Measurability Properties of Spectra

We study Borel measurability of the spectrum in topological algebras. We give some equivalences of the various properties, show that the spectrum in a Banach algebra is continuous on a dense Gs, and prove that in a Polish algebra the set of invertible elements is an FaS and the inverse mapping is a Borel function of the second class. This article has its origin in the papers [7] and [5]. We study Borel measurability of the spectrum and related sets and mappings in various classes of algebras. The best-known example is the case of Banach algebras: the spectrum is then an upper semicontinuous mapping, the set of invertible elements open, and the inverse mapping continuous. In the first part of the paper we establish relations between various measurability properties and study the set of points of measurability of the spectrum considered as a set-valued function. (This part of the paper—in particular Lemma 6 through Corollary 9—can be read independently.) We prove, among other things, the following fact. Theorem (cf. Theorem 7). If X is a Banach algebra, then the spectrum x -> o(x): X -» 2C is continuous on a dense Gs in X. The second part of the paper uses topological methods to deduce measurability results: we prove that in a Polish algebra the set of invertible elements is an FaS and the inverse mapping is a Borel measurable function of the second class. We prove in particular (Theorem 13), without any assumption of separability or local convexity, that the spectrum of a continuous linear operator acting on a complete metrizable vector space is always a GSa set. This answers a question posed by A. L. Shields [6]. Terminology. Throughout the paper X will denote a complex algebra with identity e (commutativity is not assumed), which is also a topological vector space. The continuity properties of multiplication will be specified case by case. An F-space is a complete metric vector space. A Polish space is a complete separable metric space. A topological space Z is said to be a Baire space if every nonempty open subset of Z is of second category (i.e. cannot be represented as the union of a countable family of nowhere dense subsets of Z). Received by the editors January 18,1985 and, in revised form, August 21,1985. 1980 Mathematics Subject Classification. Primary 46H05; Secondary 54H05. 1 The first author is a member of G.N.A.F.A. (C.N.R.). ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 225 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 226 S. LEVI AND Z. SLODKOWSKI Let Y and Z be topological spaces and B: Y -* 2Z a set-valued mapping. The graph of B is GrB = {(y,z) e Y X Z: z g £(>>)} and if FcZ, ^(K) = {y g F: 5(j>) n V* 0}. 5 is lower semicontinuous (lsc) at ^ S Y if for every F open in Z such that V n £(>>) 9* 0, >> G int ß-^F). B is upper semicontinuous (use) at v g y if for every V open in Z such that 5(j>) c V, there exists a neighborhood U oî y with B(t) a F for each f in Í/. B is continuous if it is both lsc and use. The maps IIY and II z on the product Y X Z denote the natural projections. The complement of a set A is denoted by CA. The inverse mapping in the algebra X is denoted by x -» x_1. Let us recall that A' is a complex algebra and a topological vector space, IF is the set of invertible elements of X, and o: A' -» 2C is the spectrum mapping: o(x) = (X e C: x -Xeé IF}. Theorem 1. The following conditions are equivalent: (i) IFw open. (ii) Gr a is closed in X X C. (iii) a « upper semicontinuous. Proof. Define 6: X x C ^> X by 8(x, X) = x Xe; 6 is continuous, and the equivalence of (i) and (ii) follows from the equalities e-l(W) = CGro and W= UX[6-\W) C\(XX {0})]. If o is use, then W is open because CW = ^({O}). Suppose, conversely, that W is open; then V = {x g X: a(x)cC\{l}}= W + e is an open neighborhood of 0 in X. Since the map (x, X) -» Xx (x g X, X g C) is continuous, there exist e > 0 and a neighborhood D of 0 in X such that B(0, e) X D c V, where 2?(0, e) = {t g C: |i| < e}. Fix x in D: then for every |f| < e, t # 0,1 € o(tx); that is, 1/t £ a(x) and o(x) c 5(0,1/e). Thus a is uniformly bounded on a neighborhood of zero. Suppose a is not use at a point x0 g X: then there exist an open neighborhood G of o(x0) in C and nets xy -» x0 and Xj g a(x7) \ G. Pick n in N such that x0 g nD and determine j0 such that V7 > 70 x^ g «D; it follows that |\y| < M for j > 70 and some M > 0. Extract a subnet {XJa } converging to X. By the equivalence of (i) and (ii), Gra is closed. Therefore X g a(xQ) c G, a contradiction. Corollary 2. 7/an>' of the conditions of Theorem 1 is verified, a(x) is compact for every x in X, and a maps compact sets into compact sets. Proof. Since a has a closed graph, it carries compact sets into closed sets. Moreover the image of a compact set is bounded since a is uniformly bounded on a balanced neighborhood of zero in X. D License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use measurability properties of spectra 227 Theorem 3. The following conditions are equivalent: (i)WisaGs. (ii) Gro is an Fa. (iii) o~1(F) is an Fa in Xfor every closed subset F of C. If X is an F-space with continuous multiplication, the next condition is also equivalent to the above three: (iv) the mapping x -» x"1 is continuous. The proof of Theorem 3 is similar to the proof of Theorem 5. The equivalence of (iv) and (i) when X is an F-space is due to Banach [3]. Remark 4. Let C(r) be the quotient field of the algebra of complex polynomials in t. It is possible to define a metrizable locally convex topology on C(r) such that multiplication is jointly continuous and the set of invertible elements is open but such that the mapping x -» x"1 is not continuous. For the details see [2, pp. 277, 279]. This shows that the completeness assumption in Theorem 3 is essential. As for Borel sets of the second class, we have Theorem 5. Suppose that the topology of X is completely metrizable. Then the following conditions are equivalent: (i) WisanFaS. (ii) Gra is a GSa. (iii) o~x(F) is a GSa for every closed subset F of the plane. Proof. Assume (ii). Thus Gro = \JmGm, where each Gm is a Gs in X X C Let (Kn) be a sequence of compact subsets of C such that C = U„ Kn. Then a-\F) = UX[(XX F) n Gra] = lJUn^[( A" x(F n K„)) n Gm). n m Note that (X X (F n K„)) n Gm is a Gs in X X C and is therefore completely metrizable. The mapping n^^x^ is closed and continuous. We use now the following result of Vainstein [8, p. 320, Theorem 6]: if / is a continuous closed mapping of a complete metric space Z onto a metrizable space W, then W is completely metrizable (and so an absolute Gs). It follows that UX[(X X (F n Kn)) n G„] is a Gs in X The formula for a'1(F) now concludes the proof. We will show below that if A' is a Polish algebra with continuous multiplication, the above conditions are fulfilled. Lemma 6. Suppose that X is a Baire space and W is a Gs. Then a is lower semicontinuous on a dense Gs in X. Proof. By Theorem 3, a'1(F) is an Fa in X for each closed subset F of the plane. Since open subsets of C are Fa's, their preimages under a are F„'s too. The mapping a is not lsc at x0 G X if there exists an open V c C such that VC\ a(x0) # 0 and x0 <£ into-_1(F). Let K = {x e X: a is not lsc at x}. Then K = {J{[o-1(V)\mto-1(V)]: V is open-in C}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 228 S. LEVI AND Z. SLODKOWSKI Each set a"1(F)\inta"1(F) is an F„ of the first category, and since C has a countable base the same conclusion holds for K. The lemma now follows because X is a Baire space. D As a consequence of Lemma 6 and Theorem 1, we have Theorem 7. Suppose X is a Baire space and W is open. Then a is continuous on a dense Gs. Remark 8. (i) By Corollary 2, a(x) is compact for each x in X. Thus a is lsc and use if and only if it is continuous, as a point-map, with respect to the Hausdorff metric in the plane. (ii) The hypotheses of Theorem 7 are verified by Banach algebras. We have, as a consequence, that the spectral radius is continuous on a dense Gs. For examples of spectral discontinuities in Banach algebras see [1]. Corollary 9. Suppose X is a Banach algebra. If the set M of points of continuity of the spectrum is closed under addition, it coincides with the whole algebra. Proof. Since the origin belongs to M, the assumption implies that M is a vector subspace. By Theorem 7, M is topologically complete and, hence, by a result of Klee [4], it is complete. Therefore M = X. D Let Z be a Polish space and A^ a subset of Z. Definition 10 [7]. A family (E¿: j, n g N) of subsets of X0 is a regular system for X0 if the following conditions hold: (i) each E¿ is relatively closed in Aq, (Ü)EJ o and yk^ B(ok) with yk ->_y in Y. We must show that v G B(o). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use measurability properties of spectra 229 Define à* g NN for every k — 1,2,.... by putting ök = sup{aTM: m > k ). Clearly ak > a* + 1 for every k, and lima* = a. Therefore B(ök + 1) c B(ök) for every k and b(o) = n^4). *»i On the other hand, am < k, and so F(am) c 5(0*) for m ^ k. This implies that for each k the closed set 5(0*) contains the sequence (ym)m > k together with its limit y. Therefore y g B(o). Proof of Theorem 11. Let F: NN -» 2Z be the set-valued mapping E(o) = DnF„a", where a = (a„)„ g NN. GrF is closed in NN X Z as a consequence of Lemma 12. Thus GrF is a Polish space. Let us put -n = nrz^GlE. tr is continuous and w(GrF) = A'0, by property (iii) of regular systems and the equality Uon„Fn°" = nnu0F„°". Let us show that -n maps open subsets of GrF onto Fa sets. Let M = {a g NN: a, = c,,..., ak = ck} be a basic open set in NN and let V be open in Z. It is enough to show that ir[(M X V) n GrF] is F„. But (M X V)C\ GrF = {(o,x): x g F(a) n F; at = c1,...,ok = ck) and tr[(M X F) n GrF] = (J (E(a) n V) = \J (fl^" n f) = ̂ n( IJ n^)=vnlf) U £n°") ' je*/ n ' V n aEM ' by property (ii) of regular systems. This shows that -n[(M X V) n GrF] is F0 in X0, since each F^* is closed in A^. By Theorem 2 of [5] we can conclude that A^ is FoS in X. D As an immediate corollary we have Theorem 13. Let T be a continuous linear operator on an F-space. Then the spectrum of T is GSa in the plane. Proof. As shown in [7], the resolvent of T admits a regular system. Thus, by the preceding theorem, it is FaS in C. Theorem 13 sharpens the conclusions of [7], where o(T) was shown to be a Borel subset of the plane and proved a GSa only in the separable or locally convex case. The preceding result naturally leads to the following open question: is every GSa of the plane the spectrum of a continuous linear operator on an F-space? Theorem 14. Let X be a Polish algebra with continuous multiplication. Then W is FaS in X, and the mapping x -* x"1 is a Borel measurable function of the second class. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 230 S. LEVI AND Z. SLODKOWSKI Proof. Let d be a complete invariant metric on X, and for each k, n in N let F* = {a g X: Vx g X, d(0,x) > 1/2" => ¿(0,ax) > 1/2* and d(0, xa) > 1/2* and a ■ Bk c Bn and Bk ■ a c B„}, where fi¿= (xG X: d(0,x) < 1/2*}, Bn= {xg A": ¿(0,x)< 1/2"}; put F„* = {a g IF: a"1 • 5, c fi„, 5, • a"1 czBn,a-Bkcz Bn, Bk-acB„}. It is easy to check that Fk — W n F*. Each F* is closed because multiplication is continuous. The family (Fk: k, n G N) is a regular system for IF: conditions (i)-(iii) have straightforward verifications. Let us prove condition (iv). Let (kn) be a sequence of natural numbers. We remark that (a-1: a g (*)„ F„*-) is an equicontinuous family—both on the right and on the left—of linear operators on X. Thus, if (as) is a Cauchy sequence in D„ Fk", the formula a'} — a'} = a'Ha » — a -)a"1shows that (a;1) is also a Cauchy sequence. Let a0 and o0 be the limits in X of (a5)and (a;1) respectively. Then, by the continuity of multiplication, a0b0 = b0aQ = eand a0 g fl„ F„*\ This proves that D„ Fk" is closed in X and (F„*: n, k G N) is aregular system for IF. By Theorem 11, IF is an FaS in AT.Let us now put f(x) = x~l for x in IF. It is clear that f(Fk) = F„* for each k andn in N. The formulae a b = b(b~l a'l)a and a-1 b'1 = b~l(b a)a~l(a, b g IF) show that for each sequence (an) = a the mapping / |Dn Fn°":D„ F„°" -»n„ F„a"is a uniform isomorphism. Let F be closed in IF; then (F n F„*: k,n e N) isa regular system for F. Moreover f(F n D„ Fn°")= /(F) n D„ F„°" is closed in X asffl n„Fn°» is complete.Therefore the mapping B: N* -» A1defined by B(a)= f(F n D„ F„°»)satisfiesthe hypotheses of Lemma 12.Since f(F) = Ux(GrB) and II^^ is open to Fa on Gr£, it follows that /(F) isan FaS in X and that /, being its own inverse, is a Borel function of the second class.D References1. B. Aupetit, Propriétés spectrales des algebres de Banach, Lecture Notes in Math., vol. 735,Springer-Verlag, Berlin and New York, 1979.2. E. Beckenstein, L. Narici, and C. Süffel, Topological algebras, Mathematics Studies, Vol. 24,North-Holland, Amsterdam, 1977.3. S. Banach, Remarques sur les groupes et corps métriques, Studia Math. 10 (1948), 178-181.4. V. Klee, invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3(1952),484-487.5. S. Levi and A. Maitra, Borel measurable images of Polish spaces, Proc. Amer. Math. Soc. 92 (1984),98-102. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use MEASURABILITY PROPERTIES OF SPECTRA231 6. A. L. Shields, The spectrum of an operator on an F-space, Proc. Roy. Irish Acad. Sect. A 74 (1974),291-292.7. Z. Slodkowski, Borel sets and the spectrum of an operator on an F-space, Proc. Roy. Soc. EdinburghSect. A (1981),257-261.8. I. A. Vainstein, On closedmappings,Dokl. Akad. Nauk SSSR 57 (1947),319-323 (Russian).Dipartimento di Matemática, Università di Pisa, Via Buonarroti 2, 56100 Pisa, Italy (Currentaddress of S. Levi)Department of Mathematics, University of California, Los Angeles, California 90024 Current address (Z. Slodkowski): Department of Mathematics, University of Illinois, Chicago, Illinois60680 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use