Global holomorphic functions in several non-commuting variables II

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1 arxiv: v [math.fa] 29 Jun 207 Global holomorphic functions in several non-commuting variables II Jim Agler U.C. San Diego La Jolla, CA July 3, 207 John E. McCarthy Washington University St. Louis, MO 6330 Abstract: We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk. Math Subject Classification: 5A54 Introduction Let M n denote the n-by-n matrices with complex entries, and let M d = n= Md n be the set of all d-tuples of matrices of the same size. An ncfunction on a set E M d is a function φ : E M that satisfies (i) φ is graded, which means that if x E M d n, then φ(x) M n. (ii) φ is intertwining preserving, which means if x,y E and S is a linear operator satisfying Sx = ys, then Sφ(x) = φ(y)s. The points x and y are d-tuples, so we write x = (x,...,x d ) and y = (y,...,y d ). By Sx = ys we mean that Sx r = y r S for each r d. See [9] for a general reference to nc-functions. Partially supported by National Science Foundation Grants DMS and DMS Partially supported by National Science Foundation Grant DMS nc is short for non-commutative

2 The principal result of [2] was a realization formula for nc-functions that are bounded on polynomial polyhedra; the object of this note is to give a simpler proof of this formula, Theorem.5 below. Let δ be an I-by-J matrix whose entries are non-commutative polynomials ind-variables. Ifx M d n, thenδ(x)canbenaturallythoughtofasanelement of B(C J C n,c I C n ), where B denotes the bounded linear operators, and all norms we use are operator norms on the appropriate spaces. We define B δ := {x M d : δ(x) < }. (.) Any set of the form (.) is called a polynomial polyhedron. Let H (B δ ) denote the nc-functions on B δ that are bounded, and H (B δ) denote the closed unit ball, those nc-functions that are bounded by for every x B δ. Definition.2. A free realization for φ consists of an auxiliary Hilbert space M and an isometry such that, for all x B δ, we have φ(x) = A + B ( C MCI C A B MC J C D δ(x) [ D ) δ(x) ] C (.3). (.4) The s need to be interpreted appropriately. If x M d n, then (.4) means φ(x) = A + B id M δ(x) id M id C I D id M δ(x) C We adopt the convention of [] and write tensors vertically to enhance legibility. The bottom-most entry corresponds to the space on which x originally acts; the top corresponds to the intrinsic part of the model on M. The following theorem was proved in [2]; another proof appears in [6]. Theorem.5. The function φ is in H (B δ) if and only if it has a free realization. It is a straightforward calculation that any function of the form (.4) is in H (B δ). We wish to prove the converse. We shall use two other results, Theorems.8 and.9 below. If E M d, we let E n denote E M d n. If K and L are Hilbert spaces, a B(K,L)-valued nc function on a set E M d is a function φ such that 2.

3 (i) φ is B(K,L) graded, which means if x E n, then φ(x) B(KC n,l C n ). (ii) φ is intertwining preserving, which means if x,y E and S is a linear operator satisfying Sx = ys, then id L S φ(x) = φ(y) id K S. Definition.6. An nc-model for φ H (B δ ) consists of an auxiliary Hilbert space M and a B(C,MC J )-valued nc-function u on B δ such that, for all pairs x,y B δ that are on the same level (i.e. both in B δ M d n for some n) φ(y) φ(x) = u(y) [ δ(y) δ(x) ] u(x). (.7) Again, the s have to be interpreted appropriately. If x,y B δ M d n, then (.7) means [ ] φ(y) φ(x) = u(y) id M u(x). id C J C n δ(y) δ(x) Theorem.8. A graded function on B δ has an nc-model if and only if it has a free realization. Theorem.8 was proved in [2], but a simpler proof is given by S. Balasubramanian in [5]. Let us note for future reference that the functions u in (.7) are locally bounded, and therefore holomorphic [2, Thm. 4.6]. The finite topology on M d (also called the disjoint union topology) is the topology in which a set Ω is open if and only if for every n, Ω n is open in the Euclidean topology on M d n. If H is a Hilbert space, and Ω is finitely open, we shall let Hol nc H (Ω) denote the B(C,H) graded nc-functions on Ω that are holomorphic on each Ω 2 n A sequence of functions u k on Ω is finitely locally uniformly bounded if for each point λ Ω, there is a finitely open neighborhood of λ inside Ω on which the sequence is uniformly bounded. The following wandering Montel theorem is proved in []. If u is in Hol nc H (Ω) and V is a unitary operator on H, define V u by n (V u) Ωn = V u Ωn. 2 A function u is holomorphic in this context if for each n, for each x Ω n, for each h M d n, the limit lim (u(x+th) u(x)) exists. t 0 t 3

4 Theorem.9. Let Ω be finitely open, H a Hilbert space, and {u k } a finitely locallyuniformlyboundedsequenceinhol nc H (Ω). Thenthereexistsasequence {U k } of unitary operators on H such that {U k u k } has a subsequence that converges finitely locally uniformly to a function in Hol nc H (B δ). Let φ H (B δ). We shall prove Theorem.5 in the following steps. I For every z B δ, show that φ(z) is in Alg(z), the unital algebra generated by the elements of z. II Provethatforevery finitesetf B δ, thereisannc-modelforafunction ψ that agrees with φ on F. III Show that these nc-models have a cluster point that gives an nc-model for φ. IV Use Theorem.8 to get a free realization for φ. Remarks:. Step I is noted in [2] as a corollary of Theorem.5; proving it independently allows us to streamline the proof of Theorem To prove Step II, we use one direction of [3, Thm.3] that gives necessary and sufficient conditions to solve a finite interpolation problem on B δ. The proof of necessity of this theorem used Theorem.5 above, but for Step II we only need the sufficiency of the condition, and the proof of this in [3] did not use Theorem All three known proofs of Theorem.5 start by proving a realization on finite sets, and then somehow taking a limit. In [2], this is done by considering partial nc-functions; in [6], it is done by using noncommutative kernels to get a compact set in which limit points must exist; in the current paper, we use the wandering Montel theorem. 2 Step I Let {e j } n j= be the standard basis for Cn. For x in M n or M d n, let x(k) denote the direct sum of k copies of x. If x M d n and s is invertible in M n, then s xs denotes the d-tuple (s x s,...,s x d s). 4

5 Lemma 2.. Let z M d n, with z <. Assume w / Alg(z). Then there is an invertible s M n 2 such that s z (n) s < and s w (n) s >. Proof: Let A = Alg(z). Since w / A, and A is finite dimensional and therefore closed, the Hahn-Banach theorem says that there is a matrix K M n such that tr(ak) = 0 a A and tr(wk) 0. Let u C n C n be the direct sum of the columns of K, and v = e e 2...e n. Then for any b M n we have tr(bk) = b (n) u,v. Let A id denote {a (n) : a A}. We have a (n) u,v = 0 a A and w (n) u,v 0. Let N = (Aid)u. This is anaid-invariant subspace, but it is not w (n) invariant (since v N, but v isnot perpendicular to w (n) u). Sodecomposing C n C n as N N, every matrix in Aid has 0 in the (2,) entry, and w (n) does not. Let s = αi N +βi N, with α >> β > 0. Then [ ] A B s s = C D [ A β α B α C D β If the ratio α/β is large enough, then for each of the d matrices z r, the corresponding s (z r id)s will have strict contractions in the (,) and (2,2) slots, and each (,2) entry will be small enough so that the whole thing is a contraction. For w, however, as the (2,) entry is non-zero, the norm of s w (n) s can be made arbitrarily large. Lemma 2.2. Let z B δ M d n, and w M n not be in A := Alg(z). Then there is an invertible s M n 2 such that s z (n) s B δ and s w (n) s >. Proof: As in the proof of Lemma 2., we can find an invariant subspace N for A id that is not w-invariant. Decompose δ(z (n) ) as a map from (N C J ) (N C J ) into (N C I ) (N C I ). With s as in Lemma 2., and α >> β > 0, and P the projection from C n C n onto N, we get δ(s z (n) s) = P id δ(z(n) ) P id 0 5 β α ]. P id δ(z(n) ) P id P id δ(z(n) ) P id. (2.3)

6 The matrix is upper triangular because every entry of δ is a polynomial, and N is A-invariant. For α/β large enough, every matrix of the form (2.3) with z B δ is a contraction, so s z (n) s B δ. But s w (n) s will contain a non-zero entry multiplied by α, so we achieve the claim. β Theorem 2.4. If φ is in H (B δ ), then z B δ, we have φ(z) Alg(z). Proof: We can assume that z B δ and that φ on B δ. Let w = φ(z). If w / Alg(z), then by Lemma 2.2, there is an s such that s z (n) s B δ and φ(s z (n) s) = s w (n) s >, a contradiction. Note that Theorem 2.4 does not hold for all nc-functions. In [4] it is shown that there is a class of nc-functions, called fat functions, for which the implicit function theorem holds, but Theorem 2.4 fails. 3 Step II Let F = {x,...,x N }. Define λ = x x N, and define w = φ(x ) φ(x N ). As nc functions preserve direct sums (a consequence of being intertwining preserving) we need to find a function ψ in H (B δ ) that has an nc-model, and satisfies ψ(λ) = w. Let P d denote the nc polynomials in d variables, and define Let I λ = {q P d : q(λ) = 0}. V λ = {x M d : q(x) = 0 whenever q I λ }. We need the following theorem from [3]: Theorem 3.. Let λ B δ M d n and w M n. There exists a function ψ in the closed unit ball of H (B δ ) such that ψ(λ) = w if (i) w Alg(λ), so there exists p P d such that p(λ) = w. (ii) sup{ p(x) : x V λ B δ }. Moreover, if the conditions are satisfied, ψ can be chosen to have a free realization. Since φ(λ) = w, by Theorem 2.4, there is a free polynomial p such that p(λ) = w, so condition (i) is satisfied. To see condition (ii), note that for all x V λ B δ. we have p(x) = φ(x). Indeed, by Theorem 2.4, there is a 6

7 polynomial q so that q(λ x) = φ(λ x). Therefore q(λ) = p(λ), so, since x V λ, we also have q(x) = p(x), and hence p(x) = φ(x). But φ is in the unit ball of H (B δ ), so φ(x) for every x in B δ. So we can apply Theorem 3. to conclude that there is a function ψ in H (B δ ) that has a free realization, and that agrees with φ on the finite set F. Remark: The converse of Theorem 3. is also true. Given Theorem 2.4, the converse is almost immediate. 4 Steps III and IV Let Λ = {x j } j= be a countable dense set in B δ. For each k, let F k = {x,...,x k }. By Step II, there is a function ψ k H (B δ ) that has a free realization and agrees with φ on F k. By Theorem.8, there exists a Hilbert space M k and a B(C,M k C J ) valued nc-function u k on B δ so that, for all n, for all x,y B δ M d n, we have [ ] ψ k (y) ψ k (x) = u k (y) u k (x). (4.) δ(y) δ(x) Embed each M k in a common Hilbert space H. Since the left-hand side of (4.) is bounded, it follows that u k are locally bounded, so we can apply Theorem.9 to find a sequence of unitaries U k so that, after passing to a subsequence, U k u k converges to afunctionv inhol nc H(Ω). Wehave therefore that φ(y) φ(x) = v(y) [ δ(y) δ(x) ] v(x) (4.2) holds for all pairs (x,y) that are both in Λ M d n for any n, so by continuity, we get that (4.2) is an nc-model for φ on all B δ, completing Step III. Finally, Step IV follows by applying Theorem.8. 5 Closing remarks One can modify the argument to get a realization formula for B(K, L)-valued boundednc-functionsonb δ, ortoproveleechtheorems(alsocalledtoeplitzcorona theorems see [0] and [8]). For finite-dimensional K and L, this was done in [2]; for infinite-dimensional K and L the formula was proved in [6], using results from [7]. 7

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