Pacific Journal of Mathematics OPERATOR VALUED ROOTS OF ABELIAN ANALYTIC FUNCTIONS FRANK LARKIN GILFEATHER Vol. 55, No. 1 September 1974 PACIFIC JOURNAL OF MATHEMATICS Vol. 55, No. 1, 1974 OPERATOR VALUED ROOTS OF ABELIAN ANALYTIC FUNCTIONS FRANK GILFEATHER In this paper, all spaces are separable Hubert spaces and all operators are bounded linear transformations. Questions involving the structure of an operator for which an analytic function of it is normal or which satisfies a polynomial with certain operator coefBcients have been considered and studied separately. Using von Neumann's reduction theory, a unified approach to these and similar questions can be given. This method yields generalizations of the cases which has been previously investigated, including structure results for n- normal operators. Through reduction theory of von Neumann algebras, the study of structural questions for a particular orerator is reduced to the properties of the often simpler, reduced operators. In all of the applications presented in this paper, the reduced operators will simply involve algebraic operators. In § 1, we introduce and study analytic functions f (z), defined on a complex domain & and taking values in a commutative von Neumann algebra Szf. Such a function will be called an abelian analytic function; and where there is any question, we shall specify the algebra Jzf. Using the direct integral decomposition of S/ into factors, we obtain the decomposition of ψ into a normal family of scalar valued analytic functions on & indexed by a real variable. The main results in this section will be to show that the zeros of the scalar valued analytic functions can be chosen to be Borel func- tions of the real variable. We shall restrict our attention to a class of abelian analytic functions, called locally nonzero, so that each scalar valued analytic function in the corresponding normal family has no subdomain on which it is identically zero. An operator T in the commutant Szff of Sf is called a root of an abelian analytic function ψ, if σ(T), the spectrum of T, is con- tained in & and φ(T) = 0 where ψ(T) is to be defined in the usual J3* algebraic manner or in an equivalent way using the direct integral decomposition of Ssf into factors. Section 2 develops the struc- ture for roots of locally nonzero abelian analytic functions. The main result, Theorem 2.1, states that the root of an abelian analytic function is "piecewise" a spectral operator of finite type. The structure theorem shows that roots of abelian analytic functions have hyperinvariant subspaces or are scalar multiples of the identity. The remaining two sections of this paper are essentially appli- 127 128 FRANK GILFEATHER cations of the structure theorem for roots of abelian analytic func- tions to several classes of operators and the further use of reduction theory in their study. In § 3, our investigation leads to theorems concerning solutions of where / is an analytic function on a domain containing o{A) and N is a normal operator. The use of reduction theory in the study of (*) was introduced by the author in [9], and solutions of (*) have been previously studied by many authors with various restrictions on /, A, or N. The most complete investigation of the solutions of (*) has been done by C. Apostol in the setting of the theory of generalized spectral operators, however, his results are of a quite different nature from those given here [1]. If we set ψ(z) = f(z) — N, then Ϋ becomes an analytic abelian function and a solution A of (*) is just a root of ψ. Hence, we may apply our methods and results; and in doing so, we are able to obtain two structure theorems for A. If there is no subdomain of on which / is identically zero, then / will be called locally nonzero. We show that whenever A is a solution of (*) where /' is locally nonzero and, of course, where σ(A) is contained in sgr, then it follows that A is the direct sum of two operators; the first, A which is algebraic and the second, A, 19 2 which is "piecewise" similar to a normal operator. In the latter situation, the summand A and the corresponding normal operator 2 have the same spectrum. Under certain conditions, we may conclude that the solution A of (*) is "piecewise" similar to a normal solution N of (*) and that A and iVo have the same spectrum. We also o give a decomposition of certain operators satisfying (*) into direct summands each of which satisfy certain operator valued polynomials. Thus, we are able to generalize results obtained previously by C. Apostol, H. Radjavi, and P. Rosenthal and others [1, 10-13, 15, 16, 18]. The structure of operators satisfying certain operator valued polynomials is studied in § 4. An important class of such operators are the ^-normal operators (n x n matrices of commuting normal operators). An ^-normal operator A satisfies a normal valued poly- nomial of degree n by virtue of the Hamilton-Cay ley Theorem; and moreover, the coefficients of the polynomial are in the center of the von Neumann algebra generated by A. N. Dunford has studied these operators primarily from the viewpoint of when they were spectral operators [6]. Since operators in a type I von Neumann algebra n are also ^-normal, they naturally occur in the study of operator algebras. Also the structure and existence of hyperinvariant sub- spaces for certain ^-normal operators have been investigated by OPERATOR VALUED ROOTS OF ABELIAN ANALYTIC FUNCTIONS 129 various authors [3-5, 12, 13, 15]. We may then apply the theorems in §1 to ^-normal operators showing that they are "piecewise" similar to spectral operators and obtaining conditions for similarity which are compatible to those given in [6]. Whenever an operator A satisfies a monic polynomial of degree less or equal to two with coefficients in the center of the von Neumann algebra generated by A, we can use reduction theory to obtain a complete structure theorem for it. This result will generalize results in [3, 16] and is closely connected to the work of A. Brown on binormal operators (2-normal) [2, 11]. Finally in § 4, we give some sufficient conditions for a root of an abelian analytic function to be a spectral operator and, more specifically, a scalar type (similar to a normal operator) operator. For the ^-normal case, our results complement those given by N. Dunford [6]. Also, we give some examples based on an example introduced by J. Stampfli of a 2-normal operator whose square is normal yet it is not similar to a normal square root of its square [18]. The essential component of von Neumann reduction theory is the concept of the direct integral decomposition of an algebra. For the details of the direct integral decomposition of a von Neumann algebra, we refer to [17]; however, we shall introduce some basic notations and results here. Let μ be the completion of a finite positive regular measure defined on the Borel sets of a separable metric space Λ, and let e, 1 <L n <Ξ: oo be a collection of disjoint Borel sets of A with union n A. Let H £ H g £ H be a sequence of Hubert spaces, with t 2 w H having dimension n and H^ being separable. By n H - ( 0 H(X)μ(dX) J A we shall denote the space of weakly /^-measurable functions from A into JSk such that f(X) e H, if λ € e , and ί ||/(λ) ||2 μ(dX) < oo. The n Λ space if is a Hubert space, and we shall denote the element feH determined by the vector valued function f(X) as I φ/(λ)«((Zλ) . An operator A on H is said to be decomposable if there exists a /^-measurable operator valued function A(X) so that A(X)f(X) for feH. The operator A is denoted by 4 =[ 0 A(λ)μ(ώλ) . A Furthermore, every von Neumann algebra J^f on a separable space is spatially isomorphic to an algebra of decomposable operators on a direct integral of Hubert spaces, such that the von Neumann algebra 130 FRANK GILFEATHER ( generated by {A(X)}, where A G J^ is a factor μ-a.e. Finally, we use the fact that if A = \ φ A(X)dμ(X) generates J^ then A(X) }Λ generates the von Neumann algebra J^f(X) μ-a.e. Whenever in our use of this decomposition, there is no confusion over the space Λ, we shall suppress it. If A is an operator, we shall denote by R(A), R(A)', and Z(A), respectively, the von Neumann algebra generated by A, the commu- tant of R{A) and the center of R{A). N. Suzuki has introduced the notion of a primary operator. One calls an operator A primary, in case R(A) is a factor; i.e., Z(A) is just the scalar multiples of the identity. Let A be defined on a separable Hubert space and let H = I φ H(X)μ(dX) be the direct integral decomposition of H related to R(A) for which the algebra R(A)(X) is a factor μ-a.e., then this decomposition is unique in the sense of [17; I. 6], Thus, the operator A is decomposed as A = 1 φ A(X)μ(dX), where A(X) is primarily JΛ μ-a.e., and we shall refer to this particular decomposition as the primary decomposition of A. We shall call a projection central for T if it is in Z(T). Finally, we shall let R(z; A) denote (zl - A)'1. 1* Abelian analytic functions* In this section, we shall develop the notion of an abelian analytic function and investigate its proper- ties. Let Szf be an abelian von Neumann algebra and ψ(z), an Jzf valued analytic function on a domain s& in the complex plane, then ψ is called an abelian analytic function with domain 3f. For the usual facts about J5* valued analytic functions, we refer to [7; III, 14]. Given an abelian von Neumann algebra J^ we may decompose it into a direct integral of factors. That is, H is unitary equivalent to a direct integral of Hubert spaces 1 φ H(X)μ(dX), and this induces a spatial isomorphism between s/ and the diagonal operators on I φ H(X)μ(dX). Thus, H= ί ®H(X)μ(dX); and for A e sf, there is a unique g e L^Λ, μ), so that A = \ φ g(X)I(X)μ(dX), where J(λ) is the identity operator on H(X) [17; I, 2.6]. Let ψ be an abelian analytic function and Sf the corresponding von Neumann algebra with \ φ H(X)μ(dX) the decomposition of H given above. Since ψ(z) belongs to J^ for each z, we have (1.1) *(*) where {fz, X) corresponds via the isomorphism mentioned above to ψ(z). We first give the relationship between the analyticity of ψ(z) and that of ψ(z, λ). OPERATOR VALUED ROOTS OF ABELIAN ANALYTIC FUNCTIONS 131 PROPOSITION 1.1. If ψ(z) is an abelian analytic function with domain £&, then ψ(z, λ), given by (1.1), is analytic on & for almost all X and \\ ψ(z, λ) j^ is uniformly bounded on compact subsets of £&. Conversely, let ψ(z, X) be a family' of functions defined on Si x A, where £& is a complex domain. If ψ(z, X) is analytic in z for almost all X on the domain £$ and if ψ(z, λ) e LJ^A, μ) with \\ψ(z, )IL uniformly bounded on compact subsets of £&, then ψ(z), given by (1.1), is an abelian analytic function with domain ϋ^. Proof. We assume that ψ is an abelian analytic function on £& and that z e 3f. The series ψ(z) — Σ N*((z ~ z)n/nϊ) converges Q o with N given by Cauchy's formula is in J^ and z is in some neigh- n borhood S of z. If JV = \ φ g(X)I(X)μ(dX) then for z fixed in o 0 n n f J Λ So, ψ(z)(X) = Σ«0»M((s - z)n/nl)I(X) for almost all λ. Hence, by o the convergence properties of power series, we may conclude that ψ(z, X) is analytic in a neighborhood of z and hence on £&μ a.e. 0 Conversely, we assume that ψ(z, X) belongs to L^A, μ) and || f{z, •) I loo is bounded for z in compact subsets of ^. For z in 3ί, let 0 ψ(z, X) = Σn^n(^)((« — z)*/nl) be the power series expansion in a o neighborhood S of z. Since the functions {g} are given by Cauchy's Q 0 n formula and ψ(z, •) is measurable, we conclude that {g} are meas- n urable. We are done if we can show that g e Ljyί, μ). That, n however, also follows from Cauchy's formula and using the hypothesis that \\ψ{z, •)!!«, are uniformly bounded on compact subsets of «£^. REMARK. If it is the case that φ(z, X) is independent of λ, then the proposition is trivial. For example, if ψ(z) = f{z)I, then ψ(z)(X) = f(z)I(X) almost everywhere. In order to save the repetitiousness of deleting a set of measure zero from every argument, whenever ψ(z) is an abelian analytic function on a domain £%r, we will always assume that ψ(z, X) is analytic on a domain containing &. The main result in this section will show that the zeros of ψ(z, X) can be chosen in a μ measurable way. Such a result consti- tutes a generalization of the key lemmas in the study of ^-normal operators by N. Dunford [6; XV, 10] and is also related to the Theorem 1 in [5]. For this problem to be well defined, we must make a restriction so that ψ(z, X) is not identically zero on some subdomain of ϋ?*. We shall call an abelian analytic function ψ locally nonzero if for every convergent sequence {z} in & with z~+z in &f then Γ\ Λ"(Ψ(z*)) = n n 0 n {0} (yV(A) denotes the nullspace of the operator A). For scalar valued functions, this is the usual definition of locally nonzero. To see this, we just let H be one dimensional, then ψ(z) is just a scalar 132 FRANK GILFEATHER valued function and Λ^(ψ(z)) Φ {0} means that f(z ) = 0. The n n following lemmas establish the relationship between ψ(z) and ψ(z, X) with respect to this property. LEMMA 1.2. An abelian analytic function ψ is locally nonzero if and only if ψ(-, X) is locally nonzero for almost all X. Proof. First assume that ψ is not locally nonzero. That is, there exists a nonzero xe H and a sequence {z } in gf converging to n z in &, so that ψ(z)x = 0. If E = {λ e A \ x(X) Φ 0} and E = 0 n 1 2 U* ίλ I TK2**, λ)a?(λ) ^ 0}, then E = E\E is a set of positive measure 2 on which ψ( , X) is not locally nonzero. Conversely, if ψ( , X) is not locally nonzero for λ in a set E of positive measure, then we can show that ψ(z) is not locally nonzero. For this, we let ψ(z, X) be zero on the subdomain 2f if λ 6 E. Since x the domain of analyticity of ψ(z, X) contains 3f, each ϋ% contains one of the subdomains of £&\ and thus, there is a subset F of E with positive measure so that Π ^ΊZD ^f a subdomain of 3f. λeF Of Therefore, ψ{z, X) = 0 for Xe F and ze £%r . Let z —*z in & and 0 % 0 0 x e H so that {λ | a (λ) ^ 0} = F, then xef) ^{ΨίzJ). This completes the proof of this lemma. Let a locally nonzero abelian analytic function ψ be decomposed as in (1.1). The following theorem shows that the zeros of the functions ψ( , λ) restricted to a compact subset of & can be made measurable. THEOREM 1.3. Let ψ(z, λ) be given by (1.1) with domain 2$ x Λ. If D is a bounded subdomain of Si with D c ϋ^, then there exist disjoint Borel sets E i = 0, 1, with the measure of Λ\Uί^o E zero u t and for λ e E the analytic function ψ(- λ) has exactly j zeros counted jf f to their multiplicities in D. Moreover, there exist Borel functions WS=i so that if Xe E , then r,(λ) 1 ^ i ^ j are those zeros. 3 Proof. Since the number of zeros of an analytic function inside a desk is given by an integral formula, it is easy to see that if %(λ) denotes the number of zeros counted to multiplicity of ψ(z, X) con- tained in Dy then S = {λ | n(X) ^ k} is Borel subset of A. Hence, if k we may set E = S\S , then E is a Borel set; and it follows k k k+1 k that Λ\\J?= Ei has measure zero. We shall fix n and define r on Q t E; and this will be clearly sufficient to complete the proof. n Henceforth, we are assuming that E — Λ, 1 <; n < oo, and, the n mapping ψ on D x A is a Borel measurable map from the product space into the complex numbers. The projection of {(z, X) | ψ(z, X) = 0} onto A is A (a.e.) and by the Principle of Measurable Choice one OPERATOR VALUED ROOTS OF ABELIAN ANALYTIC FUNCTIONS 133 finds a Borel function r: A—+D so that (r (λ), λ) is in the null space x L of ψ, that is, ^(n(λ), λ) = 0 for all XeΛ [17; I, 4.7], Consider now the function ψ(z, X)(z — r^λ))"1 = Φ(z, X). By judiciously applying Schwartz's lemma on the modulus of a complex valued function one can show that φ(z, X) is uniformly bounded in λ on compact subsets in &. Thus by Proposition 1.1 we conclude that φ is again an abelian analytic function. Moreover, it is clear that ^( , λ) has n — 1 zeros in D counted to their multiplicity almost everywhere. The propo- sition now follows with repeated application of the above argument. The motivation for introducing abelian analytic functions is to study the structure of certain of their operator roots; and in doing so, unify several previous investigations. Whenever ψ(z) is a poly- nomial with commuting normal coefficients and T is an operator commuting with those coefficients, then ψ(T) has an obvious definition. The definition of ψ(T) we shall now give will be compa table with this usual definition when ψ is a polynomial. Let ψ be an abelian analytic function on a domain £gr with values in the von Neumann algebra Stf. If H = 1 φ H(X)μ(dX) is the direct JΛ integral decomposition of H corresponding to the decomposition of Jzf into factors; and if ΓGJ/', then T is a decomposable operator. That is, T is represented as T = [ φ T(X)μ(dX) where T(X) is an operator on H. Now let Te Stf' and σ(T) c &. Since σ(T(X)) c σ(T), λ almost everywhere, the operator ψ(T(X), X) is well defined by the usual functional calculus [7, 11]. To complete the definition of ψ(T), let Γ be an admissible curve for f{T) in &. Thus ^(!Γ(λ) λ) = (2πi)~1 [ R(z; T(X))ψ(z, X)dz and f ψ(T(X), X) is clearly a measurable operator function. If we can show that it is essentially bounded, then we may define ψ(T) to be the decomposable operator given by ψ(T)(X) = ψ(T(X), X). Now let z be n a dense set on Γ. Since almost everywhere || R(z; T(X)) \\ ^ n || R(z; T) ||, we may eliminate a set E of measure zero and have on n the complement of E, \\R(z; Γ(λ))|| ^ \\R(z; T)\\ for all zeΓ. By Proposition 1.1, \\ψ(z, X) IU ^ M < °o for all z on Γ and thus \\f{z, X)R(z; T(X)) || <^ M on the complement of a set of measure zero and for all zeΓ. Hence if k = (2πi)~1\ \dz\, we have that || f (Γ(λ), λ) || ^ Mk, for almost all X and therefore ψ(T) is a bounded operator on iϊif it is the decomposable operator defined by ψ(T)(X) = f(T(X), λ). It is clear that ψ(T)ejzfr since +(Γ(λ),λ)ej/'(λ)' for each λ. We conclude our remarks on the definition of ψ(T) be noting that we have actually shown that ψ(T) satisfies the conditions of a Fubini type theorem. Alternately ψ(T) may be defined by usual J9* 134 FRANK GILFEATHER algebraic techniques as (1.2) ψ(T) = (2ττi)-1 \Ψ(z)R(z; T)dz , r where ψ(z) is a Szf valued analytic function defined on a domain containing σ(T) and with Tejzf' and the integral converging in the norm. We may conclude that z, X)R(z; T{X))dzμ(dX) (1.3) ]i ]Γ ^ ^ f(z, X)R(z; T(X))μ(dλ)dz , that is, ψ(T)(X) = ψ(T(X), X) almost everywhere. In the two applications of this theory, we wish to pursue we note that ψ(T) coincides with previously understood definitions. If ψ(z) is the polynomial ψ(z) = Nzn + + Nz + JV, with coefficients n t 0 N in an abelian von Neumann algebra, then by (1.3) we see that t ψ(T) is just NTn + + N,T + N . On the other hand, if ψ(z) is n Q a scalar valued analytic function, then by (1.3) we have established that ψ(T) is the usual operator determined by the standard functional calculus [7; VII]. Moreover, in this latter case, the fact that the definition above for ψ(T) and the usual one given by contour inte- gration are the same as a special case of Theorem 1 in [11]. 2* Roots of abelian analytic functions* We shall call T a root of the abelian analytic function ψ if ψ(T) = 0 where ψ(T) was defined in § 1. If f has domain of analyticity <& and takes values in the von Neumann algebra Jzf, then, by the definition of ψ(T), we are assuming that Γej/1 and that (J(T)CJ^ In this section, we give a structure theorem for all roots of an abelian analytic function and several applications. We shall state and prove the main theorem after which we shall restate it using the language of spectral operators. THEOREM 2.1. Let ψ be a locally nonzero abelian analytic func- tion on & taking values in the von Neumann algebra Szf and let T be a root of ψ. There exists a normal operator S in J^fr and a sequence of mutually orthogonal projections {P} in Szf with I = ΣP n n so that TP is similar to (S + L)P, where L is a nilpotent operator n n n n SL = LS and both L and the operator which induces the similarity n n n are in Proof. In assuming that T is a root of ψ(z) we have that Te Ssf. We shall give the structure of T by first decomposing T OPERATOR VALUED ROOTS OF ABELIAN ANALYTIC FUNCTIONS 135 into a direct integral of operators via the direct integral of decom- position of J^ and then determining the structure of each reduced operator in the decomposition of T. Let H = 1 φ H(X)μ(dX) be the decomposition of H corresponding J Λ to the primary decomposition of J^C Since Te J^", we may decompose T as T = \ φ T(X)μ(dX). Furthermore, by (1.3) if ψ(T) = 0, then j Λ almost everywhere ψ(T(X), X) = 0, where ψ(z, X) is an analytic function in a neighborhood of σ(T(X)). By Lemma 1.2, the analytic function ψ(z, X) is locally nonzero in 3&. In fact, by Theorem 1.3, there are disjoint Borel sets 2£<, i = 0, 1, •••, where i\US^i has measure zero, and Borel functions r^λ), % = 1, •••, so that if XeE k then r^λ), , r(X) are the zeros of ψ(z, X) in σ(T) counted to their k multiplicities. Since {£7J determine mutually orthogonal projections in J^ we may assume without loss of generality that for almost all f λ in Λ, ψ(z X) has k roots in σ(T) counted their multiplicities and y since f(A(X), X) = 0 a.e., that μ(E) = 0. 0 It follows from the measurability of {^(λ)};^, that the distinct roots of ψ(z, X) as well as their multiplicities can be chosen measurably. Thus we let ^(λ), •••, ^(λ) be the distinct roots of ψ(z, X) in σ(T) for λ in the Borel set F = {λ | ψ(z, X) has n distinct roots in σ(T)} n and let the multiplicity of z^X) be &*(λ). Define §(X) — min^ | ^^(λ) — ^•(λ)l> which is also a Borel function. For each i, we determine the algebraic projections (2.1) EAX) = (27a)-1 ( R(z; T(X))dz , M where Γ is the circle centered at z^X) of radius <5(λ)/2. Since T(X) t is an algebraic operator with σ(T(X)) c {^i(λ)}i we have =1 (2.2) T(\)/E(\)H(X) = t where JV^λ) is nilpotent of order k^X). Setting (2.3) then jβ(λ) is invertible on H(X), R(X)E(X)R(X)~1 = P«(λ) are mutually i orthogonal self-adjoint projections with I(X) = Σ ? P () f F and (2.4) UW^fiίλ)-1 - Σ «*WP,(λ) + L(λ) , where L(λ)fc = 0 and P,(λ)L(λ) = L(λ)P,(λ) for each i. The form (2.4) is what we desired as our structure theorem. The only drawback to