MULTICENTRIC HOLOMORPHIC CALCULUS FOR n TUPLES OF COMMUTING OPERATORS

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1 Adv. Oper. Theory ISSN: X (electronic) MULTICENTRIC HOLOMORPHIC CALCULUS FOR n TUPLES OF COMMUTING OPERATORS DIANA ANDREI Communicated by V. Bolotnikov Abstract. In multicentric holomorphic calculus, one represents the function φ using a new polynomial variable w = p(z), z C, in such a way that when it is evaluated at the operator T, then p(t ) is small in norm. Usually it is assumed that p has distinct roots. In this paper we aim to extend this multicentric holomorphic calculus to n tuples of commuting operators looking in particular at the case when n = Introduction Let z C n and let p be a vector of polynomials mapping z 1 p 1 (z 1 ) w 1 z = z 2 p 2 (z 2 ) = w 2 = w Cn, p n (z n ) z n where for every j = 1, 2,..., n, we have that p j (z j ) is a monic polynomial of degree d j with distinct roots λ 1,j, λ 2,j,..., λ dj,j. Consider the particular case when n = 2. In multicentric holomorphic calculus one deals with polynomials p of level below 1, thus for w = (w 1, w 2 ) C 2 we have w j 1, for j = 1, 2, i.e. D 2 = {w C 2 : w j 1, for j = 1, 2} is a unit bidisk. Therefore, by using the change of variable w j = p j (z j ) one works on unit Copyright 2019 by the Tusi Mathematical Research Group. Date: Received: Apr. 16; Accepted: Oct. 2, Mathematics Subject Classification. Primary 47A60; Secondary 46E20, 47A13, 47A25. Key words and phrases. multicentric calculus, commuting operators, lemniscates, von Neumann s inequality, homogeneous polynomials. 1 w n

2 2 D. ANDREI disks which are simple sets, and then gets results on the rather more complicated sets Ω = {z = (z 1, z 2 ) C 2 : z j p 1 j (D 2 ), j = 1, 2}. The multicentric calculus was introduced in [11] by O.Nevanlinna for single variable z, where holomorphic functions φ(z) are represented with the help of vector-valued function f = (f 1,..., f d ), where for every k = 1,..., d, f k : w f k (w) is holomorphic in w, with w = p(z). Each polynomial p(z) of degree d, with simple roots λ k, induces a unique multicentric representation of φ φ(z) = d δ k (z)f k (w) with w = p(z), k=1 where δ k denote the polynomials of degree d 1 taking the value 1 at λ k and vanishes at the other roots. In [10] we see that this multicentric representation of functions gives a simple way to generalize the von Neumann result, i.e. the unit disk is a spectral set for contractions in Hilbert spaces, or in other words, this calculus allows the preimage of the disk to be a K spectral set. In a similar way, for n = 2 the multicentric representation will be given by d 2 φ(z 1, z 2 ) = δ j,1 (z 1 )δ k,2 (z 2 )f j,k (w 1, w 2 ) k=1 j=1 with w 1 = p 1 (z 1 ), w 2 = p 2 (z 2 ). When moving to n variables, i.e. for z C n one represents the holomorphic function φ using the new variables w, and with the help of a vector f containing vector-valued functions of variables w j, mapping w f(w) C d (by d we mean d 1 d 2 d n ). The multicentric representation will then be of the form φ(z) = j δ j (z)f(w) with w = p(z), where j is a multi-index (j 1,..., j n ), with j r = 1,..., d r for all r = 1, 2,..., n. Therefore, following [11], we first extend the multicentric holomorphic calculus from one operator to a pair of commuting operators. This is possible since it is well known that the von Neumann inequality holds true for commuting contractions T = (T 1, T 2 ). Thus the functional calculus can be applied to a pair of commuting operators. All the estimates and formulae are presented in the next section of this paper, after a short introduction of basic definitions, properties and results for commuting operators. The calculus is then carried out for n tuples of commuting operators together with the main result of this paper and at the end we remind the reader what tuples of commuting matrices are.

3 MULTICENTRIC HOLOMORPHIC CALCULUS 3 2. Multicentric representations for a pair of commuting operators Basic notions about tuples of commuting operators and their spectral properties are nicely presented in papers like [3] or [8], from where we just mention a few that are relevant to our work. More useful theorems and results from other literature, such as [1], [9] and [14], are then stated so that we are comfortable when extending the multicentric holomorphic calculus from [11] to a pair of commuting operators Preliminaries. Recall that given e = {e 1, e 2,..., e n } indeterminates one defines Λ n [e] to be the exterior algebra on the generators e 1, e 2,..., e n. This is a linear space over the complex plane C endowed with an anticommutative exterior product e i e j = e j e i, for every 1 i, j n. For F = {i 1,..., i p } {1,..., n} with i 1 < < i p, we write e F = e i1 e ip. The exterior algebra over C is then given by Λ n [e] = { α F e F : e F = e i1 e ip and α F C}. F We let here e to be the identity element for the exterior product. If we denote Λ k n[e] = { F =k α F e F : α F C}, where F is the cardinal of F, then clearly dim Λ k n[e] = ( ) n k for every k n, Λ k n [e] Λ l n[e] = Λ k+l n [e] and Λ n [e] = n k=0 Λk n[e]. Given a Banach space X, the exterior algebra over X is defined to be Λ n [e, X] = { F x F e F : e F = e i1 e ip and x F X}. The subspaces Λ p n[e, X] = { F =p x F e F : x F X}, for p n are given in a similar way. Naturally Λ 0 n[e, X], Λ 1 n[e, X] and Λ n n[e, X] can be identified with X, X n and X, respectively. Since no confusion is possible Λ k n[x] and Λ n [X] can be written instead of Λ k n[e, X] and Λ n [e, X], respectively. If T B(X), (here B(X) denotes the space of all bounded linear operators on a Banach space X) one keeps the same symbol T to denote the operator defined on Λ n [X] by ( ) T x F e F = T x F e F. F F For i {1, 2,..., n}, let E i : Λ n [X] Λ n [X] be the left multiplication operator by e i : E i (e F ) = e i e F. It is usually called the creation operator. With any commuting n tuple T = (T 1,..., T n ) we associate the linear mapping defined over Λ n [X] by n δ T = T i E i : x F e F n T i x F e i e F. i=1 F F i=1 Set δ k T := δ T Λ k n[x]. We construct a co-chain complex K(T), called the Koszul complex associated with T on X as follows: K(T) : 0 δ 1 T Λ 0 n[x] δ0 T Λ 1 n[x] δ1 T δn 1 T Λ n n[x] δn T 0.

4 4 D. ANDREI The operator T is said to be non-singular, or Taylor invertible, if ker δt k = Imδ k 1 T, for k = 1,..., n, equivalently ker δ T = Imδ T. The associated Koszul complex is said to be exact in this case. The description above follows [3]. Now let H be a Hilbert space and denote by B(H) the space of all bounded linear operators acting on H. We call an n tuple of commuting operators the n tuple of Hilbert space operators T = (T 1, T 2,..., T n ) satisfying the property T i T j = T j T i for all i, j = 1, 2,..., n. In this case, T B(H) n, i.e., for every j = 1, 2,..., n we have T j B(H). The spectrum of T is then the Taylor spectrum σ T (T) = {λ C n : K(T λ) is not exact}, where K(T λ) is the Koszul complex associated with T λ. Useful results for this paper are the spectral mapping theorem and the von Neumann s inequality for a pair of commuting contractions. The spectral mapping theorem states that if p is a polynomial with complex coefficients and T B(H) then σ(p(t )) = p(σ(t )). For commuting tuples we have the following. Theorem 2.1. (Spectral Mapping Property). Let T = (T 1,..., T n ) B(H) n be a commuting n tuple of operators and let p be a polynomials in n variables. Then σ T (p(t)) = p(σ T (T)). This is a special case of Theorem 2 in [8] where a vector-valued polynomial p is used. The von Neumann s inequality [9] says that for a contraction T on a Hilbert space and a polynomial p one has p(t ) sup{ p(z) : z 1}. Andô showed in [1] that the von Neumann s inequality holds for a pair of commuting contrations T = (T 1, T 2 ) and any polynomial p in two variables, i.e. p(t 1, T 2 ) p (2.1) where by p one means the supremum norm over the bidisk D 2 in C 2. For a better understanding of this result one can also check [14]. For n tuple of commuting contractions T = (T 1, T 2,..., T n ) the von Neumann s inequality does not hold in this simple way when n > 2. Extra conditions are needed and we discuss them in the next section of this paper. Now we remind the reader of notions and results on the single variable case, following [10], where it is showed that the multicentric representation of functions gives a simple way to generalize the von Neumann result, i.e., the unit disk is a spectral set for contractions in Hilbert spaces. In other words, this calculus allows the preimage of the disk to be a K spectral set. Since the von Neumann inequality works for contractions with spectrum in the unit disk, the multicentric representation applies a suitable polynomial p to the operator T so that p(t ) becomes a contraction with spectrum in the unit disk, then the preimage becomes a spectral set and thus the usual holomorphic functional calculus holds.

5 MULTICENTRIC HOLOMORPHIC CALCULUS 5 In multicentric holomorphic calculus one represents scalar functions φ(z) in the original variable by a vector-valued function f(w) in a new polynomial variable w = p(z). With a suitable polynomial p, one can map a neighbourhood of a complicated set into a disk. To be able to estimate a holomorphic function φ effectively at an operator T, O. Nevanlinna uses in [10] the approach introduced in [11], that is, each polynomial d p(z) = (z λ j ) j=1 with simple roots λ j induces a unique multicentric representation of φ, d φ(z) = δ k (z)f k (w) with w = p(z), (2.2) k=1 where δ k denote the polynomials of degree d 1 taking the value 1 at λ k and vanishes at the other roots. In [11] is discussed the practical computation of the Taylor series of f k. In fact, the coefficients may be computed recursively if the derivatives of the original function φ are available at the local centres λ k. O. Nevanlinna in [10] states that the representation (2.2) allows an obvious avenue for analysis, estimation and computation in complicated sets. One just treats the functions f k in disks w ρ and combines the estimates for φ in the sets V p (T ), where V p (T ) = {z C : p(z) p(t ) } satisfying p(z) ρ. In [10] the author demonstrates this approach by generalizing a well-known result of von Neumann on contractions in Hilbert spaces. In order to do this an estimate of the following form is needed sup f k (w) C(ρ) sup φ(z). (2.3) w ρ p(z) ρ This would then imply that the sets V p (T ) are K spectral sets with some K. Now, let γ ρ denote the lemniscate γ ρ = {z C : p(z) = ρ}. For small ρ the lemniscate consists of d separate circular curves, while for large ρ it reduces to just one circular curve. In general the lemniscate is smooth except if it contains a critical point, where the derivative of p vanishes. Thus there are at most d 1 such exceptional values ρ. A more detailed discussion on the lemniscate and separation can be found in [2], a joint paper with O. Nevanlinna. There we look at the separation of spectrum for the case when polynomial p has simple roots and then by taking powers of p we analyze the case when we have roots with multiplicities. The key result in [10] is the following theorem, which will just be stated here. For the proof one can check [10]. Theorem 2.2. If p is a monic polynomial of degree d with distinct roots, then there exists a constant C such that if φ is holomorphic for p(z) ρ, then the

6 6 D. ANDREI functions f k in (2.2) are holomorphic for w ρ and if γ ρ does not contain any critical points of p the estimate (2.3) holds with some C(ρ) satisfying C(ρ) 1 + C s(ρ), d 1 where s(ρ) denotes the distance from γ ρ to the set of critical points Formulas and estimates. Let z = (z 1, z 2 ) C 2 and let p = (p 1 (z 1 ), p 2 (z 2 )), with p 1, p 2 polynomials of degrees d 1, d 2, respectively, with distinct roots λ 1,1,..., λ d1,1 and λ 1,2,..., λ d2,2, respectively. Let T = (T 1, T 2 ) be a pair of commuting operators on a Hilbert space H. Denote by V p (T) the set V p (T) = {(z 1, z 2 ) C 2 : p j (z j ) p j (T j ) for j = 1, 2} and note that by the spectral mapping theorem we have the following. Proposition 2.3. With the above notations we have σ T (T) V p (T). Proof. Indeed, for p(z 1, z 2 ) = (p 1 (z 1 ), p 2 (z 2 )), (z 1, z 2 ) C 2, one applies the spectral mapping theorem twice σ T (p(t)) = p(σ T (T)) σ T (p 1 (T 1 ), p 2 (T 2 )) = (p 1 (σ(t 1 )), p 2 (σ(t 2 ))) = (σ(p 1 (T 1 )), σ(p 2 (T 2 ))) since the Taylor spectrum for single operators equals the usual spectrum, i.e. σ T (T ) = σ(t ). Knowing that V pj (T j ) are K spectral sets and σ(t j ) V pj (T j ), for every j = 1, 2, the stated inclusion follows immediately. Following the same line of proof, one can state a more general result. Proposition 2.4. Let z = (z 1,..., z n ) C n and let p = (p 1 (z 1 ),..., p n (z n )), with p 1,..., p n polynomials of degrees d 1,..., d n, respectively, with distinct roots λ 1,1,..., λ d1,1,..., λ 1,n,..., λ dn,n, respectively. Let T = (T 1,..., T n ) be an n tuple of commuting operators acting on a Hilbert space H. Denote by V p (T) the set V p (T) = {(z 1,..., z n ) C n : p j (z j ) p j (T j ) for j = 1,..., n}. Then σ T (T) V p (T). When applying the multicentric calculus described above (see [11]) we treat our two variables separately. First we fix one variable, say we fix z 2, then we fix the first one, z 1, and we apply the calculus as follows. The polynomials p j are taken as new variables w 1 = p 1 (z 1 ) and w 2 = p 2 (z 2 ) and functions φ(z 1, z 2 ) are represented with the help of a vector-valued function f, mapping (w 1, w 2 ) f(w 1, w 2 ) C d 1+d 2. Recall that in the single variable case, if φ is holomorphic in a neighbourhood of w ρ then f is holomorphic in the disk, see Theorem 1.1 in [10]. When dealing with multiple variables, if φ is holomorphic in a neighbourhood of w j ρ j, for j = 1, 2,..., n then f is holomorhic in all variables, which means that all components f j,k, as functions of multiple variable, are holomorphic in all variables, see [3]. Thus when applying

7 MULTICENTRIC HOLOMORPHIC CALCULUS 7 the calculus by treating the variables separately, all the results from one case variable can be used and the calculations that follow below hold true. Note that, when applying φ to the operator T, then φ will be well defined as a polynomial with commuting T js and things hold when going to limit for analytic functions. Now, denote by δ k,1 P d1 1, δ j,2 P d2 1 the Lagrange interpolation basis polynomials at λ j,1 and λ k,2, respectively, that is δ k,1 (z 1 ) = δ j,2 (z 2 ) = 1 p 1(λ k,1 ) 1 p 2(λ j,2 ) (z 1 λ j,1 ) j k (z 2 λ k,2 ). Then, when fixing z 2, the multicentric representation of φ takes the form k j φ(z 1, z 2 ) = δ j,1 (z 1 )f j,1 (p 1 (z 1 ), z 2 ) j=1 which becomes the following when fixing z 1 d 2 φ(z 1, z 2 ) = δ j,1 (z 1 )δ k,2 (z 2 )f j,k (p 1 (z 1 ), p 2 (z 2 )). (2.4) k=1 j=1 Since we treat the two variables separately, let V p1 (T 1 ) = {z 1 C : p 1 (z 1 ) p 1 (T 1 ) } and V p2 (T 2 ) = {z 2 C : p 2 (z 2 ) p 2 (T 2 ) }, and let p 1 (z 1 ) = ρ 1, p 2 (z 2 ) = ρ 2, be lemniscates which are then mapped onto disks w j ρ j, for j = 1, 2. Note that ρ 1 and ρ 2 are so that the lemniscate does not pass through any critical points of p 1 or p 2, respectively. Inserting T 1, T 2 instead of z 1, z 2 in the equation (2.4), gives a simple way of defining φ(t 1, T 2 ). This way φ(t 1, T 2 ) is well defined since T 1 and T 2 commute, thus we set d 2 φ(t 1, T 2 ) = δ j,1 (T 1 )δ k,2 (T 2 )f j,k (p 1 (T 1 ), p 2 (T 2 )). k=1 j=1 In this formula δ j,1 (T 1 ) and δ k,2 (T 2 ) are polynomials and the key is to estimate f j,k (p 1 (T 1 ), p 2 (T 2 )). The following lemma is used in proving the main result of this section, Theorem 2.6. Lemma 2.5. Let z = (z 1,..., z n ) C n and let p = (p 1,..., p n ) be a vector of monic polynomials p j of degree d j, having distinct roots, for every j = 1,..., n. Let ρ j be the levels of p j, for every j = 1,..., n, that is p j ρ j. Let V p = {z C n : p j (z j ) ρ j, for j = 1,..., n}.

8 8 D. ANDREI Denote p j (z j ) = w j, j = 1,..., n. In the multicentric representation φ(z) = j δ j (z)f(w 1,..., w n ), where j δ j(z) = d 1 j 1 =1 d n j n =1 δ j 1 (z 1 )... δ jn (z n ), one has the following estimate sup w w n 1 f(w 1,..., w n ) C(ρ 1 )... C(ρ n ) sup φ(z 1,..., z n ). (2.5) Moreover, we have for every j = 1,..., n C(ρ j ) 1 + C j s(ρ j ) d j 1, (2.6) where C j is a constant and s(ρ j ) is the distance from the lemniscate of p j to the set of critical points of p j. Proof. We treat the variables separately and we apply the results from the previous section. First we fix all the variables except z 1 and we use the estimate (2.3) to get sup f j1 (w 1, z 2,..., z n ) C(ρ 1 ) sup φ(z 1,..., z n ). w 1 1 For this last estimate we fix all variables except for z 2, then we apply (2.3) and we get sup w 1 1 w 2 1 f j1,j 2 (w 1, w 2, z 3..., z n ) C(ρ 1 )C(ρ 2 ) sup φ(z 1,..., z n ). We continue in the same way until we fix all variables except for z n. Using then once more the relation (2.3) we get the desired result sup w w n 1 f(w 1,..., w n ) C(ρ 1 )... C(ρ n ) sup φ(z 1,..., z n ). The estimate (2.6) follows immediately from Theorem 2.2. Theorem 2.6. With the above notations one has d 2 φ(t 1, T 2 ) δ j,1 (T 1 ) δ k,2 (T 2 ) C(ρ 1 )C(ρ 2 ) sup φ(z 1, z 2 ), (2.7) k=1 j=1 where z = (z 1, z 2 ) and ρ 1 and ρ 2 are the levels of p 1 and p 2, respectively, such that they do not pass through any critical points. Proof. Since the von Neumann s inequality holds for a pair of commuting contractions on a Hilbert space H, see (2.1), one can apply it to estimate f j,k (p 1 (T 1 ), p 2 (T 2 )) provided that p 1 (T 1 ) and p 2 (T 2 ) commute. This is possible since we always assume f to be holomorphic in a compact set, as well as its components f j,k, so then f j,k are holomorphic in an open neighbourhood of that set, and hence the

9 MULTICENTRIC HOLOMORPHIC CALCULUS 9 von Neumann inequality can be applied to these holomorphic functions by an approximation argument. Thus we have for all j, k f j,k (p 1 (T 1 ), p 2 (T 2 )) Therefore we get the following φ(t 1, T 2 ) d 2 d 1 k=1 j=1 d 2 d 1 sup f j,k (w 1, w 2 ). w 1, w 2 1 δ j,1 (T 1 ) δ k,2 (T 2 ) f j,k (p 1 (T 1 ), p 2 (T 2 )) δ j,1 (T 1 ) δ k,2 (T 2 ) k=1 j=1 sup w 1, w 2 1 f j,k (w 1, w 2 ). (2.8) To estimate sup f j,k (w 1, w 2 ) where w 1 1 and w 2 1 from above by sup z Vp φ(z 1, z 2 ) we apply Lemma 2.5 with n = 2 in (2.5) to the inequality (2.8) and thus we get the desired estimate (2.7). 3. Representations for an n tuple of commuting operators Further we want to extend the above results for an n tuple of commuting contractions. Let T = (T 1,..., T n ) be an n tuple of commuting contractions on a Hilbert space H. It is known that the von Neumann s inequality fails for n 3, but it is still a question whether it holds for a constant C n, that depends on n. Due to Dixon [4], we know that for a smaller class of polynomials and some constant C n, the von Neumann s inequality holds true when applied to an n tuple of commuting contractions on a Hilbert space H. The smaller class of polynomials are the k homogeneous polynomials that we describe below together with main result on the constant C n due to [5] Homogeneous polynomials. It is an open problem in operator theory to determine whether or not there exists a constant C n that adjust von Neumann s inequality, more precisely, it is unknown if for every n there exists a constant C n such that P (T 1,..., T n ) C n sup{ P (z 1,..., z n ) : z i 1}, for every polynomial P in n variables and every n tuple (T 1,..., T n ) of commuting contractions in B(H) (the set of all bounded linear operators acting on a Hilbert space H). According to [5], Dixon in [4] gave lower estimates for the optimal C n. He did this by considering the problem in the smaller class of k homogeneous polynomials. Definition 3.1. A k homogeneous polynomial in n variables is a function P : C n C of the form P (z 1,..., z n ) = a α z α, α Λ(k,n)

10 10 D. ANDREI where Λ(k, n) := {α N n 0 : α := α α n = k}, z α = z α 1 1 z α n n a α C. Dixon s estimate on C n is n 1 2[ k 1 2 ] Ck, (n) n k 2 2, where [z] denotes the integer part of z (see [4]). He studied the asymptotic behaviour (as n tends to infinity) of the smallest constant C k, (n) such that and P (T 1,..., T n ) B(H) C k, (n) sup{ P (z 1,..., z n ) : z i 1}, (3.1) for every k homogeneous polynomial P in n variables and every n tuple of commuting contractions T. Further, Montero and Tonge in [7], for 1 q <, they consider C k,q (n), the smallest constant such that n P (T 1,..., T n ) B(H) C k,q (n) sup{ P (z 1,..., z n ) : z i q 1}, for every k homogeneous polynomial P in n variables and every n tuple of commuting contractions T with n i=1 T i q 1. Their upper and lower estimates for the growth of C k,q (n) are the following i=1 n k 1 q 2[ 1 k 2] Ck,q (n) n k 2 q for 1 q 2, n k 2 1 2([ k 2]+1) Ck,q (n) n k 2 2 for 2 q <, where q 1 denotes the conjugate of q, i.e., + 1 = 1. In all the estimates k is q q considered to be fixed. Based on the combinatorial methods from [4], the authors in [5] change the construction of the Hilbert space and the operators given there to find the exact asymptotic growth of C k, (n), answering a question posed by Dixon. Their main result in [5] is the following theorem. Theorem 3.2. For k 3, and 1 q <, let C k,q (n) be the smallest constant such that Q(T 1,..., T n ) B(H) C k,q (n) sup{ Q(z 1,..., z n ) : (z i ) i q 1}, for every k homogeneous polynomial Q in n variables and every n tuple of commuting contractions T with n i=1 T i q 1. Then (i) C k, n k 2 2 (ii) for 2 q < we have log 3/q (n)n k 2 2 C k,q (n) n k 2 2. In particular, n k 2 2 ε C k,q (n) n k 2 2 for every ε > 0. Now we move to Dixon s estimate (3.1) and extend the multicentric calculus for an n tuple of commuting contractions T = (T 1,..., T n ) and then we will comment on Theorem 3.2.

11 MULTICENTRIC HOLOMORPHIC CALCULUS Main estimates. Let z = (z 1,..., z n ) C n, p(z) = (p 1 (z 1 ),..., p n (z n )) with p j polynomial of degree d j with distinct roots, for every j = 1,... n. Denote w = p(z) with w = (w 1,..., w n ). In the multicentric decomposition φ(z) = j δ j (z)f(w) we write the components of the vector f(w) as a sum of m k homogeneous polynomials and thus φ(z) becomes a sum of m multicentric decompositions φ k, consisting of k homogeneous polynomials in variable w. For these we can apply the functional calculus, since von Neumann s inequality holds. Then the main estimate is obtained by summing up all estimates for φ k. The computations are worked out as follows. Consider the decomposition φ(z) = j δ j (z)f(p(z)) (3.2) where f(w) = m a α p(z) α = k=0 α =k k=0 m Q k,j (p(z)), that is, f is a sum of m k homogeneous polynomials in n variables, denoted by Q k,j and by j δ j(z) we mean d n δ j1 (z 1 ) δ jn (z n ). j 1 =1 j n =1 Then we can define φ k (z) = j δ j (z) α =k a α p(z) α = j δ j (z)q k,j (w). (3.3) With the above notation it is obvious that (3.2) becomes m φ(z) = φ k (z). (3.4) k=0 The main result of this paper is the following. Theorem 3.3. Let z C n, p(z) = (p 1,..., p n ) be a vector of polynomials such that for j = 1,..., n, p j is a monic polynomial of degree d j with distinct roots and let ρ j 0 satisfying p j (z j ) ρ j be such that the lemniscate contains no critical points of p j. Let k 3 and let T = (T 1,..., T n ) be an n tuple of commuting contractions. Denote V p = {z C n : p j (z j ) ρ j, for j = 1,..., n}. Then for all φ defined by (3.4) there holds φ(t) K sup φ(z), with K being a constant that depends on the levels ρ j s of the polynomials p j s, on δ j at the operator T and on the degree of φ but not on φ itself.

12 12 D. ANDREI For the poof we need to define the norm φ = max k sup φ k (z). This is indeed a norm since φ 0 and if φ = 0 then φ = 0, hence for any a C n we have aφ = a φ. It is also easy to see that the following holds φ + ψ = max k max k sup φ k (z) + ψ k (z) sup φ k (z) + max sup ψ k (z) k = φ + ψ. We denote the dimension of the space of polynomials where φ lies by (d, m) where d is the vector containing the degrees of the polynomials p j. The space where φ lies is finite dimensional and using the fact that finite dimentional spaces are norm equivalent, then there exists a constant C such that 1 φ sup φ(z) C φ. (3.5) C Since sup z Vp φ k (z) max k sup z Vp φ k (z) = φ for every k, then using the above estimate (3.5) we get sup φ k (z) C sup φ(z). (3.6) 3.3. Proof of Theorem 3.3. The goal is to estimate φ(t). Inserting T and then applying the norm to (3.4) we see that m φ(t) φ k (T), and using (3.3) we get m φ(t) k=0 j 1 =1 d n j n =1 k=0 δ j1 (T 1 ) δ jn (T n ) Q k,j (p(t)). Since Q k,j (p(t)) are k homogeneous polynomials, from (3.1) it follows that Q k,j (p(t)) C k, (n) sup p j (z j ) q 1 Q k,j (p(z)). Note that the constant C k, (n) does not depend on the degree of the polynomial d n Q k,j, thus, if for simplicity we denote δ j1 (T 1 ) δ jn (T n ) by A k, we have φ(t) j 1 =1 m A k C k, (n) k=0 j n=1 sup p j (z j ) q 1 Q k,j (p(z)). Now we can use Lemma 2.5 to each component φ k (z) to get sup Q k,j (p(z)) C(ρ 1 )... C(ρ s ) sup φ k (z), p j (z j ) q 1

13 MULTICENTRIC HOLOMORPHIC CALCULUS 13 where s n and ρ j 1 is the level of p j, such that it does not pass through any critical points and C(ρ j ) is given in Lemma 2.5. Again, for simplicity we denote C(ρ 1 )... C(ρ s ) by B k, which is a constant depending on the levels ρ j and is finite whenever the lemniscate p j (z j ) ρ j does not pass through any critical points. Hence we have φ(t) m k=0 A k C k, (n)b k sup φ k (z). (3.7) Further we need to be able to estimate sup z Vp φ k (z) from above by sup z Vp φ(z). For this we use the norm, defined above. Therefore from (3.6) and (3.7) we obtain an estimate of φ(t) from above by sup z Vp φ(z), that is, φ(t) K sup φ(z), with K being a constant that depends on the levels ρ j s of the polynomials p j s, on δ j at the operator T and on the degree of φ but not on φ itself. Hence the proof of Theorem 3.3 is completed. Remark 3.4. Note that the above estimates from Theorem 3.3 work for any q such that 1 q < due to Theorem Concluding remarks When applying the calculus in practice to n tuples of commuting operators we use n tuples of commuting matrices. In finding matrices which commute to a given set of n n matrices two tools appear to be useful, that is, a standard form for a given matrix (Jordan canonical form) and restrictions on the form of the commuting matrices. The latter is a canonical form called the H-form introduced by K.C. O Meara and C. Vinsonhaler in [13]. Definition 4.1. A set of matrices A 1, A 2,..., A n is said to commute if they commute pairwise. A property of commuting matrices is that they preserve each other s eigenspaces, so they map the same invariant subspaces. If both matrices are diagonalizable, then they can be simultaneously diagonalized. Moreover, if one of the matrices has the property that its minimal polynomial coincide with its characteristic polynomial, i.e. the characteristic polynomial has only simple roots, then the other matrix can be written as a polynomial in the first matrix. Two Hermitian matrices (or self-adjoint matrices) commute if their eigenspaces coincide. In particular, two Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. Example 4.2. (of commuting matrices) The unit matrix commute with all matrices. Diagonal matrices commute. Jordan blocks commute with upper triangular matrices that have the same values along the diagonal and superdiagonals. If the product of two symmetric matrices is symmetric, then they must commute.

14 14 D. ANDREI Now, given a set of n n matrices one can find matrices that commute with the one in the set by using the H-form. This form was introduced by K.C. O Meara and C. Vinsonhaler in [13] as a tool for preserving the upper triangularity property in commuting matrices since the Jordan canonical form fails to accommodate this. Upper triangular matrices are easier to work with in determining commuting properties. It is well known that a set of commuting matrices can be simultaneously triangularized (see, for example, R. A. Horn and C. R. Johnson Matrix Analysis, 1985). Hence the H-form for an n n matrix over an algebraically closed field will let one assume that all commuting matrices are also upper triangular. Following the description given in [13], a basic H-matrix is a blocked-matrix generalization of a basic Jordan matrix, with associated eigenvalue λ, where one replaces the eigenvalues by scalar matrices and the 1 s by full column rank matrices in reduced raw echelon form. The H-form does not allow multiple basic H-matrices for the same eigenvalue λ, as the Jordan form does. Definition 4.3. (Definition 4.1. in [13]) A basic H-matrix with eigenvalue λ is an n n matrix A of the following form: There is a partition n 1 +n 2 + +n r = n of n with n 1 n 2 n r 1 such that when A is viewed as a blocked matrix with diagonal blocks of size n 1, n 2,..., n r, the diagonal blocks are the n i n i scalar matrices λi and the first super-diagonal blocks are full rank n i n i+1 matrices in reduced echelon form (i.e. an identity matrix followed by zero rows). All other blocks of A are zero. In this case, we say that A has an H-block structure (n 1, n 2,..., n r ). Remark 4.4. When working with diagonalizable matrices which have eigenvalues with non trivial Jordan blocks one needs to assume that φ is smooth enough. For this case extra smoothness is needed and a different approach was carried out by O. Nevanlinna in [12]. The functional calculus in [12] agrees with the holomorphic functional calculus if applied to holomorphic functions, but is defined for functions that do not need to be differentiable at any point. Acknowledgments. I would like to express my special thanks of gratitude to professor Olavi Nevanlinna who gave me numerous ideas, many constructive comments and helped me throughout the process of completing this paper. References 1. T. Andô, On a pair of commuting contractions, Acta Sci. Math. (Szeged) 24 (1963), D. Apetrei and O. Nevanlinna, Multicentric calculus and the Riesz projection, J. Numer. Anal. Approx. Theory 44 (2016), no. 2, C. Benhida and E.H. Zerouali On Taylor and other joint spectra for commuting n-tuples of operators, J. Math. Anal. Appl. 326 (2007), no. 1, P. G. Dixon, The von Neumann inequality for polynomials of degree grater than two, J. London Math. Soc. (2) 14 (1976), no. 2, D. Galicer, S. Muro, and P. Sevilla-Peris, Asymptotic estimates on the von Neumann inequality for homogeneous polynomials, J. Reine Angew. Math. 743 (2018), L. Kaup and B. Kaup, Holomorphic functions of several variables: An introduction to fundamental theory, With the assistance of Gottfried Barthel, Translated from the German

15 MULTICENTRIC HOLOMORPHIC CALCULUS 15 by Michael Bridgland, De Gruyter Studies in Mathematics, 3. Walter de Gruyter & Co., Berlin, A. M. Mantero and A. Tonge, Banach algebras and von Neumann s inequality, Proc. London Math. Soc. (3) 38 (1979), no. 2, V. Müller, Taylor functional calculus, Alpay, Daniel (ed.), Operator theory, In 2 volumes, Basel: Springer, Springer Reference, (2015). 9. J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes (German), Math. Nachr. 4 (1951), O. Nevanlinna, Lemniscates and K-spectral sets, J. Funct. Anal. 262 (2012), no. 4, O. Nevanlinna, Multicentric holomorphic calculus, Comput. Methods Funct. Theory 12 (2012), no. 1, O. Nevanlinna, Polynomial as a new variable - a Banach algebra with a functional calculus, Oper. Matrices 10 (2016), no. 3, K. C. O Meara and C. Vinsonhaler, On approximately simultaneously diagonalizable matrices, Linear Algebra Appl. 412 (2006), no. 1, V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, Department of Mathematics and Systems Analysis, University of Aalto, P.O. Box 11100, Otakaari 1M, Espoo, FI Aalto, Finland. address: diana.andrei@aalto.fi