of Hi and Hz- into K will be denoted by WI and Wz, respectively. The next step is to define a certain isometry Vo in the space K. In preparation we verify that, if Xl and xz are vectors in Hi and Hi", respectively,.. then IlxI + U2'xzllo = IIUlxl + xzllo. The equality depends on the intertwining relation XTI = T2 X. We have IIUIXI + xzl16 = IIUlxllli + IIxzll~ + 2Re (XPIUIXI,XZ)z. The inner product on the right-hand side does not change if we replace Xz by Pzxz (obviously) and Xl by PIXI (since UI sends the subs pace Hi 8HI into itself). It therefore equals (XTIPIXI, PZX2)z, and hence equals (T2X PIXI, P2XZ)2. The last inner product equals (X PIX}, P2U'; PZX2)z' Here we can replace P2X2 by Xz (since U:; sends the subspace Hi" 8Hz into itself), so

5 202 Donald Sarason New Hilbert Spaces from Old 203 we finally obtain IlU1Xl + x2/15= IIUlxIili + Ilx211~+2Re(XPlxl,P2U;X2h = Ilxllli + IIU;x211~+ 2Re (X PIXI, U;X2)2 = IIxl -+- U;x211~, as desired. We now define the operator Yo. Its domain, D(Vo), is to be the subspace of K that corresponds to the subspace Hi -+- U2 Hz of Ko; in other words, D(Vo) = WIHt + W2U2 Hz. Its range, R(Vo), is to be the subspace of K that corresponds to the subspace UIHt -+- H:; of Ko; in other words, R(Vo) = WIUIHi + W2H:;. The definition is VO(WIXI + W2U;X2) = WIUIXI + W2X2 (Xl E Hi, X2 E Hz). The calculation above shows that Vo is an isometry. From the definition one sees that Wj intertwines Uj and Vo to the extent p03sible: WIUIXI = VQWIXI for Xl in Hi, and W2U2X2 = VOW2X2for X2 in U2'H:;. Next, we take a unitary extension of Vo. This is a standard construction. First we extend Vo by continuity to an isometry from the closure of D(Vo) onto the closure of R(VO). Then we take a larger Hilbert space K' containing K as a subspace such that the orthogonal complements of D(Vo) and R(VO) in K' have the same dimension. [If the orthogonal complements of D(Vo) and R(Vo) in K itself happen to have the same dimension, we can take K' = K but are not obliged to do so. If those orthogonal complements have different dimensions, then K' e K must be infinite dimensional. ] If we take any isometry of K' e D(Vo) onto K' e R(Vo) and use it to extend VO, we get the desired unitary extension, which we shall denote by V. The injections WI and W2 extend naturally to isometries of H~ and H~ into K' that intertwine V with Ul and U2, respectively. Consider first WI. Let K I be the closed invariant subspace of V* generated by WI Ht. Since WIHi is already invariant under V (by the intertwining relation connecting Vo, Ul, and Wd, the subspace Kl is V-invariant as well as V*-invariant, so the restriction V I KI is a unitary operator. In KI we consider the subspace WIHI, the natural image of HI, and the transplantation to WIHI of the operator TI' namely, the operator WITI Wi- An examination of our construction shows that V I Kl is a minimal unitary dilation of WITI wt. The uniqueness part ofsz.-nagy's theorem thus yields the desired extension of WI to an isometry, which we also denote by WI, of H~ onto KI that intertwines Ul and V: WIUl = VWl. To obtain a similar extension of W2 we apply the same reasoning, but with T2', U;;" and\!* in place of TI' UI, and V. [The switch to adjoints is made because D(VO-l) contains W2H2 while D (Vo) possibly does not.] One gets an isometric extension of W2, also called W2, from H~ into K' that intertwines U2' and V*; W2U; = V*W2. Now, believe it or not, the proof of the lifting theorem is all but finished, for the operator Y = W2'WI has all of the required properties. The inequality IIYII ;; 1 is obvious because WI and W2 are isometries; that equality actually holds will follow once the relation between Y and X has been established. The equality YUl = U2Y follows immediately from the two equalities WIUI = VWl and W2'V = U2W2'. To establish the remaining properties we take a vector Xl in Hi and a vector X2 in H:; and note that (YXl,X2)2 = (WIXI, W2X2) = (Xl + 0,0 + X2)O = (X PIXl, X2)2. If Xl is in HI and X2 is in H2, the left-hand side can be rewritten as (P2YxI,X2h and the right-hand side as (XXl,x2b, from which it follows that X = P2Y I HI. If X2 is in H:;o the right-hand side vanishes, from which it follows that Y Hi c H:J'. If finally Xl is in Hio the right-hand side again vanishes, from which it follows that Y Hio C 'Hio. The lifting theorem is completely proved.., The reader may have noticed that one can recast Arocena's proof in the language of scattering theory, referring to WI and W2 as wave operators and W; WI (= Y) as a scattering operator. The proof fits into the scatteringtheoretic approach to unitary dilations developed by Adamyan and Arov [1]. REFERENCES 1. V.M. Adamyan and D.Z. Arov, On unitary couplings of semi-unitary operators, Am. Math. Soc. Trans I. Ser. 2, 95 (1970), V.M. Adamyan, D.Z. Arov, and M.G. Krein, Infinite Hankel matrices and generalized problems of Caratheodory--Fejer and 1. Schur, Funkcional. Anal. Prilozhen. 2,4 (1968), 1~ N.!. Akhiezer, The Classical Moment Problem, Hafner, New York, R. Arocena, Unitary extensions of isometries and contractive intertwining dilations, Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, 1989, Vol. 41, pp. 13~ P.R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125~ P.R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102~1l2. 7. Z. Nehari, On bounded bilinear forms, Ann. Math. 65 (1957), 153~ 162.

6 204 Donald Sarason 8. M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, New York, D. Sarason, Moment problems and operators in Hilbert space, Moments in Mathematics, Proc. Symp. Appl. Math., Am. Math. Soc., Providence, R.I., 1987, Vol. 37, pp J.J. Schaffer, On unitary dilations of contractions, Proc. Am. Math. Soc. 6 (1955), B. Sz.-Nagy, Sur les contractions de l'espace de Hilbert, Acta Sci. Math. 18 (1953), B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Space, North-Holland, Amsterdam, Department of Mathematics University of California Berkeley, CA 94720