An Analytic Set-valued Selection and Its Applications to the Corona Theorem, to Polynomial Hulls and Joint Spectra

It is shown that for every annulus P = {z 6 Cn : 6 < \z\ < r}, b > 0, there exists a set-valued correspondence z —> K{z): P —> 2C , whose graph is a bounded relatively closed subset of the manifold {(z,w) 6 Px Cn : ziwi + ■ ■ ■ + ZnWn = 1} which can be covered by n-dimensional analytic manifolds. The analytic set-valued selection K obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation a\X\ + ■ ■ ■ + anxn = y. For example, an elementary proof is given of the following special case of a theorem due to Oka: every bounded pseudoconvex domain in C2 is a domain of holomorphy. 0. Introduction. Consider the family of complex hyperplanes Lz = {w G Cn : ZfWf + ■ ■ ■ + znwn = 1}, z G Cn \ {0}. By Hartogs's theorem there does not exist a single-valued analytic selection from {Lz} over the annulus P6,r = {zGCn:6<\z[ K(z): G —> 2C" is called analytic Received by the editors May 15, 1985. 1980 Mathematics Subject Classification. Primary 32A30, 32E20, 46J15; Secondary 32D15, 47A10. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 367 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 368 ZBIGNIEW SLODKOWSKI if for every (n + l)-dimensional complex plane L C Cfc+n, the intersection Y = Ln{(z,w): z G G, w G K(z)} has local maximum property in the following sense: there does not exist a function f(z,w) analytic in a neighbourhood of a point (z*,w*) such that |/| | Y has strict local maximum at (z*,w*). The reader is referred for background and further information on analytic multifunctions to [3-4, 8-10 and 11-15]. 1. An analytic set-valued selection. Similarly as in [11, §4] we are looking for a set-valued selection of the form (1.1) K(z) = {xGCn: (z,w) = l,\w z\z\-2[ < e?^1^, where z G Ps,r and p: (logé,logr) —► R is a smooth function. ((z,w) denotes the bilinear form ZfWi + ■ ■ ■ + znwn.) The motivation is that both the manifold (z, w) = 1 and the graph of K, (1.2) Y = {(z,w)GPs,rxCn:wGK(z)}, are preserved by the group G consisting of biholomorphic maps (1.3) (z,w)-> {Az,A~w): C2n^C2n, where A is any unitary n x n matrix. We will prove now the lemma which gives a sufficient condition for the multifunction (1.1) to be analytic. LEMMA 1.1. Let 0 < 6 < r and let p: (log6,logr) —> R be a convex, C^smooth function satisfying the inequality (1.4) p"(x)(e2p^+2x(p'(x) 1) 2) > 2(p'(x) + l)2. Then the graph Y of the multifunction K: P$¡r —> 2e", defined by (1.1) can be covered by n-dimensional analytic manifolds. PROOF. The case n = 2 of this lemma was proved in [11, Theorem 4.3 and Lemma 4.4]. (The reader will note a misprint in equation (4.7) in [11].) We have shown that the set {(z, w) G C4 : 6 < [z\ < r, w & K(z)} is strictly pseudoconvex in {(z, w) : 6 < [z\ < r} and so, as is well known, its complement can be covered by (open pieces of) two-dimensional analytic manifolds. Decompose C2n = H © V, where V = {zf = z2 = Wf = w2 = 0} and H is its orthogonal complement. Since Y C\ H is the graph of the multifunction (1.1) for n = 2, for every p G H H Y there is a two-dimensional complex manifold M such that p G M C H D Y and M is relatively compact in Y. ASSERTION. For fixed M there is e > 0 such that M x ßn_2(e) C Y where ßn_2(£) = {ueCn-2: |u| 0, (z,w) = l, 6<|z| 0 for u G C"-2, with |u| small enough. Set fa,b(u) = F(a,u,b,0) = R(\a\2 + \u\2) \b\2 + (\a\2 + H2)"1. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AN ANALYTIC SET-VALUED SELECTION 369 Since R(t) = R(\a\2) + R'([a\2)(t ]a[2) + o(t \a\2), (1.7) /0,6(«) = (R(\a\2) |6|2 |a|"2) + (£'(|a|2) |a|-4)|U|2 + o(\u\2). The scalar R'(\a\2) \a[~4 is positive. (Indeed, i?'(|a|2) |a|-4 = |a|-2(e2"(lo8lal)p'(log|a|) |a|~2) = e-2x(e-2xD + e2p(x) +e-2x), for x = log |a|, and D = (p'(x) — 1) exp(2p(x) + 2x) — 2 is positive by assumption (1.4).) Thus the function fa,b(u) is convex in some ball Z?n_2(e), and by smoothness of F and relative compactness of M in Y, £ > 0 can be chosen independently on (a, b) G M. Furthermore by (1.7) grad/a,b(0) = 0. Therefore /a,6(tt)>/a,b(0)=í,(o,0,6)0)>0, for (a, b) G M, \u\ < £, and so (a, u, b, 0) G Y, which settles the assertion. To complete the proof of the lemma observe that every point q G Y is the image by a biholomorphic map gA £ G (cf. (1.3)) of some point p G H H Y and so q G gA(M x B^2(e)) C Y. Q.E.D. REMARK. The lemma can also be checked directly by parametrizing the manifold (z,w) = 1 (for z\ t¿ 0) by z\,... ,zn,w2,..., wn and substituting w\ = Zf1 Zf1(z2w2-\-\-znwn) into (1.5). Direct computation of the Hessian of the resulting function at points with z2 = ■ ■ ■ = zn = w$ = ■ ■ ■ = wn = 0 yields fairly quickly condition (1.4). REMARK 1.2. We have to give an example of the function p(-). By substituting p(x) = \f(4x) x we reduce inequality (1.4) to the simple form r(e/(/'-l)-l)>¿(/')2. The function f(x) = \ (x a)2 + (x — a) + 2 satisfies this inequality in (a, oo). The relation of Lemma 1.1 to Definition 0.1 and to another notion of analyticity of set-valued functions is discussed in §3. 2. A proof of the corona theorem for n generators. The following theorem was proven independently and in different ways by Alexander and Wermer [2] and by the author [15]. THEOREM 2.1. Let X be a compact subset of 3D x Cn (D denotes the unit disc). Assume that for every z G 3D the section Xz = {w G Cn : (z,w) G X} is convex. Then for every point (z*,w*) in the polynomially convex hull h(X) of X there exists a bounded analytic function g: D —> CTM such that g(z*) = w* and (z,g(z)) G h(X) for all z G D. This theorem and the next proposition (cf. [11, Theorem 3.2] for n = 1 and [13, Proposition 2.3] for arbitrary n) are the basis of the recent approach to the corona theorem due to Berndtsson and Ransford [6]. PROPOSITION 2.2. Let L:G -> 2C", where G is a bounded planar domain, be an upper semicontinuous multifunction whose restriction to G is analytic. Then the graph of L is contained in the polynomial hull of the set X = {(z, w) : z G dG, w G L(z)}. (The proof is sketched at the end of this section.) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 370 ZBIGNIEW SLODKOWSKI COROLLARY 2.3 [6]. Let L: D —> 2C" be an upper semicontinuous multifunction whose restriction to D is analyticc. Then there exists a bounded analytic function g: D —> C" whose nontangential limit g(e%e) belongs to L(e10) for almost all 9. Berndtsson and Ransford [6] made an important observation that this fact, which is a direct consequence of Theorem 2.1 and Proposition 2.2, is all that is needed to get a proof of the corona theorem, provided a suitable multifunction L is constructed. In this way they obtained a new proof of Wolff's ô-theorem (and thereby a new proof of the corona theorem). After being informed about this proof, the author has pointed out that Corollary 2.3 and [11, Theorem 4.3] yield a direct proof of the corona theorem without the <3-equation (in case of two generators). Subsequently Berndtsson and Ransford obtained another direct proof (in the same case) [6, §2], and the author applied his argument to an arbitrary number of generators, as presented here. (See also final remarks in Berndtsson and Ransford [6, §3].) Let /i,...,/„ê H°°(D) satisfy the inequality (2.1) ¿2<|/1(*)|2+--+ |/„(2)|2<1, zGD. We are looking for gi,...,gn€ H°°(D) satisfying (2.2) gi(z)fi(z) + --+ gn(z)fn(z) = l, zGD, and a uniform estimate sup(\gi(z)\2 + --+ [gn(z)\2)<C¡, \z\<l where Cs depends only on 6. We can assume without loss of generality that /i,...,/„ are continuous on D. Set f(z) = (ff(z),... ,fn(z)). Then f(D) C P6 = {zGCn: 6<\z\< 1}. LEMMA 2.4. Let K: Ps —> 2C" be any multifunction satisfying the conditions of Lemma 1.1. Then the multifunction L(z) = K(f(z)), z G D, is analytic in D. PROOF. Fix z* G D and w* G L(z). Then (f(z*),w*) G Y (= gr(K)). By Lemma 1.1 there exists an n-dimensional analytic manifold M such that (f(z*), w*) G M c Y, M is locally given by n analytic equations gi(u,w) = 0, i = 1,... , n, (u,z) GC2n. Let N = {(z,w) gCxC": gz(f(z),w) = 0,1 < i < n}. Then N is an analytic variety such that (z*,w*) G N C gr(L) and has dimension > 1 at (z*,w*). Observe then that for no analytic function h(z,w) defined in a neighborhood of (z*,w*) can [h[ restricted to gr(L) have strict maximum at (z*,w*), for this would violate the open mapping theorem for the function h\N (by [7, Proposition 10, p. 54]). Q.E.D. (A more elementary but longer proof follows from [11, Proposition 5.1]; cf. also discussion in §3.) PROOF OF THE CORONA THEOREM. We construct the multifunction L as in Lemma 2.4, assuming in addition that the function p is continuous in [logé,0]. Then L satisfies the uniform bound sup{|u;|: z G D,w G L(z)} < Cs. Applying License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AN ANALYTIC SET-VALUED SELECTION 371 Corollary 2.3 to L, we obtain a bounded analytic function g: D —> Cn such that g(e%e) G L(et6) for a.a. 6. By construction of K (cf. (1.1)) this means that (f(e^),~g(ee)) = l a.a. 0, which implies (2.2). Of course sup \g(z)\ < sup \g(el9)[ < C8. Q.E.D. zGD 9 Proof of Proposition 2.2 (Sketch). If not, then there is a polynomial p(z,w) and e > 0 such that maxu|gr(L) > maxu[X, where u(z,w) = |p(2,w)|2 + s(\z\2 + ]w\2). Let (z*,w*) be any point at which u|gr(L) attains its maximum; z* G G. Using the complex Taylor formula at (z*,w*) we can write u(z,w) = Rep(z,w) + h(z,w) + r2(z,w), where p(z,w) is an analytic polynomial with degp < 2, h is a strictly positive homogeneous function in z z*, w w* of degree 2 (i.e., Hessian) and r2 = o([z z*\2 + \w w*|2). Therefore the restriction of Rep = u h — r2 to gr(L) has strict local maximum at (z*,w*). Thus Definition 0.2 fails for exp(p(z)). Q.E.D. 3. Approximation of analytic multifunctions by projections of polynomial hulls. In this section we apply the set-valued selection of §1 to prove the following result. THEOREM 3.1. Let G be a bounded planar domain and K: G —> 2C" an analytic multifunction with bounded graph. Then there exists a sequence of upper semicontinuous multifunctions Ks : G —► 2e , s > 1, analytic in G and such that (3.1) Ks+l(z)cKs(z); K(z) = f] K°(z);