PARTS OF ADJOINT WEIGHTED SHIFTS

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1 J. OPERATOR THEORY 74:2(205), doi: /jot.204jun Copyright by THETA, 205 PARTS OF ADJOINT WEIGHTED SHIFTS ANDERS OLOFSSON In memory of Ulri Tibell Communicated by Niolai K. Niolsi ABSTRACT. We characterize the parts of a vector-valued adjoint shift operator using operator inequalities and a stability condition. The characterization applies in a class of eighted shifts containing the classical Hardy shift operator as ell as the full scale of standard eighted Bergman shift operators on the unit disc. Related operator models form an integral part in recent or on generalized mathematical systems theory and Bergman inner functions. KEYWORDS: Operator model theory, eighted shift operator, standard eighted Bergman space. MSC (200): Primary 47A5, 47B32; Secondary 30H20, 46E22.. INTRODUCTION Let E be an auxiliary Hilbert space and let = { } be a positive eight sequence such that 0 = and lim inf =. We consider the space A (E) of all E-valued analytic functions / (.) f (z) = a z, z D, in the open unit disc D ith finite norm (.2) f 2 = a 2. It is straightforard to chec that the space A (E) is a Hilbert space of E-valued analytic functions in D in the usual sense of bounded point evaluations A (E) f f (ζ) E at points ζ D. For notational simplicity e rite A (D) = A (C) in the case of complex-valued analytic functions.

2 250 ANDERS OLOFSSON The reproducing ernel function for the space A (E) is the operator-valued function K : D D L(E) defined by the reproducing property that K (, ζ)e A (E) and f (ζ), e = f, K (, ζ)e for all ζ D, e E and f A (E). Here the symbol L(E) denotes the space of all bounded linear operators on the Hilbert space E. Notice that existence of the ernel function K is ensured by existence of bounded point evaluations as follos by Riesz representation theorem. It is straightforard to chec that the ernel function has the form K (z, ζ) = (ζz)i E, (z, ζ) D 2, here the function is given by the formula (z) = z, z D, and I E is the identity operator on E. Another canonical object often associated ith the space A (E) is the shift operator hich is the operator S = S defined by (S f )(z) = z f (z) = a z, z D, for f A (E) given by (.). A straightforard calculation using (.2) shos that S 2 = sup +. We shall be concerned ith the case hen the operator S is a contraction hich equivalently means that the eight sequence = { } is decreasing: + for 0. Observe also that the operator S is a contraction if and only if the function ( ( z) (z) = + ) z, z D, has non-negative Taylor coefficients. A calculation shos that the adjoint shift operator S acts as (.3) (S f )(z) = + a + z, z D, on functions f A (E) given by (.). In particular, the adjoint shift acts on reproducing elements by (.4) S K (, ζ)e = ζk (, ζ)e for ζ D and e E. A special property possessed by the adjoint shift operator is that lim S f = 0 in A (E) for every f A (E). It is also evident that the operator S is a contraction since S is. We shall be concerned in this paper ith a characterization of the parts of the adjoint shift S. Recall that an operator A

3 PARTS OF ADJOINT WEIGHTED SHIFTS 25 L(H) is called a part of an operator B L(K) if the operator A is unitarily equivalent to the restriction of B to one of its invariant subspaces. Observe that A L(H) is part of B L(K) if and only if there exists an isometry V L(H, K) from H into K such that VA = BV. The simplest instance of the above construction is the vector-valued Hardy space H 2 (E) hich corresponds to the constant eight sequence = for 0. The ernel function for the space H 2 (E) is the function S(z, ζ) = ζz I E, (z, ζ) D 2, hich is often called the Szegö ernel. The shift operator S on H 2 (E) is non as the Hardy shift. A special property of the Hardy shift is that it is an isometry in the usual sense that S f 2 = f 2 for f H 2 (E). The shift operator is also pure in the sense that S (H 2 (E)) = {0}. The so-called Wold decomposition asserts that these to properties determine the Hardy shift operator up to unitary equivalence. The Wold decomposition then leads to a parametrization of the shift invariant subspaces of H 2 (E) using operator-valued inner functions as as pointed out by Halmos [5]. An interesting property of the adjoint Hardy shift S is that it constitutes a universal model for contractions in the class C 0 : A bounded linear Hilbert space operator T L(H) is a contraction such that lim T x = 0 in H for every x H if and only if it is unitarily equivalent to the restriction of S to a coinvariant subspace of H 2 (E) for some multiplicity E as above. To put this result in explicit form e need some more terminology. By a contraction e understand a bounded Hilbert space operator T L(H) such that Tx 2 x 2 for all x H. A contraction T is said to belong to the class C 0 if lim T = 0 in the strong operator topology in L(H), that is, lim T x = 0 in H for every x H. The defect operator for a contraction T is the operator D T defined by the formula D T = (I T T) /2 in L(H), here the positive square root is used. The defect space D T is the closure in H of the range D T (H) of D T. For x H e consider the D T -valued analytic function (.5) V x(z) = D T (I zt) x = (D T T x)z, z D, in D. Let T L(H) be a contraction. For x H, the sequence { T x 2 } is decreasing and non-negative. By polarization this gives that the operator limit

4 252 ANDERS OLOFSSON lim T T exists in the ea operator topology in L(H) and, by a classical result, therefore also in the strong operator topology in L(H) (see Halmos, Problem 20 of [6]). We no introduce the positive operator Q L(H) characterized by the formula (.6) Q 2 = lim T T in L(H). Notice that Q = 0 if and only if T belongs to the class C 0. The space Q is defined as the closure in H of the range Q(H) of Q. Using (.6) it is straightforard to chec that Qx 2 = QTx 2 for x H. This last norm equality gives that the map U : Qx QTx is ell-defined and extends uniquely by continuity to an isometry U L(Q) such that UQ = QT. A version of the universal model property of the adjoint Hardy shift S no says that the map [ ] [ ] V V x V = : x Q Qx defined by (.5) and (.6) is an isometry mapping H into the space H 2 (D T ) Q such that [ ] S 0 VT = V, 0 U here the operator U L(Q) is as above. Results of this type goes bac to Rota [29], de Branges and Rovnya [], Sz.-Nagy and Foias ([33], Section I.0), and others, and have been of interest in several branches of operator theory such as dilation theory, characteristic operator functions and mathematical systems theory (see for instance the survey Ball and Cohen [0]). Related operator models form an integral part in or on constrained von Neumann inequalities by Badea and Cassier [7]. We mention here also the St. Petersburg school and the related study of so-called K Θ model subspaces (see for instance Niolsi [22]). An interesting scale of spaces much studied over the past thirty years is the scale of so-called standard eighted Bergman spaces for the unit disc. For α > e denote by A α (D) the space of all analytic functions f in D ith finite norm f 2 α = f (z) 2 da α (z), here D da α (z) = (α + )( z 2 ) α da(z), z D, and da is usual Lebesgue area measure normalized so that the unit disc D is of unit area. Notice that A 0 (D) is the uneighted Bergman space and that the Hardy space H 2 (D) is the limit case of spaces A α (D) as α. For an account of Bergman space theory e refer to the boo Hedenmalm, Korenblum and Zhu [8].

5 PARTS OF ADJOINT WEIGHTED SHIFTS 253 The standard eighted Bergman space A α (D) can be put on the above form A α (D) = A α+2 (D) for α > ith the eight sequence α = { α; } given by α; = ( +α ) = Γ(α)!, = 0,, 2,..., Γ(α + ) for α > 0, here Γ denotes the usual Gamma function. Observe that the eight sequence α+2 is the sequence of moments α+2; = z 2 da α (z), = 0,,..., D for the measure da α appearing above. Notice also that the ernel function for the space A α (D) has the form K α (z, ζ) = ( ζz) α, (z, ζ) D2, as follos by standard paraphernalia for the binomial series. A celebrated result by Agler [], [2] concerns the parts of the adjoint standard eighted Bergman shift operator for integer eight parameter. Let n Z + be a positive integer. An operator T L(H) is called an n-hypercontraction if (.7) m =0( ) ( m ) T T 0 in L(H) for m n. Notice that in this terminology a -hypercontraction is a contraction, hereas for n 2 the class of n-hypercontractions is a subclass of the class of contractions. For an n-hypercontraction T L(H) e consider the defect operator D n,t = ( n ( n ( ) =0 ) T T ) /2 in L(H), here the positive square root is used. The defect space D n,t is the closure in H of the range D n,t (H) of the operator D n,t. For x H e consider the D n,t -valued analytic function (.8) V n x(z) = D n,t (I zt) n x = (D n,t T x)z, z D, n; in D. THEOREM. (Agler). Let S n = S n, here n Z +. Let T L(H). Then the folloing holds: (i) The operator T is part of an operator of the form S n U ith U an isometry if and only if T is an n-hypercontraction. (ii) The operator T is part of an operator of the form S n if and only if T is an n-hypercontraction in the class C 0.

6 254 ANDERS OLOFSSON Furthermore, hen T is an n-hypercontraction, then the map [ ] [ ] Vn Vn x V = : x Q Qx defined by formulas (.6) and (.8) above is an isometry mapping H into the space A n (D n,t ) Q such that [ ] S VT = n 0 V, 0 U here the operator U L(Q) is as above. An important addition to Theorem. is the result that an operator T L(H) is an n-hypercontraction if it is a contraction and (.7) is satisfied ith m = n. Another interesting property of the class of n-hypercontractions is that of dilation invariance: An operator T L(H) is an n-hypercontraction if and only if rt is for 0 < r <. Notice that Theorem. ith n = yields the characterization of the parts of the adjoint Hardy shift operator discussed above. To further emphasize its importance, e mention that Theorem. forms the basis of recent progress on operator-valued Bergman inner functions, characteristic operator functions and related mathematical systems theory by the author [23], [25], [26] and Ball and Bolotniov [8]. A detailed and straightforard discussion of Theorem. and related matters can be found in [24]. We ish to mention here that a rather big branch dealing also ith multivariable generalizations of Theorem. has emerged in recent years ith contributions by several authors, see for instance [2], [3], [20], [2], [28], [34]. Of particular mention here are the contributions by Engliš and collaborators [4], [5] and more importantly the recent or Ball and Bolotniov [9] hich points also in a direction of interesting applications. Hoever, none of these results is sufficiently strong to allo for a characterization of the parts of an adjoint standard eighted Bergman shift operator in terms of a finite number of operator inequalities and a C 0 condition that goes beyond Theorem.. The importance of Theorem. stems from the explicit form of the isometry V hich describes the embedding. The proof of Theorem. boils don to establishing the identity x 2 = D n,t T x 2 + lim T x 2 n; for x H. The purpose of the present paper is to carry out the analogous constructions in the context of a more general eight sequence = { } satisfying some additional properties that e shall introduce as e go along. In particular, our results ill apply in the full scale of eight sequences α ith α and thereby extend the validity of Theorem. and its addition to the full scale of standard eighted Bergman spaces in D.

7 PARTS OF ADJOINT WEIGHTED SHIFTS 255 Let us return to the ernel function corresponding to a positive decreasing eight sequence = { } such that 0 = and lim =. We / assume in addition that the ernel function is non-vanishing in D: (z) = 0 for z D. The reciprocal function / has no the poer series expansion (.9) (z) = c z, z D, convergent in the unit disc D. Let T L(H) be a contraction and consider the operator quantities D,T (r) = r c T T in L(H) for 0 < r <. We say that a contraction T L(H) is a -hypercontraction if it has the property that D,T (r) 0 in L(H) for 0 < r < (see Definition 2.). Specializing to the adjoint shift T = S = S in L(A (E)) e chec that the operator S is a -hypercontraction if and only if the ernel function has a certain Property 2.4 (see Corollary 6.3). Subject to Property 2.4 e sho that for T L(H) a -hypercontraction, the limit D,T () = lim r D,T (r) exists as a decreasing limit in the strong operator topology in L(H) (see Proposition 2.6). We next define the higher order defect operator by D,T = D,T () /2 in L(H), here the positive square root is used (see Definition 2.7). Subject to also an additional Property 3.4 of the ernel function e establish the operator formula (.0) x 2 = D,T T x 2 + lim T x 2 for x H (see Theorem 3.7). With (.0) at hand the characterization of the parts of S in the spirit of Theorem. above is more of a standard matter (see Theorem 7.2 and Corollary 7.3). We establish also a certain universal mapping property of the embedding provided (see Theorem 7.6). Another easily derived byproduct is a formula for a related ernel function (see Theorem 7.4). Subject to also an additional Property 4.2 of the ernel function hich is a stronger version of Property 3.4 e sho that the number of inequalities needed for an operator to be a -hypercontraction can be considerably reduced: A contraction T L(H) is a -hypercontraction if and only if c T T 0 in L(H), here the c s are as in (.9) (see Theorem 4.5). We point out that Property 4.2 is a slightly stronger form of the assumption of absolute summability of the coefficients {c } in (.9) hich as used in Ball and Bolotniov [9].

8 256 ANDERS OLOFSSON The results discussed above point at a need to better understand the Properties 2.4, 3.4 and 4.2 for a ernel function appearing in our results. First e ish to point out that non-vanishing of the ernel function is a non-trivial requirement (see for instance Section 2 of [24] for an example). Property 2.4 has the nature of a strengthened form of the positive definiteness property of a ernel function and seems the easiest to verify (see Proposition 5.). Properties 3.4 and 4.2 concern estimates of a reciprocal ernel / and seem harder to efficiently chec. By formula (.9) e have that c 0 / 0 = and c j = 0 j=0 j for. Solving these equations for the c s e obtain that c 0 = and c n = n = ( ) n + +n =n n n2 for n, here the rightmost sum is taen over all -tuples (n,..., n ) of positive integers such that n + + n = n. Thus ( (z) = + ( ) ) z n. n n + +n =n n n2 n These latter formulas suggest that the reciprocal ernel / carries interesting arithmetic information related to the sequence = { }. As indicated above Properties 2.4, 3.4 and 4.2 are satisfied in the case of standard eight sequences = α ith α > 0 (see Corollary 5.5). More generally, e sho that Properties 2.4, 3.4 and 4.2 hold hen the ernel function is a finite product of (radial complete) Nevanlinna Pic ernels (see Proposition 5.4). n 2. POSITIVITY CONDITIONS AND DEFECT OPERATORS In this section e introduce the class of -hypercontractions and prove existence of higher order defect operators. Recall that by a contraction e mean a bounded Hilbert space operator T L(H) such that Tx 2 x 2 for all x H. The defect operator for a contraction T is the operator D T defined by the formula D T = (I T T) /2 in L(H), here the positive square root is used. Observe that D T x 2 = x 2 Tx 2 for x H. For bacground information on contractions e refer to [33]; see also [4].

9 PARTS OF ADJOINT WEIGHTED SHIFTS 257 As before, let = { } be a positive decreasing eight sequence ith 0 = such that lim = and assume that the function is non-vanishing / in D. For a contraction T L(H) e shall consider the operator quantities D,T (r) = r c T T, 0 < r <, in L(H), here the c s are as in (.9). Notice that the sum defining D,T (r) is absolutely summable in operator norm since is non-vanishing in D. DEFINITION 2.. By a -hypercontraction e mean a contraction T L(H) such that D,T (r) 0 in L(H) for 0 < r <. An important property inherent in the definition of -hypercontractions is that of dilation invariance. PROPOSITION 2.2. Let = { } be a positive decreasing eight sequence ith 0 = such that lim = and assume that the function is non-vanishing / in D. If T L(H) is a -hypercontraction, then so is rt for 0 r <. The result is evident by Definition 2.. Assume that the function is non-vanishing in D and let r, s [0, ]. We shall mae use of poer series expansions of the form (2.) (rz) (sz) = a (r, s)z, z < min( r, s ). Notice that a 0 (r, s) =. A calculation shos that (2.2) a (r, s) = j=0 r j j c j s j for 0, here the c s are as in (.9). Operator formulas of the folloing type ill play an important role in our analysis. LEMMA 2.3. Let = { } and be as in Proposition 2.2. Let T L(H) be a contraction and r, s [0, ). Then (2.3) D,T (r) = a (s, r)t D,T (s)t in L(H), here the a (s, r) s are as in (2.). Proof. Identifying Taylor coefficients in (2.) e have that c r = a j (s, r)c j s j j=0

10 258 ANDERS OLOFSSON for 0. A calculation shos that a (s, r)t D,T (s)t = a (s, r)c j s j T (j+) T j+ j, = m 0( m j=0 a m j (s, r)c j s j) T m T m = c r T T = D,T (r), hich yields (2.3). Observe that changes of order of summation are permitted by absolute convergence. Observe that (2.3) is equivalently formulated that, for x H, D,T (r)x, x = a (s, r) D,T (s)t x, T x. We thin of the operator formula (2.3) as the result of so-called hereditary functional calculus applied to the function identity (rz) = (sz) (rz) (sz), see for instance [8], [9]. We denote by,r the dilated function defined by,r (z) = (rz) for z D. We shall be concerned ith non-vanishing ernels having the folloing additional property of positive definiteness: PROPERTY 2.4. The function /,r has non-negative Taylor coefficients for 0 < r <. In terms of the coefficients in (2.) Property 2.4 means that a (, r) 0 for all 0 and 0 < r <. In vie of Schur s theorem on Schur products of positive matrices e can thin of Property 2.4 as a strengthened form of the positive definiteness property of a ernel function (see Aronszajn, Section I.8 of [6]; see also Shimorin [30]). We no return to the operator quantities D,T (r). LEMMA 2.5. Let = { } be a positive decreasing eight sequence ith 0 = such that lim = and assume that the function is non-vanishing in / D and satisfies Property 2.4. Let T L(H) be a contraction such that D,T (s) 0 in L(H) for some 0 < s <. Then D,T (r) D,T (s) in L(H) for 0 < r < s. Proof. Recall Lemma 2.3. Property 2.4 gives that the coefficients in the poer series expansion (2.) are non-negative hen r > s: a (r, s) 0 for 0 if 0 < s < r. Observe also that T D,T (s)t 0 in L(H) for 0 since D,T (s) 0 in L(H) by assumption. The conclusion of the lemma no follos by (2.3). We no return to -hypercontractions.

11 PARTS OF ADJOINT WEIGHTED SHIFTS 259 PROPOSITION 2.6. Let = { } be a positive decreasing eight sequence ith 0 = such that lim = and assume that the function is non-vanishing / in D and satisfies Property 2.4. Let T L(H) be a -hypercontraction. Then the operators D,T (r) decrease ith r (0, ). As a consequence the limit (2.4) D,T () = lim r D,T (r) exists in the strong operator topology in L(H). Proof. By Lemma 2.5 e have that D,T (r) D,T (s) 0 in L(H) for 0 < r < s <. Passing to the limit e have that lim D,T (r)x, x exists for r every x H, and a polarization argument gives (2.4) ith convergence in the ea operator topology. By a classical result, monotonicity allos us to conclude that (2.4) holds ith convergence in the strong operator topology (see Halmos, Problem 20 of [6]). We remar that part of the conclusion of Proposition 2.6 is that 0 D,T (s) D,T (r) I in L(H) if 0 < r < s <. Proposition 2.6 leads to a natural notion of higher order defect operator for a -hypercontraction. DEFINITION 2.7. For T L(H) a contraction such that the limit (2.4) exists and is positive e set D,T = D,T () /2 in L(H), here the positive square root is used. We observe that 0 D,T I in L(H) under the assumptions of Proposition 2.6. Notice that D T = D,T in the terminology of Definition 2.7. We notice also that Lemma 2.5 simplifies the positivity conditions for a - hypercontraction. COROLLARY 2.8. Let = { } and be as in Proposition 2.6. Let T L(H) be a contraction such that D,T (r ) 0 in L(H) for some sequence r, 0 < r <. Then the operator T L(H) is a -hypercontraction, that is, D,T (r) 0 in L(H) for all 0 < r <. The result is a straightforard consequence of Lemma 2.5. Proposition 2.6 restates as follos using the defect operators from Definition 2.7. COROLLARY 2.9. Let = { } and be as in Proposition 2.6. Let T L(H) be a -hypercontraction. Then D,rT 2 D2,T in L(H) for 0 < r < and lim D,rT = D,T in the strong operator topology in L(H). r

12 260 ANDERS OLOFSSON Proof. Recall that the operator rt is a -hypercontraction for 0 < r < (see Proposition 2.2). The inequality D,rT 2 D2,T is a restatement of Proposition 2.6. The last limit assertion that lim D,rT = D,T SOT follos from Proposition 2.6 r by an approximation argument (see Halmos, Problem 26 of [6]). 3. STRIPPED ISOMETRIC EMBEDDING The purpose of the present section is to establish our basic isometry result (.0) in Theorem 3.7 belo. The proof of Theorem 3.7 is accomplished by first expressing the first order defect operator D T in terms of the higher order defect operator D,T in Lemma 3.6. We no proceed to details. LEMMA 3.. Let = { } be a positive decreasing eight sequence ith 0 = such that lim = and assume that the function is non-vanishing in D / and satisfies Property 2.4. Let T L(H) be a -hypercontraction. Then, for x H, ( D,T x 2 + ) D,T T x 2 D T x 2. Proof. Let x H and 0 < r <. The function identity rz = ( rz) (rz) (rz), leads by hereditary functional calculus to the operator formula ( (3.) x 2 r Tx 2 = D,T (r)x, x + ) r D,T (r)t x, T x (compare Lemma 2.3). Passing to the limit in (3.) as r using Proposition 2.6 and Fatou s lemma e have that ( D,T x 2 + ) D,T T x 2 D T x 2. This completes the proof of the lemma. The folloing result is ell-non but included here for the sae of completeness. LEMMA 3.2. Let T L(H) be a contraction. Then, for x H, (3.2) x 2 = D T T x 2 + lim T x 2. Proof. Let x H. Observe that D T T x 2 = T x 2 T + x 2 for 0. Summing these equalities e have n =0 D T T x 2 = x 2 T n x 2

13 PARTS OF ADJOINT WEIGHTED SHIFTS 26 by cancellation. A passage to the limit as n yields the conclusion of the lemma. We mention that a classical reference for Lemma 3.2 is Section I.0 of [33]. Notice that Lemma 3.2 says that the map V in Theorem. is an isometry for n =. LEMMA 3.3. Let = { } and be as in Lemma 3.. Let T L(H) be a -hypercontraction. Then, for x H, D,T T x 2 + lim T x 2 x 2. Proof. Let x H. By Lemma 3. e have that ( D,T T j x 2 + ) D,T T +j x 2 D T T j x 2 for j 0. Summing these inequalities for j 0 e have by a change of order of summation that D,T T m x 2 m 0 m D T T j x 2. j 0 An application of Lemma 3.2 no yields the conclusion of the lemma. We shall no mae use also of the folloing additional property of a nonvanishing ernel function in D: PROPERTY 3.4. The quotients,r / have uniformly bounded Taylor coefficients for 0 < r <. In terms of the coefficients in (2.) Property 3.4 means that a (r, ) C for all 0 and 0 < r <, here C is a finite positive constant. LEMMA 3.5. Let = { } be a positive decreasing eight sequence ith 0 = such that lim = and assume that the function is non-vanishing / in D and satisfies Properties 2.4 and 3.4. Let T L(H) be a -hypercontraction and 0 < r <. Then (3.3) ( r D T x 2 + a (r, ) D T T x 2 = D,T x 2 + r ) D,T T x 2 for x H, here the a s are as in (2.). Proof. Let r < s <. The function identity (rz) (sz) ( z) = ( z) (rz) (sz)

14 262 ANDERS OLOFSSON leads by hereditary functional calculus to the operator identity (3.4) ( r a (r, s) D T T x 2 = D,T (s)x, x + for x H. Indeed, by calculation e have that r ) ( r D,T (s)x, x + r ) D,T (s)t x, T x = =,j 0 r D,T (s)t x, T x r s j c j T +j x 2,j 0 D,T (s)t x, T x r D,T (s)t + x, T + x r s j c j T +j+ x 2 r = s j c j D T T +j x 2 =,j 0 a m (r, s) D T T m x 2 m 0 for x H, here the last equality follos by a change of order of summation using formula (2.2). We shall next pass to the limit in (3.4) as s. Notice first that Property 3.4 gives that the coefficients a (r, s) in (2.) are uniformly bounded for 0 < r s : a (r, s) C for and 0 < r s, here C is a finite positive constant. Recall also Lemma 3.2. By dominated convergence e no have that for x H. Observe also that lim a (r, s) D T T x 2 = a (r, ) D T T x 2 s ( r lim D,T(s)x, x + r ) D,T (s)t x, T x s ( r = D,T x 2 + r ) D,T T x 2 for x H by Proposition 2.6 and dominated convergence. The conclusion of the lemma no follos by letting s in (3.4). We are no ready for our ey lemma. LEMMA 3.6. Let = { } and be as in Lemma 3.5. Let T L(H) be a -hypercontraction. Then, for x H, ( D T x 2 = D,T x 2 + ) D,T T x 2.

15 PARTS OF ADJOINT WEIGHTED SHIFTS 263 Proof. We shall pass to the limit as r in the result (3.3) of Lemma 3.5. Let x H and recall Lemma 3.3. By dominated convergence e have that lim ( r r ) ( D,T T x 2 = r ) D,T T x 2. By Property 3.4 e have that the coefficients a (r, ) in (3.3) are uniformly bounded for 0 and 0 < r < : a (r, ) C for all 0 and 0 < r <, here C is a finite positive constant. Observe also that lim a (r, ) = 0 for hich easily follos from (2.). Lemma 3.2 allos us to apply dominated convergence to r conclude that lim a (r, ) D T T x 2 = 0. r The conclusion of the lemma no follos by letting r in (3.3). We can no establish the stripped isometric embedding. THEOREM 3.7. Let = { } be a positive decreasing eight sequence ith 0 = such that the function is analytic and non-vanishing in D. Assume also that the function has Properties 2.4 and 3.4. Let T L(H) be a -hypercontraction. Then, for x H, x 2 = D,T T x 2 + lim T x 2. Proof. Let x H and recall Lemma 3.2. By Lemma 3.6 e have that ( (3.5) D T T j x 2 = D,T T j x 2 + ) D,T T +j x 2 for j 0. Summing equalities (3.5) for j 0 e have that D T T j x 2 = j 0 D,T T x 2 by a change of order of summation. By Lemma 3.2 this yields the conclusion of the theorem. We mention that the method of proof of Theorem 3.7 using Lemma 3.6 is adapted from our previous paper ([24], Proposition 7.2). 4. REDUCTION OF POSITIVITY CONDITIONS In this section e sho that the number of inequalities needed for an operator to be a -hypercontraction can be considerably reduced provided the ernel function is sufficiently regular.

16 264 ANDERS OLOFSSON LEMMA 4.. Let = { } be a positive decreasing eight sequence ith 0 = such that the function is non-vanishing and analytic in D. Let T L(H) be a contraction and 0 < r <. Then, for x H, lim D,T(r)T x, T x = (r) lim T x 2. Proof. Let x H and notice that D,T (r)t x, T x = c j r j T +j x 2. j 0 Passing to the limit as e have that ( lim D,T(r)T x, T x = c j r j) lim T x 2 = j 0 (r) lim T x 2 by (.9), hich yields the conclusion of the lemma. We denote by A + (D) the space of all analytic functions f (z) = a z, z D, in D ith finite norm f A + = a. The space A + (D) has the structure of a commutative Banach algebra under pointise multiplication of functions and is commonly called the Wiener algebra in vie of a classical result by Norbert Wiener (see for instance Theorem XXX.2.5 of [4]). We shall no mae use also of the folloing additional property of a nonvanishing ernel function in D: PROPERTY 4.2. The reciprocal ernel / belongs to A + (D) and the quotients,r / for 0 < r < form a uniformly bounded family in A + (D). In terms of the coefficients in (2.) Property 4.2 means that a (r, ) C for 0 < r <, here C is a finite positive constant. Notice that Property 4.2 implies Property 3.4. Let us denote by l the space of all sequences a = {a } of complex numbers ith finite norm a l = a.

17 PARTS OF ADJOINT WEIGHTED SHIFTS 265 Folloing usual practice e denote by c 0 the space of all sequences a = {a } of complex numbers such that lim a = 0 equipped ith the norm a c0 = max a. Recall that the space l is the dual of the space c 0 by means of the standard pairing (a, b) = a b, here a c 0 and b = {b } l. In this ay the space l becomes naturally equipped by a ea topology. Recall the coefficients a (r, s) from (2.). LEMMA 4.3. Let = { } be a positive decreasing eight sequence ith 0 = such that the function is analytic and non-vanishing in D. Assume also that the function has Properties 2.4 and 4.2. Then lim s {a (r, s)} = {a (r, )} in the ea topology of l for 0 < r <. Furthermore, lim {a (r, )} = δ r in the ea topology of l, here δ = {δ } is the sequence given by δ 0 = and δ = 0 for. Proof. Recall the ell-non result that a sequence {b n } n in l, here b n = {b n }, converges to b 0 = {b 0 } ea in l if and only if sup b n l < n + and lim b n = b 0 for. The lemma is an easy consequence of this fact. n We can no prove our ey lemma. LEMMA 4.4. Let = { } be a positive decreasing eight sequence such that the function is analytic and non-vanishing in D. Assume also that the function has Properties 2.4 and 4.2. Assume that T L(H) is a contraction such that in L(H). Then, for x H and 0 < r <, D,T () = c T T 0 D,T (r)x, x (r) lim T x 2 a (, r) D,T ()T x, T x. Proof. Let 0 < r < t < s <. The function identity (tz) (sz) (rz) = (tz) (rz) (sz)

18 266 ANDERS OLOFSSON leads by hereditary functional calculus to the operator formula (4.) a (t, s) D,T (r)t x, T x = a (t, r) D,T (s)t x, T x for x H. We shall next pass to the limit in (4.) as s. Observe first that (4.2) lim s a (t, r) D,T (s)t x, T x = a (t, r) D,T ()T x, T x for x H since D,T (s) D,T () SOT in L(H) as s and the coefficients a (t, r) in (4.2) decay rapidly as. We shall next turn our attention to the left hand side in (4.). Let x H and let L = (r) lim T x 2 be the limit from Lemma 4.. We rerite the left hand side in (4.) as a (t, s) D,T (r)t x, T x = a (t, s)( D,T (r)t x, T x L) + L a (t, s). Observe first that a (t, s) = (t) (s) 0 as s for t [0, ) fixed, hich easily follos by (2.) since () = / = +. By Lemma 4. the sequence { D,T (r)t x, T x L} belongs to c 0. Passing to the limit as s using Lemma 4.3 e have that (4.3) lim s a (t, s) D,T (r)t x, T x = a (t, )( D,T (r)t x, T x L). By (4.), (4.2) and (4.3) e conclude that, for x H, (4.4) a (t, )( D,T (r)t x, T x L) = a (t, r) D,T ()T x, T x. We shall next pass to the limit in (4.4) as t. Let x H. Notice that lim t a (t, )( D,T (r)t x, T x L) = D,T (r)x, x L by Lemma 4.3 since the sequence { D,T (r)t x, T x L} belongs to c 0 by Lemma 4.. Observe also that a (t, r) 0 for t > r and 0 by Property 2.4. An application of Fatou s lemma to (4.4) as t no yields the conclusion of the lemma. We can no simplify the defining property of a -hypercontraction.

19 PARTS OF ADJOINT WEIGHTED SHIFTS 267 THEOREM 4.5. Let = { } be a positive decreasing eight sequence ith 0 = such that the function is analytic and non-vanishing in D. Assume also that the function has Properties 2.4 and 4.2. Assume that T L(H) is a contraction such that (4.5) c T T 0 in L(H), here the c s are as in (.9). Then T is a -hypercontraction. Proof. By Lemma 4.4 e have that D,T (r) 0 in L(H) for 0 < r <. This yields the conclusion of the theorem. We remar that the sum in (4.5) is absolutely summable since / A + (D) and T is a contraction. 5. EXAMPLES OF KERNEL FUNCTIONS The purpose of this section is to provide some examples of ernel functions satisfying Properties 2.4, 3.4 and 4.2 or combinations thereof. In particular, e shall chec that Properties 2.4, 3.4 and 4.2 are satisfied in the case of standard eight sequences = α ith α > 0 (see Corollary 5.5). Notice first that Properties 2.4 and 4.2 are both stable ith respect to pointise product of functions: If j is a non-vanishing analytic function in D satisfying Property 2.4 (Property 4.2) for j =, 2, then so is the product = 2. An easy criteria for checing Property 2.4 goes as follos. PROPOSITION 5.. Let be a non-vanishing analytic function in D ith (0) = such that the function s = log has non-negative Taylor coefficients in D. Then / r has non-negative Taylor coefficients in D for 0 < r <. Proof. Let 0 < r <. Since = e s e have that (z) (rz) = exp(s(z) s(rz)) = (s(z) s(rz))n n! n 0 for z D. It is straightforard to chec that the function s s r has non-negative Taylor coefficients. This yields the conclusion of the proposition. For applications of Proposition 5. it is useful to have available the logarithm of the Szegö ernel: ( ) log = z n zn n for z D. As a consequence of Proposition 5. e see that all ernels of the form α (z) = ( z) α, z D,

20 268 ANDERS OLOFSSON ith α > 0 satisfy Property 2.4. Let be a non-vanishing analytic function in D ith non-negative Taylor coefficients normalized by (0) =. We say that is of NP type if its reciprocal has a poer series expansion of the form (5.) (z) = b z for some numbers b 0 for. Kernels of NP type have attracted reneed interest because of their use in the study of Nevanlinna Pic interpolation, see for instance the boo Agler and McCarthy [3]. Solving for in (5.) e see that a function is of NP type if and only if it has the form (5.2) (z) = ϕ(z), z D, for some analytic function ϕ in D ith non-negative Taylor coefficients such that ϕ(0) = 0 and ϕ(z) < for z D. As a consequence of the representation formula (5.2) e have that the class of NP type functions is invariant under pullbac ith analytic functions ϕ in D having non-negative Taylor coefficients such that ϕ(0) = 0 and ϕ(z) < for z D. An old theorem of Th. Kaluza says that a ernel function (z) = z, z D, ith 0 = and > 0 for is of NP type if the eight sequence = { } is log-concave, that is, + 2 for (see Hardy, Theorem IV.22 of [7] or Szegö, Subsection.2 of [32]). Applications of Kaluza s theorem sho that the ernels α for 0 < α and the ernel function for the Dirichlet space 0 (z) = ( ) z log = z n + zn, z D, n 0 are all of NP type. We mention in passing that the above ernel α is not of NP type hen α >. PROPOSITION 5.2. Let be an NP type function. Then / A + (D) and = 2 2, A + () here () = lim (x) along positive real numbers. x Proof. Passing to the limit in (5.) e have () = b, hich yields the conclusion of the proposition.

21 PARTS OF ADJOINT WEIGHTED SHIFTS 269 We next observe that the class of NP type functions is preserved under the quotient operation / r. PROPOSITION 5.3. Let be an NP type function and 0 < r <. Then the quotient / r is an NP type function. Proof. An application of Proposition 5. using the representation formula (5.2) shos that the function / r has non-negative Taylor coefficients. We next verify that the function r / has non-positive Taylor coefficients. By (5.) e have that (rz) (z) = (rz) ( (z) (rz) ) = (rz) ( r )b z, hich proves the claim since the property of non-negativity of coefficients is preserved under pointise multiplication of functions (see Schur s theorem, Section I.8 of [6]). We next provide examples of eight sequences satisfying Properties 2.4 and 4.2. PROPOSITION 5.4. Let = { } be a positive eight sequence such that the associated ernel function has the form (5.3) (z) = (z) n (z), z D, here j is an NP type function for j n. Then the function has Properties 2.4 and 4.2. Proof. A straightforard application of Proposition 5. shos that has Property 2.4. By Propositions 5.2 and 5.3 e have that,r / A + 2 n for 0 < r < hich follos by the Banach algebra property of A + (D). This shos that has Property 4.2. We record that the Bergman ernels for standard eights satisfy Properties 2.4 and 4.2. COROLLARY 5.5. The functions α for α > 0 all have Properties 2.4 and 4.2. Proof. The function α is a finite product of NP type functions. The result follos by Proposition 5.4. We close this section by giving an example of a positive decreasing eight sequence = { } ith 0 = and lim / = such that the ernel function satisfies Properties 2.4 and 3.4 but / A + (D) hich violates the regularity assumption from Ball and Bolotniov [9]. Notice that yet Theorem 3.7 applies for such a eight sequence = { }. Consider the function (z) = + z z, z D,

22 270 ANDERS OLOFSSON hich is the ernel function for the eight sequence = {} given by 0 = and = /2 for. An application of Proposition 5. shos that satisfies Property 2.4. It is evident that the function / has a single pole at the point z =, so that / A + (D). Moreover, a calculation shos that (rz) (z) = + 2 r + r (( ) r )z for z D and 0 < r <. As a consequence, the function has Property 3.4. We mention that further elaborations on Kaluza s theorem can be found in Shimorin [3]. 6. THE OPERATOR MODEL Let E be an auxiliary Hilbert space and let = { } be a positive decreasing eight sequence ith 0 = such that the ernel function is analytic and non-vanishing in D. The purpose of this section is to discuss some basic properties of the adjoint shift S = S on A (E). In particular, e shall sho that S is a -hypercontraction if and only if the ernel function has Property 2.4 (see Corollary 6.3). Recall the action of poers of the adjoint shift (6.) S f (z) = j 0 j+ j a j+ z j, z D, on functions f A (E) given by (.). Formula (6.) is straightforard to chec from (.3) or (.4). As a consequence of (6.) e have that S f = 0 henever f A (E) is a polynomial of degree less than. By approximation e have that lim 0 S f = 0 in A (E) for every f A (E) since S is a contraction. We conclude that the operator S belongs to the class C 0 in the sense that lim S = 0 in the strong operator topology. Recall formula (.9) defining the numbers {c }. Homogeneity considerations mae evident that the defect operators D,S (r) = r c S S for 0 < r < act as Fourier multipliers on A (E) (see for instance [9], [27]). We next calculate this Fourier multiplier action in more detail. PROPOSITION 6.. Let = { } be a positive decreasing eight sequence ith 0 = such that the ernel function is analytic and non-vanishing in D. Let S = S be the shift operator on A (E). Then ( D,S (r) f (z) = c j r j ) a z, z D, j=0 j

23 PARTS OF ADJOINT WEIGHTED SHIFTS 27 for f A (E) given by (.) and 0 < r <. Proof. By (6.) e have that for 0. No 2 S f 2 j+ = a j+ 2 j 0 j 2 D,S (r) f, f = r c S f 2 = r j+ c a j+ 2,j 0 j = 2 m m 0 ( m =0 c r ) a m 2, m and a polarization argument yields the conclusion of the proposition. Observe that the result of Proposition 6. can be stated as (6.2) D,S (r) f (z) = a (, r)a z, z D, for f A (E) given by (.) using the coefficients {a (, r)} from (2.). We next record the action of D,S (r) on reproducing elements. COROLLARY 6.2. Let, and S = S be as in Proposition 6.. Then D,S (r)k (, ζ)e, K (, z)e = (ζz) (rζz) e, e for 0 < r <, ζ, z D and e, e E. The result follos by Proposition 6. using the explicit form of the reproducing ernel. We mention that Corollary 6.2 is alternatively proved using the standard formula (.4) for the action of the adjoint shift on reproducing elements. COROLLARY 6.3. Let, and S = S be as in Proposition 6.. Then the adjoint shift S is a -hypercontraction if and only if the ernel function has Property 2.4. Proof. Let 0 < r <. By Proposition 6. e have that D,S (r) f, f = a (, r) 2 a 2 for f A (E) given by (.). Varying f A (E) e see that D,S (r) 0 if and only if a (, r) 0 for 0. This yields the conclusion of the corollary. We next discuss the operator limit lim r D,S (r). COROLLARY 6.4. Let, and S = S be as in Proposition 6.. Then D,S (r) = sup a (, r)

24 272 ANDERS OLOFSSON for 0 < r <. Furthermore, hen these operator norms stay bounded, e have that the operator limit D,S () = lim r D,S (r) exists in the strong operator topology and equals the orthogonal projection of A (E) onto the subspace of constant functions. As a consequence D,S f = f (0) for f A (E). Proof. The formula for the norm of D,T (r) is evident by the Fourier multiplier formula (6.2) for D,T (r) provided by Proposition 6.. Recall that lim a (, r) r = 0 for by (2.), hich gives that lim D,S (r) f = f (0) for every polynomial f A (E). By approximation e have that lim D,S (r) f = f (0) in A (E) r r for every f A (E). We use the notation H() for the Hilbert space of analytic functions in D ith ernel function K(z, ζ) = (ζz) for z, ζ D. Notice that Property 2.4 of ensures that the space H( /,r ) exists for 0 < r < (see Aronszajn, Section I.2 of [6]). PROPOSITION 6.5. Let = { } be a positive decreasing eight sequence ith 0 = such that the ernel function is analytic and non-vanishing in D and satisfies Property 2.4. Then the space H( /,r ) is contractively embedded into A (D) for 0 < r <. Proof. Let 0<r <. By Corollary 6.3 the operator S is a -hypercontraction, hich by Proposition 2.6 leads to the operator inequality that D,S (r) I in L(A (D)). By the Fourier multiplier formula (6.2) from Proposition 6. this gives that a (, r) for 0. These latter inequalities yield the conclusion of the proposition (see for instance Shimorin [30] or Aronszajn, Section I.7 of [6] for related matters). REMARK 6.6. Observe that D,S (r) = for 0 < r < if in addition the ernel function has Property 2.4. Indeed, this follos from Corollary 6.3 and Proposition 2.6 by standard spectral theory. We record also that ( λ S A r S ) f (z) = ( j=0 for f A (E) given by (.) and 0 < r <, here are the Abel summability operators. A r f (z) = r a z, z D, r j ) λ j a z, z D, j

25 PARTS OF ADJOINT WEIGHTED SHIFTS EMBEDDING IN THE OPERATOR MODEL Let = { } be a positive decreasing eight sequence ith 0 = such that the ernel function is analytic and non-vanishing in D and satisfies Property 2.4. Let T L(H) be a -hypercontraction, and recall the notion of higher order defect operator D,T from Proposition 2.6 and Definition 2.7. We no define the higher order defect space D,T as the closure in H of the range D,T (H) of the operator D,T. For x H e shall consider the D,T -valued analytic function V x defined by the folloing formula, for z D: (7.) V x(z) = D,T (zt)x = (D,T T n x)z n. n 0 n PROPOSITION 7.. Let = { } be a positive decreasing eight sequence ith 0 = such that the ernel function is analytic and non-vanishing in D and satisfies Property 2.4. Let T L(H) be a -hypercontraction. Then the map V : x V x defined by (7.) is a contraction mapping H into the space A (D,T ) such that V T = S V, here S = S is the shift operator on A (D,T ). Furthermore, for x H, V x 2 + lim T x 2 x 2. Proof. The norm bound for the map V : H A (D,T ) is a restatement of Lemma 3.3. We proceed to prove the intertining relation V T = S V. Let x H. By formula (.3) for the action of the adjoint shift e have that, for z D, S V x(z) = + + (D,T T + x)z = (D,T T + x)z = V Tx(z). Recall the preliminaries about contractions from Section. We no inoe Property 3.4. THEOREM 7.2. Let = { } be a positive decreasing eight sequence such that the ernel function is analytic and non-vanishing in D and satisfies Properties 2.4 and 3.4. Let T L(H) be a -hypercontraction. Then the map V = [ V Q ] : x [ V x Qx defined by (7.) and (.6) is an isometry mapping H into the space A (D,T ) Q such that [ ] S 0 VT = V, 0 U here S = S is the shift operator on A (D,T ) and the operator U L(Q) is as in Section. ]

26 274 ANDERS OLOFSSON Proof. The first part that the map V : H A (D,T ) Q is an isometry is a restatement of Theorem 3.7. The intertining relations S V = V T and QT = UQ are proved in Proposition 7. and Section, respectively. Recall the terminology that an operator T L(H) is said to belong to the class C 0 if lim T = 0 in the strong operator topology in L(H) (see Section II.4 of [33]). COROLLARY 7.3. Let = { } and be as in Theorem 7.2. Let T L(H) be a -hypercontraction in the class C 0. Then the map V : x V x defined by (7.) is an isometry mapping H into the space A (D,T ) such that V T = S V, here S = S is the shift operator on A (D,T ). Proof. By (.6) the operator Q vanishes since T L(H) is a contraction in the class C 0. The result follos by Theorem 7.2. Recall that the reproducing ernel function for a Hilbert space H of E- valued analytic functions in D is the operator-valued function K H : D D L(E) defined by the reproducing property that K H (, ζ)e H and f (ζ), e = f, K H (, ζ)e for all e E, f H and ζ D. We next consider the ernel function K V (H) for the subspace V (H) of the space A (D,T ). THEOREM 7.4. Let = { } and be as in Theorem 7.2. Let T L(H) be a -hypercontraction in the class C 0. Then the ernel function K V (H) for the subspace V (H) of A (D,T ) has the folloing form, for z, ζ D: K V (H)(z, ζ) = D,T (zt) (ζt) D,T. Proof. Let f A (D,T ) be an element of the form f = V x for some x H. Let e D,T. By Corollary 7.3 e have that f (ζ), e = x, (ζt) D,T e = V x, V (ζt) D,T e = f, V (ζt) D,T e for ζ D. This yields the conclusion of the theorem. We mention that formulas for ernel functions similar to the result of Theorem 7.4 play an important role in recent or by Ball and Bolotniov [8], [9]. We shall next turn our attention to canonical features of the map V : H A (D,T ) Q constructed in Theorem 7.2 above. First e need a lemma.

27 PARTS OF ADJOINT WEIGHTED SHIFTS 275 LEMMA 7.5. Let = { } be a positive decreasing eight sequence such that the ernel function is analytic and non-vanishing in D. Let E and Q be Hilbert spaces and U L(Q ) an isometry. Let T L(H) be a bounded operator, and assume that there exists an isometry [ ] W W = : H A (E) Q W 2 from H into the space A (E) Q such that [ S 0 WT = 0 U here S = S is the shift operator on A (E). contraction such that ] W, (7.2) lim T x 2 = W 2 x 2 for x H. Furthermore, for x H and 0 < r <, Then the operator T L(H) is a (7.3) D,T (r)x, x = D,S (r)w x, W x + (r) W 2x 2. Proof. It is evident that T is a contraction since it is part of a contraction. Let x H. Since the map W is an isometry e have that T x 2 = WT x 2 = S W x 2 + U W 2x 2, here the last equality follos using the intertining relation. Since U L(Q ) is an isometry e conclude that (7.4) T x 2 = S W x 2 + W 2 x 2 for x H and 0. Letting in (7.4) e obtain (7.2) since S belongs to the class C 0, see Section 6. We consider next the operator quantities D,T (r). By (7.4) e have that D,T (r)x, x = c r T x 2 = c r S W x 2 + c r W 2 x 2 = D,S (r)w x, W x + (r) W 2x 2 for x H and 0 < r <, hich yields (7.3). We next inoe also Property 2.4 of the ernel function. THEOREM 7.6. Let = { } be a positive decreasing eight sequence such that the ernel function is analytic and non-vanishing in D and satisfies Property 2.4. Let E and Q be Hilbert spaces and U L(Q ) an isometry. Let T L(H) be a bounded operator, and assume that there exists an isometry [ ] W W = : H A (E) Q W 2

28 276 ANDERS OLOFSSON from H into A (E) Q such that [ S 0 WT = 0 U ] W, here S = S is the shift operator on A (E). Then T L(H) is a -hypercontraction and there exist isometries Ŵ : D,T E and Ŵ 2 : Q Q such that the operators W and W 2 have the form W x(z) = Ŵ V x and W 2 x = Ŵ 2 Qx for x H, here the map V is given by (7.) and Q is as in Section. Proof. We first sho that T is a -hypercontraction and derive formula (7.5) belo. Recall from Lemma 7.5 that T is a contraction satisfying (7.3). Since has Property 2.4 e have by Corollary 6.3 that S is a -hypercontraction, that is, D,S (r) 0 for 0 < r <. By (7.3) e conclude that T is a -hypercontraction. Passing to the limit in (7.3) as r using Proposition 2.6 e obtain that D,T x 2 = D,S W x 2 for x H since lim r (r) = +. By Corollary 6.4 e have that, for x H, (7.5) D,T x 2 = W x(0) 2. The map Ŵ : D,T E is defined by Ŵ D,T x = W x(0) for x H. By (7.5) this gives a ell-defined map hich by continuity extends uniquely to an isometry Ŵ : D,T E. We next derive the representation formula for the map W. Let x H and consider the function f = W x in A (E). Consider also the poer series expansion of f given by (.) and recall the action of poers of the adjoint shift given by (6.). By the intertining relation for W e have W T x = S f, and an evaluation at the origin gives that Ŵ D,T T x = (W T x)(0) = a for 0. We no solve for the function f to obtain that, for z D, f (z) = a z = (Ŵ D,T T x)z = Ŵ V x(z). The map Ŵ 2 : Q Q is defined by Ŵ 2 : Qx W 2 x for x H. By (7.2) the map Ŵ 2 is ell-defined and extends by continuity uniquely to an isometry from the space Q into Q such that W 2 x = Ŵ 2 Qx for x H. This completes the proof of the theorem. We point out that the result of Theorem 7.6 has the interpretation of a universal mapping property for the map V : H A (D,T ) Q from Theorem 7.2

29 PARTS OF ADJOINT WEIGHTED SHIFTS 277 above. In fact, Theorem 7.6 says that the diagram is commutative, here H V A (D,T ) Q Ŵ = W A (E) Q Ŵ [ Ŵ 0 0 Ŵ 2 ith Ŵ extended to functions f in A (D,T ) by the formula (Ŵ f )(z) = Ŵ f (z) for z D. COROLLARY 7.7. Let = { }, and T L(H) be as in Theorem 7.6. Then the map [ ] [ ] V V x V = : x Q Qx defined by (7.) and (.6) is an isometry from H into the space A (D,T ) Q. Proof. In the notation of Theorem 7.6 e have that x 2 = Wx 2 = Ŵ V x 2 + Ŵ 2 Qx 2 = V x 2 + Qx 2 for x H since Ŵ : D,T E and Ŵ 2 : Q Q are isometries. ] 8. BACK TO STANDARD WEIGHTS Let us return to the scale of standard eights. The purpose of this section is to further comment on the case of intermediate positivity conditions for hypercontractions. Recall from Section 5 that standard eight sequences = α satisfy Properties 2.4, 3.4 and 4.2 for α > 0. Furthermore, a calculation using Stirling s formula and the reflection formula for the Gamma function shos that lim ( )+ ( α ) α+ = for α > 0 (see for instance Proposition.2 of [27]). Γ(α + ) sin(πα) π THEOREM 8.. Let α >. Let T L(H) be a contraction such that ( ) ( ) α T T 0 in L(H). Then in L(H) for < β < α. ( ) ( ) β T T 0

30 278 ANDERS OLOFSSON Proof. Let < β < α and 0 < r <. The function identity ( rz) β = ( rz)α ( rz) α β leads by hereditary functional calculus to the operator formula ( ) ( ) ( ) β m+α β r T T = r m T m( m m 0 ( ) ( ) α (8.) r T T ) T m. By Theorem 4.5 and Corollary 5.5 the operator T is a α -hypercontraction, so that ( ) ( ) α r T T 0 in L(H). Passing to the limit in (8.) as r using Fatou s lemma and dominated convergence e conclude that ( ) ( ) ( ) β m + α β T T T m( m m 0 ( ) ( ) α T T ) T m in L(H). This last inequality yields the conclusion of the theorem. Acnoledgements. Research supported by GS Magnusons fond of the Royal Sedish Academy of Sciences. The author thans the referee for helpful comments. REFERENCES [] J. AGLER, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5(982), [2] J. AGLER, Hypercontractions and subnormality, J. Operator Theory 3(985), [3] J. AGLER, J.E. MCCARTHY, Pic Interpolation and Hilbert Function Spaces, Graduate Stud. in Math., vol. 44, Amer. Math. Soc., Providence, RI [4] C.-G. AMBROZIE, M. ENGLIŠ, V. MÜLLER, Operator tuples and analytic models over general domains in C n, J. Operator Theory 47(2002), [5] J. ARAZY, M. ENGLIŠ, Analytic models for commuting operator tuples on bounded symmetric domains, Trans. Amer. Math. Soc. 355(2003), [6] N. ARONSZAJN, Theory of reproducing ernels, Trans. Amer. Math. Soc. 68(950), [7] C. BADEA, G. CASSIER, Constrained von Neumann inequalities, Adv. Math. 66(2002), [8] J.A. BALL, V. BOLOTNIKOV, Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems, Integral Equations Operator Theory 76(203),