Int. Journal of Math. Analysis, Vol. 8, 2014, no. 20, 967-971 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4388 Finite Codimensional Invariant Subspace and Uniform Algebra Tomoko Osawa Mathematical and Scientific Subjects Asahikawa National College of Technology Asahikawa 071-8142, Japan Copyright c 2014 Tomoko Osawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Finite codimensional invariant subspaces in the abstract Hardy space defined by a unique representing measure on a uniform algebra are described. Mathematics Subject Classification: Primary 47B48 Keywords: uniform algebra, invariant subspace, finite codimension, Gleason part Let X be a compact Hausdorff space, C(X) the algebra of all complexvalued continuous functions on X, and A a uniform algebra on X. Fix a nonzero complex homomorphism ϕ on A and a representing measure m for ϕ whose support is X. The abstract Hardy space H p = H p (m), 1 p, determined by A is defined as the closure of A in L p = L p (m) when p is finite and as the weak star closure of A in L = L (m) when p =. We set A ϕ = { f A : ϕ(f) =0} and Hϕ p = { f Hp : fdm =0}. The set M(A) denotes the maximal ideal space of A and X is regarded as a closed subset of M(A). The set G(ϕ) denotes the Gleason part of ϕ, that is, G(ϕ) ={ ψ M(A) : ψ ϕ < 2 }. Φ is called a point derivation at ϕ, when Φ is a linear functional on A and Φ(fg)=ϕ(f)Φ(g)+Φ(f)ϕ(g) (f,g A ). For a subset M in L p, M q = M L q and [ M ] is the norm closed linear p
968 Tomoko Osawa span of M. For a subset M in L, [ M ] denotes the weak star closed linear span of M. When Hϕ p = ZH p and Z = 1, put H p = the closure ( or weak star closure ) of polynomials of Z in H p and put I p = n=1 Z n H p. Then H p = H p I p. This is known in [4, Lemma 5]. Throughout this paper, we assume m is the unique representing measure of ϕ. Then we prove the following theorem. Theorem. Let 1 p and M is a non-trivial closed subspace in H p such that AM M and dim H p /M = n<. Then the following are valid. (1) G(ϕ) ={ ϕ } if and only if M = H p ϕ. (2) G(ϕ) { ϕ } if and only if M = BH p and B = n j=1 (Z a j )/(1 a j Z) where H p ϕ = ZHp, Z =1a.e. and a j < 1(1 j n). We need several lemmas in order to prove Theorem. Lemma 1. Let M be a closed subspace ( or weak star closed ) of H p and AM M. Ifq>pthen dim H q /M q = dim H p /M and AM q M q.ifq<p then dim H q / [ M ] q = dim Hp /M and A [ M ] q [ M ] q. Proof. It is enough to prove when q>p. For if q<pthen [ M ] q Hp = M by [3, Theorem 6.1]. Suppose q>p. Then AM q M q and M q is dense in M by [3, Theorem 6.1]. Suppose dim H q /M q = n< and H q =[f q j : 1 j n ] q M q. If f is nonzero in H p then there exists α jl C and f l M q such that f ( ) nj=1 α jl f q p j + f l 0asl. Hence there exists nonzero g L p (1/p +1/p = 1 ) such that g annihilates M q and n j=1 α jl f q j gdm fgdm ( l ). Then for any fixed 1 j n we can chose g such that f q t gdm=0(t j ) and f q j gdm 0. This implies α jl converges α j.thus there exist α 1,α 2,,α n C and f M such that f = n j=1 α j f q j + f. Therefore dim H p /M = n. Conversely suppose dim H p /M = n< and H p =[f p j : 1 j n ] p M. If dim H q /M q = l n then by what was proved above, we get n = l. If n<l then H q =[f q j : 1 j l ] q M q. Then f q j = n t=1 α tj f p t + f j where α tj C and f j M. Hence there exists { β j } n+1 j=1 in C such that ( β 1,,β n+1 ) (0,, 0 ) and n+1 j=1 β j ( f q j f j )= n+1 j=1 ( n t=1 α tj f p t ) M. Hence n+1 j=1 β jf q j belongs to M H q = M q. This contradicts H q =[f q j : 1 j l ] q M q.
Finite codimensional invariant subspace and uniform algebra 969 Lemma 2. If Theorem is valid for p =2, so it is for all 1 p. Proof. Let M be a closed (or weak star closed) subspace in H p and AM M. Suppose p<2. If G(ϕ) ={ϕ}, then by hypothesis and Lemma 1 M 2 = Hϕ 2. Since M 2 and Hϕ 2 are dense in M and Hϕ, p respectively, M = Hϕ. p Conversely if M = Hϕ p then by Lemma 1 and hypothesis M 2 = Hϕ 2 and so by hypothesis G(ϕ) ={ ϕ }. This shows (1). The statement (2) can be shown similarly. Suppose p>2. If G(ϕ) { ϕ } then by hypothesis and Lemma 1 [ M ] = 2 BH 2 where B = n j=1 (Z a j )/(1 a j Z). By Lemma 1 M = BH p. Conversely if M = BH p then by Lemma1 [ M ] = 2 BH2. By hypothesis this shows G(ϕ) { ϕ } and so (2). The statement (1) can be shown similarly. Lemma 3. There exists a unique extention ϕ of ϕ to H such that ϕ M(H ). If G(ϕ) ={ ϕ } then G( ϕ) ={ ϕ } and there does not exist a weak star continuous point derivation at ϕ. Proof. By the definition of m, ϕ is continuous on A with respect to the weak star topology σ(l,l 1 ) and so ϕ has a unique extension ϕ to H. Then it is easy to see that ϕ M(H ) and G( ϕ) ={ ϕ }. If there exists a weak star continuous point derivation at ϕ then [ ] Hϕ Hϕ Hϕ. By [3, Chapter V, Theorem 7.2], Hϕ = ZH for some unimodular function Z. Since G( ϕ) ={ ϕ }, ψ(z) = 1 for any ψ in M(H ) \{ ϕ }. Put Q =(Z a)/(1 az) and 0 < a < 1. Then ϕ(q) = a and ψ(q) =1 for any ψ in M(H ) \{ ϕ }. This implies Q and Q belong to H and so Q is constant. This contradiction shows there does not exist a weak star continuous point derivation at ϕ. Lemma 4. For j = 1, 2, let M j be a closed subspace in H 2 such that AM j M j. If M 1 M 2 and > dim(m 1 M 2 ) 2 then there exists M 3 such that M 2 M 3 M 1 and AM 3 M 3. Proof.Forf in A, put S f (x) = Q(fx)(x M 1 M 2 ) where Q is the orthogonal projection from H 2 to M 1 M 2. Then there exists a function F in A such that { 0 } KerS F M 1 M 2. Since S F S f = S f S F (f A), f(kers F + M 2 ) KerS F + M 2 (f A). Put M 3 = KerS F + M 2. Proof of Theorem. By Lemma 2 it is enough to prove it only for p =2. (1) Suppose G(ϕ) ={ ϕ }. By Lemma 1, M is a weak star closed ideal of H and dim H /M = n. Hence there exists τ M(H ) such that M Kerτ and Kerτ is weak star closed and A Kerτ Kerτ. By Lemma 1, there exists a nonzero function k in H 2 which is orthogonal to Kerτ. Hence τ(f) = fkdm (f H ). This implies τ belongs to G( ϕ) by [1, Theorem 2.1.1] and so τ = ϕ. Therefore M Hϕ 2 and dim H2 ϕ M n 1. By Lemma 4, there exists M such that M M Hϕ 2 and dim Hϕ 2 M = 1. Hence
970 Tomoko Osawa by Lemma 1, dim H / M = 2 and M Hϕ. Then M is contained in a kernel of a bounded point derivation at ϕ because there does not exist a complex homomorphism τ on H such that M Kerτ and τ ϕ ( see [1, Theorem 1.6.1] ). This contradicts Lemma 3. Thus M = Hϕ 2. Conversely suppose M = 2. If G(ϕ) { ϕ } then H2 ϕ = ZH2 by [1, Lemma 4.4.3]. Hence Z 2 H 2 Hϕ, 2 dim Hϕ 2 Z 2 H 2 = 1 and AZ 2 H 2 Z 2 H 2. This contradiction shows G(ϕ) ={ ϕ }. (2) Suppose G(ϕ) { ϕ }. Then Hϕ 2 = ZH 2 by [1, Lemma 4.4.3]. If M = ZM then M l=1 Z l H 2 = I 2 and so dim H 2 M =. Hence M [ ZM ] 2 = M [ A ϕ M ] 2 { 0 } and so by [3, Chapter V, Theorem 6.2], M = qh 2 for some unimodular function q. We shall prove it only when n 2 because (2) is clear when n = 1. By Lemma 4, there exists a unimodular function q 0 such that q 0 H 2 qh 2 and dim (H 2 q 0 H 2 ) = 1. By Lemma 1, dim H /q 0 H = 1. Hence there exists τ in M(H ) such that Kerτ is weak star closed, τ G( ϕ) and q 0 =(Z a)/(1 az) where τ(z) =a and a < 1. If q 0 q, again, by Lemma 4, there exists a unimodular function q 1, such that q 0 H 2 q 1 H 2 qh 2 and dim (q 0 H 2 q 1 H 2 ) = 1. Then q 0 q 1 H and so H 2 q 0 q 1 H 2 q 0 qh 2 and dim (H 2 q 0 q 1 H 2 ) = 1. This shows q 0 q 1 =(Z b)/(1 bz) and b < 1. Hence q 1 =(Z a)(z b)/(1 az)(1 bz). Repeating this process, we get q = n j=1 (Z a j )/(1 a j Z) and a j < 1 (1 j n). Now we will prove the converse. Since Hϕ 2 = ZH2, put dμ a =(1 a 2 )/ 1 az 2 dm where a < 1. Then dμ a is a representing measure of ϕ a in G(ϕ) and ϕ a ϕ. Corollary 1. Let M be a non-trivial closed subspace in H p such that AM M and dim H p /M = n<. (1) If G(ϕ) ={ϕ} then M = H p ϕ. (2) If G(ϕ) {ϕ} then M = BH p I p where B = n j=1 (Z a j )/(1 a j Z) and a j < 1 (1 j n). For 1 <p<, it is known [3, Chapter IV, Theorem 6.2] that a bounded projection P from L p onto H p. For a function k in L, a Hankel operator H k is defined as follows : H k f =(I P)(kf) (f H p ). When p =2,ifG(ϕ) ={ϕ} and H k is compact then k belongs to H [2, Theorem 1], and if G(ϕ) {ϕ} and H k is compact then k belongs to the weak * closure of n=1 Z n H [2, Theorem 2]. As a result of Corollary 1, we describe the symbol of finite rank Hankel operator for arbitrary 1 <p<. Lemma 5. If KerH k is the kernel of H k then A(KerH k ) KerH k.
Finite codimensional invariant subspace and uniform algebra 971 Proof. Since KerH k = { f H p : kf H p (k L ) }, the proof is clear. Corollary 2. Let k be in L. (1) When G(ϕ) ={ϕ}, ifh k is of finite rank then kh H or kh ϕ H. (2) When G(ϕ) {ϕ}, ifh k is of finite rank n ( 0)then n 1 a k = j Z k 0 + k 1 j=1 Z a j where a j < 1(1 j n), k 0 is a nonzero function in H and k 1 is a function in I. Proof. (1) By Lemma 5, A(KerH k ) KerH k and so by (1) of Corollary 1, KerH k = H p ϕ or Hp. This shows (1). (2) By Lemma 5 and (2) of Corollary 1, KerH k = BH p I p where B = nj=1 (Z a j )/(1 a j Z) and a j < 1 (1 j n). Hence k(bh p I p ) H p and so k belongs to BH p. Since ZI p = I p, it is easy to see BI p = I p. Therefore k belongs to BH = BH I. This shows (2). When p = 2 and G(ϕ) ={ϕ}, it is shown [2, Theorem 1] that if H k is compact and so of finite rank then k belongs to H. Hence in (1) of Corollary 2, we would like to show that k belongs to H. Unfortunately we could not prove it. References [1] A. Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam. (1969). [2] R. E. Curto, P. S. Muhly, T. Nakazi, J. Xia, Hankel operators and uniform algebras, Arch. Math. 43(1984), 440-447. [3] T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, New Jersey(1969). [4] S. Merrill, N. Lal, Characterization of certain invariant subspaces of H p and L p spaces derived from logmodular algebras, Pacific. J. Math. 30(1969), 463-474. Received: March 25, 2014