INSTITUTE of MATHEMATICS. ACADEMY of SCIENCES of the CZECH REPUBLIC. Commutative dilation theory. Calin Ambrozie Vladimír Müller. Preprint No.

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1 INSIUE of MAHEMAICS Academy of Sciences Czech Republic INSIUE of MAHEMAICS ACADEMY of SCIENCES of the CZECH REPUBLIC Commutative dilation theory Calin Ambrozie Vladimír Müller Preprint No PRAHA 2014

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3 Commutative dilation theory C. Ambrozie and V. Müller Abstract Dilation theory of single Hilbert space contractions is an important and very useful part of operator theory. By the main result of the theory, every Hilbert space contraction has the uniquely determined minimal unitary dilation. In many situations this enables to study instead of a general contraction its unitary dilation, which has much nicer properties. he present paper gives a survey of dilation theory for commuting tuples of Hilbert space operators. he paper is organized as follows: 1. Introduction 2. Dilation theory of single contractions 3. Regular dilations 4. Ando s dilation and von Neumann inequality 5. Spherical dilations 6. Analytic models 7. Further examples 8. Concluding remarks 1 Introduction he starting point of dilation theory is the following result of Sz.-Nagy (1953): heorem 1.1. Let be a contraction acting on a Hilbert space H. hen there exists a unitary dilation of, i.e., there exist a Hilbert space K H and a unitary operator U acting on K such that k = P H U k H (k = 0, 1,... ), where P H is the orthogonal projection onto H. Moreover, under the minimality condition K = k Z U k H, the unitary dilation U is determined by uniquely up to the unitary equivalence. Since unitary operators have a simple and well-understood structure, in many situations dilation theory enables one to reduce problems for general contractions to simpler problems for unitary operators. So dilation theory became an important part of operator Keywords: commuting multioperator, dilation, model, von Neumann inequality. MSC: Primary 47A13, Secondary 47A20, 47A45,

4 theory as an efficient tool for studying Hilbert space operators and as a subject of considerable independent interest. he most comprehensive reference for dilation theory is the monograph (Sz.-Nagy and Foiaş, 1970), or its recent new edition (Sz.-Nagy et al., 2010). he goal of this paper is to give a brief survey of more recent extensions of dilation theory to the setting of n-tuples of mutually commuting operators. he paper uses the standard multiindex notation. Denote by Z + the set of all nonnegative integers. Let α = (α 1,..., α n ), β = (β 1,..., β n ) Z n +. hen write α = n i=1 α i, supp α = {i : α i 0}, α! = α 1! α n!, α + β = (α 1 + β 1,..., α n + β n ). Write α β if α i β i for all i = 1,..., n. For 1 j n write e j = (0,..., 0, 1, 0,..., 0) with 1 on the j-th position. All spaces in this paper are complex Hilbert spaces. Denote by B(H) the algebra of all bounded linear operators on a Hilbert space H. Let = ( 1,..., n ) B(H) n be an n- tuple of mutually commuting operators. For α = (α 1,..., α n ) Z n + write α = n i=1 α i i. Let H, K be Hilbert spaces, let = ( 1,..., n ) B(H) n and S = (S 1,..., S n ) B(K) n be n-tuples of operators. he n-tuples and S are unitarily equivalent, for short u S, if there exists an invertible isometry V : H K such that S j = V 1 j V for j = 1,..., n. Let H, K be Hilbert spaces, H K, n 1, let = ( 1,..., n ) B(H) n and V = (V,..., V n ) B(K) n be n-tuples of commuting operators. hen V is called a dilation of if α = P H V α H for all α Z n +. Of special interest are dilations consisting of mutually commuting isometries or unitary operators. Both of these concepts are closely related because any n-tuple of commuting isometries can be extended to commuting unitaries, cf. (Sz.-Nagy and Foiaş, 1970, p.22). heorem 1.2. Let V = (V 1,..., V n ) B(H) n be an n-tuple of commuting isometries. hen there exist a Hilbert space K H and commuting unitary operators U 1,..., U n B(K) such that U j H H and V j = U j H (j = 1,..., n). hus, if an n-tuple has a dilation consisting of commuting isometries, then has also a dilation consisting of commuting unitaries. An n-tuple V = (V 1,..., V n ) B(H) n is called doubly commuting if V i V j = V j V i and Vi V j = V j Vi for all i, j {1,..., n}, i j. Recall that a commuting n-tuple of unitary operators is automatically doubly commuting by the Fuglede-Putnam theorem. 2 Dilation theory of single contractions he multivariable dilation theory is inspired by the dilation theory of single contractions. he existence of a unitary dilation of a contraction can be proved in many ways, see (Sz.- Nagy and Foiaş, 1970; Sz.-Nagy et al., 2010). his section contains a simple geometrical approach which will be then generalized to the multivariable case in subsequent sections. Let L be a Hilbert space. Denote by l 2 (Z +, L) the Hilbert space of all functions f : Z + L satisfying f 2 := f(i) 2 <. i=0 2

5 he backward shift (of multiplicity dim L) is the operator S acting on l 2 (Z +, L) defined by (Sf)(i) = f(i + 1). Equivalently, l 2 (Z +, L) may be interpreted as the Hardy space H 2 (D, L) of all vectorvalued analytic functions f : D L defined on the open unit disc D, f(z) = i=0 f(i)zi 2π f(re it ) 2 dt <. hen S is the ad- 0 satisfying f 2 := i=0 f(i) 2 1 = lim r 1 2π joint of the multiplication operator M z : H 2 (D, L) H 2 (D, L) defined by (M z f)(z) = z f(z) ( f H 2 (D, L), z D). Consider the following simple question: which operators are unitarily equivalent to the restriction of a backward shift of some multiplicity to an invariant subspace? Certainly such an operator must be a contraction. So let H be a Hilbert space and B(H) a contraction. Let L be a Hilbert space and S : l 2 (Z +, L) l 2 (Z +, L) the backward shift. It is necessary to find an isometry V : H l 2 (Z +, L) satisfying V = SV. If such an isometry is found, its range V H will be the required subspace invariant for S and the corresponding restriction S V H will be unitarily equivalent to. Suppose such a V exists. Let V 0, V 1,... be the corresponding coordinate functions, V i h = (V h)(i) (i Z +, h H). he condition V = SV can be then rewritten as V i+1 = V i for all i 0. herefore by induction V i = V 0 i. hus it suffices to find only the operator V 0 : H L, the remaining operators V i (i 1) will be determined by the equations V i = V 0 i. Moreover, since V should be an isometry, h 2 = V h 2 = V i h 2 = V 0 h 2 + V 0 h 2 + V 0 2 h 2 + i=0 for all h H. In particular, for the vector h H this gives h 2 = V h 2 = V 0 h 2 + V 0 2 h 2 + By subtracting, one gets V 0 h 2 = h 2 h 2 for all h H. he latter inequality is satisfied by the defect operator D = (I ) 1/2 B(H) and this is essentially the only possible choice of V 0. In general, V 0 = JD where J : D H L is any isometry. he simplest choice for L is L = D := D H and V 0 := D. hen for h H, one has V h 2 = V 0 i h 2 = i=0 i=0 ( i h 2 i+1 h 2 ) = h 2 lim k h 2. Hence V is an isometry if and only if k 0 in the strong operator topology (SO). hus it was proved: heorem 2.1. Let B(H) be a contraction satisfying k 0 (SO). hen is unitarily equivalent to the restriction of the backward shift S of multiplicity dim D to an invariant subspace. he condition k 0 (SO) is clearly also necessary. If the condition k 0 (SO) is not satisfied, then V constructed above is not an isometry and satisfies only V h 2 = h 2 lim k h 2 (h H). In this case, it is possible to extend it to an isometry. 3

6 heorem 2.2. Let B(H) be a contraction. hen there exist a Hilbert space H, a unitary operator W B(H ) and a subspace M l 2 (Z +, D ) H invariant for S W such that u (S W ) M. (1) Proof. By the previous construction, there exists an operator V : H l 2 (Z +, D ) satisfying V = SV and V h 2 = h 2 lim k h 2 (h H). Define a new seminorm on H by h 2 = lim k h 2. Let N = {x H : x = 0}. It is easy to see that N is a subspace invariant for. Let H be the completion of (H/N, ). hen H is a Hilbert space. Define : H/N H/N by (h + N) = h + N. hen is an isometry and can be extended uniquely to an isometry (denoted by the same symbol ) on H. Since any isometry can be extended to a unitary operator, there exist a Hilbert space H H and a unitary operator W B(H ) such that = W H e. Let V : H H be defined by V h = h + N H/N H H. Let V : H l 2 (Z +, D ) H be defined by V h = V h V h (h H). hen V = (S W )V and V h 2 = h 2 lim k h 2 + h 2 = h 2. So V is an isometry and its range V H is the subspace invariant for S W satisfying u (S W ) V H. Remark 2.3. Note that the operator S W is a coisometry. So every contraction has a coisometric extension. Moreover, the backward shift S is a compression of the bilateral shift U acting on the space l 2 (Z, D ) defined by (Uf)(i) = f(i + 1) (f l 2 (Z, D ), i Z). Clearly U W is the unitary dilation of. Of course, all the dilation theory can be formulated equivalently for rather than for. It is easy to see that (1) is equivalent to = P H (M z W ) H, where M z W is an isometrical dilation of. 3 Regular dilations he closest multivariable analogy of the single-contraction case are n-tuples of commuting contractions having a regular dilation. he role of the backward shift is played by the backward multishift. Let L be a Hilbert space and let n N be fixed. Let l 2 (Z n +, L) denote the Hilbert space of all vector-valued functions f : Z n + L such that f 2 := α Z f(α) 2 <. Consider the operators S n j : l 2 (Z n +, L) + l 2 (Z n +, L) (j = 1,..., n) defined by (S j f)(α) = f(α + e j ). Equivalently, l 2 (Z n +, L) can be interpreted as the Hardy space H 2 (D n ) of all analytic functions f : D n C, f(z) = α Z f(α)z α (z D n ) satisfying f := n + α Z f(α) 2 <. he operators n + 4

7 S j (j = 1,..., n) are then the adjoints of the multiplication operators M zj : H 2 (D n ) H 2 (D n ) defined by M zj ( f)(z) = z j f(z) (z D n ). he n-tuple S = (S 1,..., S n ) will be called the backward multishift (of multiplicity dim L). Note that S 1,..., S n are doubly commuting coisometries. Let H be a Hilbert space and = ( 1,..., n ) B(H) n a commuting n-tuple of contractions. As in the previous section, one tries to find out when is unitarily equivalent to the restriction of the backward multishift to an invariant subspace. Suppose that a Hilbert space L and an isometry V : H l 2 (Z n +, L) satisfying V j = S j V (j = 1,..., n) have been found. For α Z n + let V α : H H be the corresponding coordinate function, V α h = (V h)(α). he intertwining relations V j = S j V mean that for each α Z n + and h H one has V α j h = (V j h)(α) = (S j V h)(α) = V α+ej h. Hence V α+ej = V α j for all α Z n + and j = 1,..., n. By induction, this gives V α = V 0,...,0 α (2) for all α Z n +. hus one can choose only V 0,...,0 : H H, the remaining operators V α are already given by (2). Moreover, since V should be an isometry, it must satisfy h 2 = V h 2 = α Z n + V α h 2 = α Z n + In particular, for each F {1,..., n} one has F h 2 = α Z n + V 0,...,0 α h 2 for all h H. V 0,...,0 α F h 2, where F = j F j (in particular, = I H ). Denote by F the cardinality of F. It follows from (2) that ( 1) F F h 2 = ( 1) F V 0,...,0 α F h 2 F {1,...,n} = V 0,...,0 β h 2 β Z n + α β max{β j α j } 1 F {1,...,n} α Z n + ( 1) β α = V 0,...,0 h 2. Hence must satisfy for all h H, or equivalently, ( 1) F F h 2 0 (3) F {1,...,n} ( 1) F F F 0. (4) F {1,...,n} 5

8 ( ) 1/2 If (4) is satisfied then set D := F {1,...,n} ( 1) F F F and D := D H. he simplest choice for L is L = D, V 0,...,0 = D and V α = D α (α Z n +). hen for each h H one has V h 2 = α Z n + = lim V α h 2 = lim max α j k 1 max α j k 1 F {1,...,n} = lim max β j k β h 2 V 0,...,0 α h 2 ( 1) F α F h 2 α β,max(β j α j ) 1 max α j k 1 ( 1) β α. Note that α β,max(β j α j ) 1 max α j k 1 ( 1) β α = 0 unless {β 1,..., β n } {0, k}. If {β 1,..., β n } {0, k}, then the sum is equal to ( 1) supp β. So V h 2 = lim ( 1) F F k h 2 = h 2 + lim ( 1) F F k h 2. F {1,...,n} =F {1,...,n} So V will be an isometry if (SO) lim j k theorem was proved: = 0 for j = 1,..., n. hus the following heorem 3.1. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of contractions satisfying F {1,...,n} ( 1) F F F 0 and (SO) lim k j = 0 for all j = 1,..., n. hen there exists a subspace M l 2 (Z n +, D ) invariant for all S 1,..., S n such that u S M. If the conditions (SO) lim j k = 0 (j = 1,..., n) are not satisfied then a weaker statement can be proved: heorem 3.2. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of contractions satisfying F {1,...,n} ( 1) F F F 0. hen there exists an operator V : H l 2 (Z n +, D ) satisfying V j = S j V (j = 1,..., n) and for all h H. V h 2 = lim F {1,...,n} ( 1) F k F h 2 As in the single variable case one would like to complete V to an isometry. his is little bit more complicated than before. he starting point is the following lemma. Lemma 3.3. Let G {1,..., n}, m = G, G = {i 1,..., i m }. Set G = {1,..., n} \ G. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of contractions satisfying F G ( 1) F F F 0 such that the operators j for j G are isometries. hen there 6

9 exist a Hilbert space H G, commuting unitary operators W G,j B(H G ) mapping V : H l 2 (Z m +, H G ) such that V h 2 = lim F G ( 1) F k F h 2 (h H), (j G) and a and V ji = S i V V j = W G,j V (i = 1,..., m), (j / G), where W G,j is lifted to l 2 (Z m +, H G ) by (W G,j f)(α) = W G,j f(α) (α Z m +, f l 2 (Z m +, H G )). Moreover, the n-tuple S 1,..., S m, W G,j (j G) is doubly commuting. Proof. Let D G := ( F G ( 1) F F F ) 1/2. Clearly N(D G ) = { x H : } F G( 1) F F x 2 = 0. Let j G. Since j is an isometry commuting with all F (F G), one has j N(D G ) N(D G ). Define a mapping j : D G H D G H by j D G h = D G j h (h H). Since j N(D G ) N(D G ), the definition is correct. Moreover, j D G h 2 = F G ( 1) F F j h 2 = F G( 1) F F h 2 = D G h 2, so j is an isometry and can be extended uniquely to an isometry (denoted also by j ) on the space D G := D G H. Clearly j (j G) are commuting isometries and can be extended to commuting unitary operators W G,j (j G) acting on a Hilbert space H G D G. Lift W G,j (j G) to the space l 2 (Z m +, H G ) by (W G,j f)(α) = W G,j f(α). Applying the previous theorem for the G -tuple ( j : j G), one concludes that there exists an operator V : H l 2 (Z m +, D G ) l 2 (Z m +, H G ) such that V ji = S i V (i = 1,..., m) and V h 2 = F G( 1) F lim F k h 2 (h H). For α = (α 1,..., α m ) Z m + define α = ( α 1,... α n ) Z n + by α ji = α i (i = 1,..., m) and α j = 0 (j G). For j G and α Z m + one has (W G,j V h)(α)=w G,j (V h(α))=w G,j V α h=w G,j D G eα h=d G j eα h=v α j h=(v j h)(α). So V j = W G,j V (j / G). Clearly the n-tuple S 1,..., S m, W G,j (j G) is commuting and Si S j = S j Si (1 i, j m, i j). By the Fuglede-Putnam theorem, the n-tuple S 1,..., S m, W G,j (j G) is doubly commuting. Corollary 3.4. Let G {1,..., n}, G = {i 1,..., i m }. Let G = {1,..., n} \ G. Let = ( 1,..., n ) B(H) n be a commuting tuple of contractions with F G ( 1) F F F 0. 7

10 hen there exist a Hilbert space H G, commuting unitary operators W G,j B(H G ) G) and a mapping V G : H l 2 (Z m +, H G ) such that V G h 2 = lim F G ( 1) F k G k F h 2 (h H), (j and V G ji = S i V G V G j = W G,j V G (i = 1,..., m), (j G), where as above W G,j is lifted to l 2 (Z m +, H G ). Moreover, the n-tuple S 1,..., S m, j (j G) is doubly commuting. Proof. Define a new seminorm on H by h = lim k G h 2. Let N = {x H : x = 0}. hen N is a subspace invariant for all j (j = 1,..., n). Let L be the completion of H/N with the norm. hen L is a Hilbert space. For j = 1,..., n define j : H/N H/N by j (h+n) = j h+n. hen j extends uniquely to an operator acting on L. he operators 1,..., n are commuting contractions, j is an isometry for j G and the m-tuple j (j G) satisfies the condition F G ( 1) F F x 2 0 (x L). By Lemma 3.3, there exist a Hilbert space H G, commuting unitary operators W G,j B(H G ) (j G) and a mapping V : L l 2 (Z m +, H G ) satisfying and V ji = S i V (i = 1,..., m), V j = W G,j V (j G), V h 2 = lim F G ( 1) F k F h 2. Let V : H L be defined be V h = h + N H/N L and let V G := V V : H l 2 (Z m +, H G ). hen and V G ji = S i V G (i = 1,..., m), V G j = W G,j V G (j G) V G h 2 = lim Clearly the n-tuple S 1,..., S m, j F G ( 1) F k G k F h 2. (j G) is doubly commuting. Let = ( 1,..., n ) B(H) n be commuting contractions. One says that satisfies the Brehmer conditions if ( 1) F F F 0 (5) F G for all G {1,..., n}, see (Sz.-Nagy and Foiaş, 1970, Section I.9) or (Brehmer, 1961). heorem 3.5. Let 1,..., n B(H) be commuting contractions satisfying (5). hen for each G {1,..., n} there exist a Hilbert space X G, doubly commuting coisometries U G,1,..., U G,n B(X G ) and an isometry V : H G X G such that ( V j = U G,j )V h (j = 1,..., n). G {1,...,n} 8

11 Moreover, for each G {1,..., n} the G -tuple U G,j and the operators U G,j (j / G) are unitaries. (j G) is the backward multishift Proof. Let X G = l 2 (Z G +, H G ) and V G : H X G be as in Corollary 3.4. Let V = G {1,...,n} V G. hen V h 2 = = G {1,...,n} A {1,...,n} V G h 2 = G {1,...,n} F G lim Ah k 2 ( 1) F. F A ( 1) F lim k k G F h 2 For each A one has F A ( 1) F = 0. Hence V h 2 = h 2 for each h H, and so V is an isometry. By heorem 1.2, any n-tuple = ( 1,..., n ) B(H) n satisfying the Brehmer conditions has a dilation consisting of commuting unitaries. However, a stronger result is true. An n-tuple U = (U 1,..., U n ) B(K) n is called a regular dilation of if α β = P H U α U β H for all α, β Z n + with supp α supp β =. Equivalently, α h, β h = U α h, U β h for all α, β Z n +, supp α supp β = and all h, h H. heorem 3.6. Let = ( 1,..., n ) B(H) n be commuting contractions. hen the following statements are equivalent: (i) satisfies the Brehmer conditions (5); (ii) there exists an extension of consisting of doubly commuting coisometries; (iii) has a regular unitary dilation. Proof. (i) (ii) was proved in the previous theorem. (ii) (iii): Let V = (V 1,..., V n ) be doubly commuting coisometric extensions of and let U = (U 1,..., U n) be the unitary extension of V. For α, β Z n + with disjoint supports and h, h H one has U α U β h, h = U α h, U β h = V α h, V β h = V α V β h, h = V β h, V α h = β h, α h = α β h, h. (iii) (i): Let U = (U 1,..., U n ) be a regular dilation of, i.e., U α h, U β h = α h, β h for all α, β Z n +, supp α supp β = {0}, h, h H. Let G {1,..., n} and x H. hen ( 1) F U G\F F x 2 0. (6) F G he left hand side of (6) is equal to ( 1) F + F U G\F F x, U G\F F x = F,F G F,F G 9 ( 1) F + F U (F F )\F F x, U (F F )\F F x

12 where c(l) = = F,F G( 1) F + F F F x 2 = L G F,F L,F F =L( 1) F + F = M L F,F :M=F F F F =L c(l) L x 2, Let M L {1,..., n}, M = m, L = l. hen ( 1) F \M + F \M = ( 1) l m 2 l m, and so c(l) = F,F :M=F F F F =L l m=0 ( 1) F \M + F \M. ( ) l ( 1) l m 2 l m = ( 1) l (2 1) l = ( 1) l. m his together with (6) gives the Brehmer conditions. he fact that Brehmer s conditions imply the existence of a regular dilation is already classical, cf. (Sz.-Nagy and Foiaş, 1970, p. 32). he structure of the regular dilation was studied in more details in (Curto and Vasilescu, 1993; 1995; Gaspar and Suciu, 1997). heorem 3.5 was formulated explicitly in (imotin, 1998). Examples 3.7. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of contractions. he Brehmer conditions are satisfied, and so has a regular unitary dilation, in particular in the following cases: (i) the operators 1,..., n are doubly commuting. Indeed, in this case ( 1) F F F x, x = (I i i )x, x = D i x 2 0. i G F G i G (ii) the operators 1,..., n are isometries. Indeed, F G ( 1) F F x 2 = 0 for all G {1,..., n} and x H. (In fact, it is sufficient to assume that all operators j but one are isometries.) (iii) Suppose that n i=1 ix 2 x 2 for all x H (such n-tuples are called spherical contractions). hen it is easy to show that F x 2 F x 2 F G, F =k+1 F G, F =k for all G {1,..., n}, k = 0,..., G 1 and x H. Consequently, satisfies the Brehmer conditions. Remark 3.8. In general it is not sufficient to assume only that (3), namely ( 1) F F x 2 0, F {1,...,n} holds for all x H. Indeed, this condition is automatically satisfied if one of the operators 1,..., n is an isometry, but the Brehmer conditions may be false. On the other hand, if additional conditions (SO) lim j k = 0 for j = 1,..., n are assumed, then has a regular dilation by heorem 3.1, and so satisfies Brehmer s conditions (5). 10

13 4 he Ando dilation and von Neumann inequality For n = 2 the situation is better. Any pair of commuting contractions has a unitary dilation, cf. (Sz.-Nagy and Foiaş, 1970, p. 20). heorem 4.1. (Ando) Let 1, 2 B(H) be commuting contractions. hen there exist a Hilbert space K H and commuting isometries V 1, V 2 B(K) such that for all i, j Z +. i 1 j 2 = P H V i 1 V j 2 H Proof. Let K = i=0 H. Define operators W i : K K (i = 1, 2) by W i (h 0, h 1,... ) = ( i h 0, D i h 0, 0, h 1, h 2,... ) (i = 1, 2). Clearly W 1, W 2 are isometries since i h D i h 0 2 = h 0 2. However, in general W 1 and W 2 do not commute. In order to obtain commuting isometries, one can modify the operators W 1, W 2 in the following way. Set L = H H H H and identify K with H i=1 L by the natural identification (h 0, h 1, h 2,... ) = ( h 0, (h 1, h 2, h 3, h 4 ), (h 5, h 6, h 7, h 8 ),... ). Let G : L L be a unitary operator. Define Ĝ : K K by Ĝ(h 0, h 1, h 2,... ) = ( h 0, G(h 1, h 2, h 3, h 4 ), G(h 5, h 6, h 7, h 8 ),... ). Clearly Ĝ is also unitary and V 1 := ĜW 1 and V 2 := W 2 Ĝ 1 are isometries. One is looking for G such that V 1, V 2 will be commuting. A direct calculation gives V 1 V 2 (h 0, h 1, h 2,... ) = ( 1 2 h 0, G(D 1 2 h 0, 0, 2 h 0, 0), (h 1, h 2, h 3, h 4 ), (h 5, h 6, h 7, h 8 ),... ) and V 2 V 1 (h 0, h 1, h 2,... ) = ( 2 1 h 0, (D 2 1 h 0, 0, 1 h 0, 0), (h 1, h 2, h 3, h 4 ), (h 5, h 6, h 7, h 8 ),... ). Since 1 2 = 2 1, the isometries V 1 and V 2 will be commuting if and only if G(D 1 2 h 0, 0, 2 h 0, 0) = (D 2 1 h 0, 0, 1 h 0, 0) (h 0 H). (7) Let L 1 = {(D 1 2 h 0, 0, 2 h 0, 0) : h 0 H} and L 2 = {(D 2 1 h 0, 0, 1 h 0, 0) : h 0 H}. It is easy to verify that D 1 2 h h 0 2 = D 2 1 h h 0 2 and dim L L 1 = dim L L 2, so it is possible to define a unitary operator G : L L satisfying (7). Define V 1 = ĜW 1 and V 2 = W 2 Ĝ 1. It is easy to see that the pair (V 1, V 2 ) is a commuting isometric dilation of ( 1, 2 ). By heorem 1.2, the isometries V 1, V 2 can be extended to commuting unitary operators, so any pair of commuting contractions has a unitary dilation. his implies immediately the following important von Neumann type inequality. Corollary 4.2. Let 1, 2 B(H) be commuting contractions. hen p( 1, 2 ) p D 2 := sup{ p(z 1, z 2 ) : (z 1, z 2 ) D 2 } 11

14 for all polynomials p in two variables. Proof. By heorem 4.1, there exist a Hilbert space K H and commuting unitary operators U 1, U 2 B(K) dilating. So p( ) = P H p(u) H p(u) = r(p(u)) sup{ p(z 1, z 2 ) : (z 1, z 2 ) D 2 }. An alternative proof follows from (Agler and McCarthy, 2005), where it was shown that any pair of commuting contractive matrices has a commutative coisometric-pair extension. For 3 or more commuting contractions the above von Neumann type inequality is not true in general. he following example was constructed by (Crabb and Davie, 1975). Example 4.3. Let H be the 8-dimensional Hilbert space with an orthonormal basis e, f 1, f 2, f 3, g 1, g 2, g 3, h. Define operators 1, 2, 3 B(H) by j e = f j, j f j = g j, i f j = g k (for i, j, k all different), j g i = δ i,j h, j h = 0. It is easy to verify that the operators 1, 2, 3 are mutually commuting and that they are contractions (in fact they are partial isometries). Let p(z 1, z 2, z 3 ) = z z z 3 3 z 1 z 2 z 3. hen p( 1, 2, 3 ) p( )e, h = 4. However, p D 3 < 4. Indeed, clearly p D 3 4. If p D 3 = 4 then there are η 1, η 2, η 3 D such that η 3 1 = η 3 2 = η 3 3 = η 1 η 2 η 3. hus (η 1 η 2 η 3 ) 3 = (η 1 η 2 η 3 ) 3 and so 1 = 1, a contradiction. Hence p D 3 < 4. A similar example was given by (Varopoulos, 1974). He constructed 3 commuting contractions on a 5-dimensional Hilbert space and a homogeneous polynomial of degree 2 such that the von Neumann inequality is not true. he following central problem is still open. Problem 4.4. Let n 3. Does there exist a constant K(n) such that p( 1,..., n ) K(n) p D n for all n-tuples of commuting contractions 1,..., n and all polynomials p in n variables? Let c n be the supremum of the norms p( 1,..., n ) taken over all commuting n- tuples ( 1,..., n ) of contractions and all polynomials p in n variables with p D n = 1. Clearly c 1 c 2. he dilation theory for a single contraction gives c 1 = 1. he Ando dilation gives c 2 = 1. Not much is known about the values of c n for n 3. he above example gives c 3 > 1 but it is not known even whether c 3 <. It is known (Varopoulos, 1974) that lim n c n =. Moreover, by (Dixon, 1976), c n grows asymptotically faster than any power of n. he best explicit estimate seems to be c n > 11 1 n for all n, see (Dixon, 1976). Of course, if = ( 1,..., n ) B(H) n is a commuting n-tuple of operators satisfying the Brehmer conditions, then has a regular dilation, and so the von Neumann inequality p( ) p D n is satisfied for all polynomials p. 12

15 5 Spherical dilations Regular { dilations considered in Section 3 are closely connected with the polydisc D n = (z1,..., z n ) C n : z j < 1 (j = 1,..., n) }. his section outlines a parallel theory connected with the unit ball B n = { (z 1,..., z n ) C n : n j=1 z j 2 < 1 }. For more details see (Müller and Vasilescu, 1993). For an operator A B(H) denote by M A : B(H) B(H) the operator defined by M A (X) = A XA (X B(H)). Let = ( 1,..., n ) B(H) n be a commuting n-tuple of operators. he Brehmer conditions can be rewritten in this notation as ( ) IB(H) M j (IH ) 0 j G for all G {1,..., n}. For spherical dilations one can consider the operator M B(H), i.e., M (X) = n j=1 j X j (X B(H)). Note that = n j=1 M j : B(H) M m (X) = α Z n + α m m! α! α X α for each m N. A commuting n-tuple of operators = ( 1,..., n ) B(H) n is called a spherical contraction if M (I H ) I H. Equivalently, n n I H, or n i=1 ix 2 x 2 for all x H. Write (1) = I H 1 1 n n. So is a spherical contraction if and = 0, i.e., if n n = I H, then is called a spherical isometry. More generally, it is possible to consider also conditions (m) 0, where only if (1) 0. If (1) (m) := (I B(H) M ) m (I H ) = ( 1) α m! α!(m α )! α α. α Z n + α m he most interesting are the cases where m = 1 and m = n. he role of the backward multishift for regular dilations will be played by certain weighted backward multishifts. As in the previous sections let l 2 (Z n +, H) be the Hilbert space of all functions f : Z n + H satisfying f 2 := α f(α) 2 <. Let m N be a fixed parameter. For α Z n + write (m + α 1)! ρ m (α) =. α!(m 1)! he weighted multishift S (m) = (S (m) 1,..., S n (m) ) acting on the space l 2 (Z n +, H) is defined by ( (S (m) ρm (α) ) 1/2f(α j f)(α) = + ej ). ρ m (α + e j ) Following the same way as in Section 3 it is possible to prove heorem 5.1. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of operators and m N. hen the following conditions are equivalent: 13

16 (i) (m) 0 and (SO) lim M k (I H) = 0; (ii) is unitarily equivalent to the restriction of S (m) to an invariant subspace. It is interesting to note that (m) 0 and M k(i H) 0 (SO) implies (1) 0. Moreover, (1) 0 and (m) 0 implies that (s) 0 for all s, 1 s m, see (Müller and Vasilescu, 1993). Again if the condition (SO) lim M k(i H) = 0 is not satisfied it is possible to complete the model. heorem 5.2. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of operators and m N. hen the following conditions are equivalent: (i) (1) 0 and (m) 0; (ii) is unitarily equivalent to the restriction of S (m) W to an invariant subspace, where W is a spherical isometry on some Hilbert space H ; (iii) is unitarily equivalent to the restriction of S (m) N to an invariant subspace, where N = (N 1,..., N n ) is an n-tuple of commuting normal operators on some Hilbert space H satisfying N 1 N N nn n = I H. Implication (ii) (iii) is based on the following result of (Athavale, 1990): any spherical isometry can be extended to an n-tuple of commuting normal operators N = (N 1,..., N n ) satisfying N1 N 1 + +NnN n = I. Note that such an n-tuple of normal operators satisfies σ(n) B n. As mentioned above, the most interesting case is the case m = n. hen the weighted multishift S (n) has an additional property that (1) = 0, i.e., S (n) is a spherical isometry. S (n) So S (n) can be extended to commuting normal operators. hus one has heorem 5.3. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of operators satisfying (1) 0 and (n) 0. hen there exist a Hilbert space K H and commuting normal operators N = (N 1,..., N n ) B(K) n such that N1 N NnN n = I K and for all α Z n +. α = P H N α H So in this case there is a complete analogy with the dilation theory of single contractions. Corollary 5.4. Let = ( 1,..., n ) B(H) n be a commuting n-tuple of operators satisfying (1) 0 and (n) 0. hen for any polynomial p in n variables. p( ) p Bn Let = ( 1,..., n ) be a spherical contraction, i.e., n i=1 i i I. hen satisfies the Brehmer conditions and so it has the regular dilation. hus the von Neumann inequality p( ) p D n is satisfied for all polynomials p. However, for spherical contractions it is more natural to consider the ball norm p Bn instead of the polydisc norm p D n. 14

17 By heorem 5.2, any spherical contraction = ( 1,..., n ) is a restriction of S (1) N to an invariant subspace, where S (1) = (S (1) 1,..., S n (1) ) is the backward weighted multishift defined above and N = (N 1,..., N n ) is an n tuple of commuting normal operators with σ(n) B n. his implies the following result, see (Drury, 1978): heorem 5.5. Let = ( 1,..., n ) B(H) n be a spherical contraction. hen for each polynomial p in n variables. p( ) p(s (1) ) However, p(s (1) ) may be bigger than p Bn. By (Varopoulos, 1974), heorem 5.6. For each K > 0 there exist n N and a commuting n-tuple of operators = ( 1,..., n ) satisfying n j=1 j 2 1 and a polynomial p of degree 3 such that (Consequently, p(s (1) ) > K p Bn ). p( ) > K p Bn. 6 Analytic models In previous sections one always started with a nice n-tuple of operators - (weighted) backward multishifts - and found a class of operators for which this multishift served as a model. he model multishift was closely related with some Hardy/Bergman space of analytic functions. Another approach is presented in this section. One starts with a nice space of analytic functions with reproducing kernel and builds a dilation theory connected with this function space. Only an outline of the proofs is given, for details see (Ambrozie et al. 2002). Let D be an open domain in C n. A Hilbert space H of functions analytic on D is called a D-space if conditions (i) (iii) below are satisfied: (i) H is invariant under the operators Z i, i = 1,..., n of multiplication by the coordinate functions, (Z i f)(z) := z i f(z), f H, z = (z 1,..., z n ) D. It follows from the next assumption and the close graph theorem that the operators Z j are, in fact, bounded. (ii) For each z D, the evaluation functional f f(z) is continuous on H. By the Riesz theorem there exists a vector C z H such that f(z) = f, C z for all f H. Let D = { z : z D}. Define the function C(z, w) := C w (z) for z D, w D. (he function C(z, w) is known as the reproducing kernel of H). It is easy to see that C is analytic on D D. (iii) C(z, w) 0 for all z D, w D. Let H be a D-space and H an abstract Hilbert space. Denote by H H the (completed) Hilbertian tensor product. Elements of H H can be viewed upon as H-valued functions analytic on D. 15

18 Consider the multiplication operators M zj on H H defined by M zj = Z j I H (j = 1,..., n) and write M z = (M z1,..., M zn ). Let us study commuting n-tuples = ( 1,..., n ) B(H) n for which Mz serves as a model. he first step is to consider the n-tuples having the joint aylor spectrum σ( ) contained in D. hen one deals, under slightly stronger assumptions on H, with n-tuples whose spectrum lies only in D. he basic prototype of a D-space is the Hardy space H 2 on the open unit disc D C. In this case C(z, w) = (1 zw) 1 and Mz is the backward shift. Let = ( 1,..., n ) be a commuting n-tuple of operators on an abstract Hilbert space H with the aylor joint spectrum σ( ) D. Consider the (2n)-tuple (L, R ) = (L 1,..., L n, R 1,..., R n ) of operators acting on B(H) defined by L j (X) = j X, R j (X) = X j (X B(H), j = 1,..., n). It is easy to see that (L, R ) is a commuting (2n)-tuple. Moreover, it is possible to show that the joint aylor spectrum of this (2n)- tuple satisfies σ(l, R ) σ(l ) σ(r ) = σ( ) σ( ) D D. he hereditary functional calculus f f(, ) defined for functions analytic on a neighborhood of σ(l, R ) by f(, ) = f(l, R )(I H ) is a generalization of the aylor functional calculus, see (Agler, 1988). Note that if p(x, y) = α,β Z c n α,β x α y β is a polynomial (or a power series) then p(, ) = α,β Z + c n α,β α β, where x = (x 1,..., x n ) + and y = (y 1,..., y n ) are commuting n-tuples, but the variables x i and y j do not commute. In the situation considered above, one can define 1 (, ) = 1 C C (L, R )(I H ). Assume that 1 (, ) 0 and let the defect operator of be defined by C D = ( 1 C (, )) 1/2. Note that in the basic example H = H 2 the positivity condition 1 C (, ) 0 reduces to I 0, i.e., is a contraction, and this notation agrees with the classical one, D = (I ) 1/2. Also, define below a bounded linear operator C : H H H. o this aim, let k u (w) denote the Martinelli kernel in n variables (Kordula and Müller, 1995; Vasilescu, 1978), that is a differential form of degree n 1 in dw 1,..., dw n and degree n in dw 1,..., dw n. he definition of C is motivated by the reproducing kernel property of the Martinelli kernel, for use in the proof of (8) below. hat is, for every analytic function f and point u in its domain one has f(u) = f(w)k u(w) where is a bounded open domain with smooth boundary, such that is included in the domain of f. For D and σ( ), set C h := C w k (w)h, (h H) where k is the operator version of the Martinelli kernel, with coefficients in B(H), the exact form of which can be found in (Kordula and Müller, 1995), for Hilbert space operators see (Vasilescu, 1978). Following the lines of the formal identities f( ) = f(w)k (w) and C (z) = C(z, ) = C(z, w)k (w) = C w (z)k (w), 16

19 after some computation using also the reproducing kernel property f(w) = f, C w of C, one can obtain the equality C h, f h = h, f( )h (8) for all h, h H and f H. his shows in particular that C choice of. Moreover, C is a bounded operator. does not depend on the Define a mapping V : H H H by the formula V = (I H D )C. (9) Within this context it is possible to prove the following result. heorem 6.1. Let H be a D-space and a commuting n-tuple of operators on a Hilbert space H such that σ( ) D and 1 C (, ) 0. hen the mapping V : H H H defined by (9) is an isometry and V j = M z j V (j = 1,..., n). Hence is unitarily equivalent to the restriction of Mz u j Mz j V H (j = 1,..., n). to an invariant subspace, namely In many interesting cases D is bounded and H is isometrically embedded into a space L 2 (m), where m is a finite nonnegative Borel measure on D. hen Z = (Z 1,..., Z n ) is a subnormal n-tuple. More precisely, it extends to the (bounded and normal) n-tuple M of multiplications by the coordinate functions on L 2 (m). Note that σ(m) = supp m D. Corollary 6.2. Let H be a D-space, a commuting n-tuple of operators on a Hilbert space H such that σ( ) D and 1 C (, ) 0. Suppose that H L 2 (m) isometrically, where m 0 is a finite Borel measure with support supp m D. hen has a normal dilation N with spectrum σ(n) supp m. Proof. Indeed, j u M z j V H, j = 1,..., n, whence α u M α z V H, and so α u P V H M α z V H for any multiindex α, where all the operators M zj considered in the space L 2 (m) H are normal. Corollary 6.3. Let H be a D-space and a commuting n-tuple of operators on a Hilbert space H such that σ( ) D and 1 C (, ) 0. Suppose that H L 2 (m) isometrically, where m 0 is a finite Borel measure with supp m D. hen von Neumann s inequality holds for all polynomials p in n variables. p( ) sup z D p(z) Consider now n-tuples such that σ( ) D. o this aim, assume the following additional hypotheses: (iv) he polynomials are dense in H, and the function 1 C is a polynomial. 17

20 Arrange the monomials z α H, α Z n + in some order. By the Gram-Schmidt orthogonalization one can find an orthonormal sequence of polynomials (ψ k (z)) k 1. Define the polynomial functions f m (z, u) = 1 m 1 k=1 ψ k (z) 1 C (z, u)ψ k(u). Note that in the case where n = 1 and H is the Hardy space H 2 (D) on the unit disc, then f m (z, u) = z m u m. Let = ( 1,..., n ) be a commuting n-tuple of operators such that 1 C (, ) 0 and sup m 1 f m (, ) <. Define now V : H H H by V h = k 1 ψ k D ψ k ( ) h. (10) hen V turns out to be well-defined, bounded and moreover this definition agrees with the previous definition (9) of V, as is shown by the following proposition. Proposition 6.4. Let D, H and be as above. Let V be given by (10) and h H. hen I = f 0 (, ) f 1 (, ) f 2 (, ) and V h 2 = h 2 lim m f m(, )h, h. Moreover, for any g H and polynomial f H, Hence definitions (9) and (10) agree. V (f g) = f( )D g. Proof. For any k 1 one has ψ k ( ) 1 1 C (, )ψ k ( ) h, h = C (, )ψ k ( ) h, ψ k ( ) h 0, and so f m (, ) f m+1 (, ) for every m. hen the limit lim m f m (, )h, h exists. Moreover for any j, m, with j < m, = m 1 ψ k D ψ k ( ) h 2 = k=j m 1 m 1 k=j D ψ k ( ) h 2 ψ k ( ) 1 C (, )ψ k ( ) h, h = (f j f m )(, )h, h, k=j and hence the partial sums of the right hand side of (10) form a Cauchy sequence, and so convergent, sequence. hus V given by (10) is well-defined. Letting j = 0 and m above one obtains, since f 0 (, ) = I, that V h 2 = lim m (f 0 f m )(, )h, h = h 2 lim m f m (, )h, h. Since the functions ψ j were obtained by Gram-Schmidt orthogonalization, any polynomial is a linear combination of such functions, and so it is sufficient to verify the second equality for f = ψ j and then use (8) to check that (9) and (10) agree. For any h H, h, V (ψ j g) = V h, ψ j g = ψ k D ψ k ( ) h, ψ j g k = D ψ j ( ) h, g = h, ψ j ( )D g (8) = C h, ψ j D g = (I H D )C h, ψ j g. 18

21 Hence definitions (9) and (10) agree. Following these lines, the following results can be obtained. heorem 6.5. Let H be a D-space such that the polynomials are dense and 1 C is a polynomial. Let be a tuple of commuting operators on a Hilbert space H. he following statements are equivalent: (i) is unitarily equivalent to the restriction of M z to an invariant subspace; (ii) 1 C (, ) 0 and lim m f m (, )h = 0 for each h H. Corollary 6.6. Let and H satisfy the previous hypotheses and condition (ii) from heorem 6.5. Suppose moreover that H L 2 (m) where m 0 is a finite Borel measure with supp m D. hen the tuple M z = (M z1,..., M zn ) of multiplications by coordinate functions on L 2 (m, H) is a normal dilation for and for any polynomial p. 7 Further examples p( ) sup z D p(z) his section lists several cases (1-8) of analytic models for commuting tuples of Hilbert space operators, providing the existence of normal dilations and von Neumann s inequality. 1. he basic example is that of contractions : H H, 1 (Sz.-Nagy and Foiaş, 1970; S.-Nagy et al., 2010), the model for which is the multiplication by the variable on the Hardy space H 2 (D). In this case C h = j zj j h (h H), and V : H H 2 (D) H is given by V h = j 0 z j (I ) 1/2 j h. hen it is possible to recover the well-known fact that V is an isometry if and only if m h 0 for all h H, in which case u M z V H, cf. Section Another example is the case when H is the Bergman space on the unit disc D. his space consists of those analytic functions that are square-integrable with respect to the planar Lebesgue measure, and its reproducing kernel is C(z, w) = (1 zw) 2. hen 1 (, ) = I and f C m (, ) = (m + 1) m m m m+1 m+1. he condition f m (, ) 0 (SO) turns out to be equivalent to m 0 (SO). Hence if I and m 0 (SO), then is equivalent to the restriction of Mz to an invariant subspace, where M z is the shift on the Bergman space, see (Agler, 1985). More generally, let H be the k-bergman space, in which case the reproducing kernel is C(z, w) = (1 zw) k. Models for operators satisfying the positivity condition 1 C (, ) = k ( ) k ( 1) j j j 0 (11) j j=0 19

22 were introduced in (Agler, 1985). Contractions satisfying (11) are called k-hypercontractions. 3. If D := B n is the Euclidean unit ball in C n and H its Hardy space, then C(z, w) = (1 z 1 w 1 z n w n ) n and H L 2 ( D, m) isometrically, where m is the rotation invariant probability measure on the unit sphere D. More generally, let H be the k-bergman space, where k > n and m is the Lebesgue measure on D. In this case the k-bergman space is isometrically contained in L 2 ((1 z 2 ) k n+1 m), C(z, w) = (1 z 1 w 1 z n w n ) k and the results in Sections 5 and 6 apply. If 1 k < n then the k-bergman space is also a D-space and the results of Section 6 apply. hus one obtains a model for n-tuples satisfying the corresponding positivity condition, see also Section 5. However, in this case the k-bergman space is not of L 2 -type, so it is not possible to obtain the von Neumann inequality. 4. Also, if D := D n is the unit polydisc and H its Hardy space, then C(z, w) = n i=1 (1 z iw i ) 1 and H L 2 ( 0 D, m) isometrically; here m is the normalized Lebesgue measure on the Shilov boundary 0 D = {z = (z 1,..., z n ) : z 1 = = z n = 1} of D. Note that the existence of a regular dilation of an n-tuple is equivalent to Brehmer s conditions ( 1) α α α 0 0 α β for all β (1,..., 1), and the inequality 1 (, ) 0 is Brehmer s condition of maximal C degree for, see Section 3. A more general notion of γ-contractions for γ Z n + was studied in (Curto and Vasilescu, 1993; 1995). 5. One can similarly consider also more general models, over certain domains D given by inequalities of the form { D = (z 1,..., z n ) C n : } c i,j z j 2 < 1, i = 1,..., l, j where all c i,j 0 (Vasilescu, 1995), or of the form p( z 1 2,..., z n 2 ) < 1 where p is a polynomial with nonnegative coefficients (Pott, 1999). 6. Another interesting case is when D is a Cartan domain, and H a generalized Bergman space. For the definitions and general properties of such spaces see (Hua, 1979; Upmeier, 1996). he unit disc in C and unit ball in C n are particular cases of such domains. Other examples are provided by the operator matrix balls D p,q consisting of all p q (p q) complex matrices z = (z i,j ) 1 i p,1 j q : C q C p of supremum operator norm z = sup h C q, h <1 z h < 1, endowed with suitable Hilbert spaces H of analytic { ( ) functions. For instance if D = D 2,2 = z = (z 1, z 2, z 3, z 4 ) C 4 : z1 z 2 < 1}, its z 3 z 4 Hardy space H 2 has the reproducing kernel ( ( z1 z C(z, w) = det I 2 2 z 3 z 4 ) ( )) 2 w1 w 3, w 2 w 4 20

23 see (Hua, 1979; Upmeier, 1996). his can provide models (Ambrozie et ( al., 2002) ) for certain commuting 4-tuples = ( i,j ) 2 11 i,j=1 such that the operator matrix 12 is contractive on H H. In the more general case of D = D p,q, for any λ q the λ-bergman space Hλ 2(D p,q) is a D-space with reproducing kernel C(z, w) = det (1 p zw ) λ. Moreover, there exists a probability measure ν = ν λ on D p,q such that H λ (D p,q ) L 2 (ν) isometrically, and the pq-tuple M z = (M zi,j ) i,j is subnormal, see (Bagchi and Misra, 1996; Hua, 1979; Upmeier, 1995). Subsequent developments of such models over general Cartan domains are given in (Arazy and Engliš, 2003), where the generalized Bergman spaces H = H λ are considered for the parameter λ in the continuous Wallach set, and moreover the multiplication M z is replaced by M z W, W being a certain generalization of a unitary tuple. his provides a multivariable generalization of the well-known result by Sz.-Nagy and Foiaş stating that is a contraction if and only if is unitarily equivalent to the restriction to an invariant subspace of the direct sum S W of a backward shift S of infinite multiplicity and a unitary operator W, see heorem If H is a regular coanalytic model of (Agler, 1982), then it is a D-space in the present sense, where D is the unit disc in C. hus the model described in Section 6 generalizes the results in (Agler, 1982). 8. A model for commuting row contractions : n i=1 ii I was given by (Arveson, 1998), by means of the Hilbert space H := H 2 (B n ) of analytic functions on the unit ball B n the reproducing kernel of which is C(z, w) = (1 n i=1 z iw i ) 1 (z, w B n ). Let B(H) denote the oeplitz C -algebra generated by operators M z1,..., M zn of the multishift M z = (M z1,..., M zn ) and A be the subalgebra of generated by M z. hen by (Arveson, 1998, heorem 6.2) for every such on a Hilbert space H there exists a unique completely positive unital map φ : B(H) such that φ(m zi ) = i for all i = 1,..., n and φ(ax) = φ(a)φ(x) for all A A and X. A map φ : B(H) on a C -algebra is m-positive if I m φ : M m M m B(H) is positive, namely (I m φ)x 0 for every x 0, where M m is the set of all complex m m matrices. φ is completely positive if it is m-positive for all m 1. hen (Arveson, 1998, heorem 8.1) for every polynomial p in n variables one has von Neumann s inequality p( ) p M where M denotes the multiplier operator norm on H. his case can be recovered by the model described in Example 3 in this section. he map φ is given by φ(x) = V XV for X, where V is the isometry provided by heorem 6.1 (for ). For another approach see Section 5. Note also (Arveson, 1998) that p M = p(s), where S is the n-tuple of symmetric left creation operators on the symmetric Fock space. Similar results appeared in (Müller and Vasilescu, 1993), and a result for the tuples r with 0 < r < 1 has been established by (Drury, 1978). his approach is related to the matrix-valued von Neumann inequality, namely, by the general dilation theorem of Arveson, an n-tuple of commuting contractions on a Hilbert space H has a unitary dilation if and only if for any k 1 and matrix-valued polynomial p = α Z c n α z α with c α M k one has p( ) sup + z D n p(z), where p( ) = α c α α M k B(H) B(H k ), see also (Kalyuzhnyi, 1998). he results for commuting row contractions can also be recovered from the noncommuting setting by letting the variables commute, see (Popescu, 1989b). 21

24 8 Concluding remarks 1. here are other directions of generalizing Sz. Nagy-Foiaş dilation theory to systems of linear operators. One of them is Agler s approach to model theory (Agler, 1988), see also (Athavale, 1987), providing models by means of liftings to algebra representations. he general idea is that two classes B F of bounded linear operators on Hilbert spaces are given, where the set B is closed with respect to direct sums and unital representations. As an example, take the case when F consists of all contractions while B is the set of all coisometries. A typical conclusion of a lifting theorem has the following form: if F, then there exist an operator S B, S B(H) and a closed linear subspace L of H, invariant under S, such that u S L. In interesting cases, there exists a universal element a in a unital C algebra A such that B B B(H) if and only if there is a unital representation π : A B(H) with π(a) = B. Denote by P the set of all hereditary polynomials p = α,β c α,βy α x β with complex coefficients c α,β, where α, β Z n + are multiindices α = (α 1,..., α n ), β = (β 1,..., β n ) and x = (x 1,..., x n ), y = (y 1,..., y n ) are n-tuples, each of them commuting but not commuting to each other. Note for later use that for any integer k 1, one may consider as well the set M k P of all k k matrix-valued hereditary polynomials p = (p s,t ) k s,t=1 where each p s,t P. Set also p( ) = α,β c α,β α β for = ( 1,..., n ). Similarly, this makes sense as well for matrix-valued hereditary polynomials p = (p s,t ) k s,t=1, by setting p( ) = (p s,t ( )) k s,t=1 (k 1). hen results of the following form can be proved (for simplicity we state below heorem 1.1 of (Agler, 1988), concerning the case n = 1). heorem 8.1. Let B(H), let A be a unital C -algebra and a A. he following statements are equivalent: (i) is unitarily equivalent to π(a) L, where π : A B(K) is a unital representation on some Hilbert space K and L K is invariant under π(a); (ii) p( ) 0 for every matrix-valued hereditary polynomial p k=1 M k P such that p(a) 0. In order to make better use of such a statement, note, for instance, that given a hereditary homomorphism φ : {p(a)} p P L(H), namely a linear map φ such that (φ(p(a)) = p(φ(a)) for all p P, if φ(a) is n-cyclic, then the complete positivity of φ is equivalent to its n-positivity (Agler, 1988, heorem 1.20). In this case φ stands for the hereditary homomorphism given (and uniquely determined) by φ(a) :=, and (ii) from above is indeed a complete positivity condition. he definitions and various applications of such lifting theorems can be found in (Agler, 1988). 2. An important result in the dilation theory of single contractions is the commutant lifting theorem, see (Sz.-Nagy and Foiaş, 1970) and for more details (Foiaş and Frazho, 1990): heorem 8.2. Let B(H) be a contraction, V B(K) its minimal isometric dilation, let X B(H) such that X = X. hen there exists Y B(K) such that Y V = V Y, Y = X and X = P H Y H. A similar result for n-tuples of contractions is not true even for n-tuples having a regular unitary dilation, see (Müller, 1994). 22

25 Nevertheless, there is a multivariable version of this result, called the functional commutant lifting theorem. Let L be a Hilbert space, H 2 (D, L) the Hardy space, M z the operator of multiplication by the variable acting on this space, let H H 2 (D, L) be a subspace invariant for Mz and let = P H M z H (by heorem 2.2, such a representation is possible if and only if k 0 (SO)). he commutant lifting theorem in this particular situation states that if X B(H), X = X then there exists a bounded analytic function f : D B(L) such that f D = X and X = P H M f H, where M f if the operator of multiplication by f. he situation is more complicated for n arbitrary, as is shown below. Note that for n 3 it is necessary to replace the usual norm f D n of f by the (Agler-)Schur norm f S := sup Z f(z), where the supremum is taken over all commuting n-tuples Z = (Z 1,..., Z n ) of strict contractions (i.e., Z j < 1). he result also provides the characterization of the functions f from the unit ball f S 1 as transfer functions of linear systems with the unitary connection matrix (Ball et al., 1999). heorem 8.3. Let E be a Hilbert space and H H 2 (D n ) E be a M z -invariant closed linear subspace. For j = 1,..., n, set j = P H M zj H and let = ( 1,..., n ). Let X B(H) satisfy X j = j X for all j = 1,..., n. he following two statements are equivalent: (i) here exists an analytic function f : D n B(E) such that X = P H M f H and f S 1; (ii) here exist positive semidefinite operators G i B(H) such that I XX = n j=1 G j and j i (I B(H) M j ) ( G i ) 0 for each i = 1,..., n. In this case, there exist a Hilbert space K and a unitary operator ( a b c d ) : K n E K n E such that f(z) = d + c (I z a) 1 z b where z : K n K n is the diagonal operator z = diag (z 1,..., z n ) of multiplications by the coordinate functions. For n = 2 this reduces to G 1 2 G and G 2 1 G 2 1 0, and f S = f. For further generalizations of the commutant lifting theorem see for instance (Ball et al., 1997; Davidson and Le, 2010; Ambrozie and Eschmeier, 2005). We mention the following version of the commutant lifting theorem for spherical contractions (Davidson and Le, 2010), see also the previous work (Ball et al., 1997) concerned only with the case of the weighted multishift (when N = 0): heorem 8.4. Let = ( 1,..., n ) B(H) n be a (commuting) spherical contraction, X B(H), X j = j X for all j. Let S (1) N be the extension described in Section 5 (i.e., S (1) is the weighted multishift described there and N = (N 1,..., N n ) is an n-tuple of commuting normal operators with σ(n) B n ). hen there exists a dilation Y of X commuting with S (1) j N j for all j such that Y = X. 3. he multivariate functional commutant lifting has applications to H -type interpolation for various problems of Nevanlinna-Pick and Carathéodory-Fejér type on domains 23