NOTES ON INVARIANT SUBSPACES OF OPERATORS ON A HILBERT SPACE REBECCA G. WAHL 1. Introduction Invariant subspaces are central to the study of operators and the spaces on which they are defined. Up to similarity, the Jordan Normal Form expresses any matrix on a finite dimensional vector space as a direct sum of restrictions to invariant sub- spaces (eigenspaces). The analog for a bounded operator T in a Hilbert space H with M ∈ LatT gives an upper triangular matrix representation of T with respect to the decomposition H = M⊕ M⊥: M M⊥ M A B T = M⊥ 0 C ⊥ ⊥ ⊥ where A = T |M, B : M →M and C : M →M . The utility of this result is demonstrated by its application to approximation and to a wide variety of problems in physics (quantum theory), computer science (data mining), chemistry (lattice theory of crystal analysis) and automorphic graph theory. 2. Preliminaries In this section we present the necessary background and notation for what follows. An excellent source of this material is the comprehensive book by H. Radjavi and P. Rosen- thal, [8]. We denote an arbitrary Hilbert space by H and the Hardy Hilbert space of the unit disk by H2(D). Definition A subspace M of H is invariant (alternatively, A−invariant) under A ∈B(H) if Ax ∈M whenever x ∈M. The collection of all subspaces of H invariant under A is denoted LatA. For collections of bounded operators L ⊆ B(H), we define Lat(L)= A∈L LatA. For A ∈ B(H), LatA is closed under intersections and spans and is completeTin the sense that both ∅ and H are in LatA. Theorem 1. If A ∈B(H) and P = P |M is the projection onto M, then M ∈ LatA if and only if AP = P AP . Theorem 2. For any A ∈B(H), LatA∗ = {M : M⊥ ∈ LatA}. Date: October 2010. 1991 Mathematics Subject Classification. Primary: 47B32; Secondary: 47B33, 47B38, 47A15. Key words and phrases. composition operator, weighted composition operator, invariant subspaces, shift-invariant subspace, weighted Hardy space, weighted Bergman space. 1 2 REBECCA G. WAHL Theorem 3. For operators A, B ∈B(H), LatA ⊆ LatB implies LatA∗ ⊆ LatB∗. Definition For all L ⊆ H, the span of L, ∨{L}, is the intersection of all subspaces of H containing L, i.e., ∨{L} = L⊆M M. Definition A subspace M isT called a reducing subspace for A if both M and M⊥ are in LatA. Theorem 4. Let M be a subspace of H. Then (a) M reduces A if and only if M ∈ ∗ (LatA) ∩ (LatA ), and (b) M reduces A if and only if AP = PA, where P |M is the projection on M. We now consider some important examples of invariant subspaces for composition 2 operators Cϕ on the Hardy Hilbert space, H (D). Example 5. If ϕ is any analytic map of the disk into itself, 1 ∈ LatCϕ. For self-maps ∗ fixing 0, its easy to see that 1 ∈ LatCϕ as well, so {1} is a reducing subspace for Cϕ. Example 6. If ϕ is any non-constant, non-automorphicW analytic map of the disk D into itself such that ϕ(0) = 0, then m 2 z H ∈ LatCϕ so by Theorem 2, m 2 ⊥ m j ∗ (z H ) = ∨j=0{z } ∈ LatCϕ for all natural numbers m. This can be used to show that if a ∈ D, ϕ(a)= a the spectrum 0 n of Cϕ includes 1 and ϕ (a) for n = 1, 2,.... Recall that the presence of an invariant subspace M simplifies the analysis of an op- erator by decomposing H = M⊕M⊥. The operator from Example 6 has a particularly m 2 ⊥ nice expression relative to the finite dimensional invariant subspace Mm ≡ (z H ) = m j ∨j=0{z } giving the decomposition 2 D m 2 D m j j 2 D H ( )= Mm ⊕ z H ( )= ∨j=0{z }⊕ z H ( ) This case is well understood (Prop. 7.32, [3]) and is a nice example of the use of such a decomposition so we reproduce some of the details here after a needed lemma that is from [3] as well. Lemma 7. Suppose H is Hilbert space with H = K ⊕ L where dim(K) < ∞, and C is a bounded operator on H that leaves K or L invariant. If C has matrix representation X Y X 0 C = or C = 0 Z Y Z with respect to the decomposition, then σ(C)= σ(X) ∪ σ(Z). NOTES ON INVARIANT SUBSPACES OF OPERATORS ON A HILBERT SPACE 3 The proof of the lemma follows after reducing to the first case (by taking taking adjoints) and noting that it is enough to show that C is invertible if and only if X and Z are. Now we return to Example 6. First, we can assume that the fixed point of ϕ is 0 since the spectrum of a Hilbert space operator is preserved under similarity and Cϕ is similar − −1 −1 a z to Cξ ◦ϕ◦ξ = Cξ CϕCξ where ξ(z) is the automorphism ξ(z)= 1−az¯ . The Taylor series 0 2 j m expansion for ϕ at 0 is ϕ(z)= ϕ (0)z + c2z + ··· . The monomials {z }j=0 form a basis for Cϕ|Mm so we have the matrix for the restriction of Cϕ to Mm: 1 0 ··· 0 . 0 ϕ0(0) 0 ··· . . . ∗ ϕ0(0)2 0 ··· Cϕ|Mm = . .. .. . ∗ . . . . 0 0 (m−1) 0 ∗ ∗ ··· ∗ ϕ (0) By noting that m is an arbitrary integer and appealing to the previous lemma it is 0 n trivial to see that the spectrum of Cϕ includes 1 and ϕ (a) for n = 1, 2,.... Definition If F is a chain (ordered by inclusion) of subspaces of H containing M, then M denotes the span of all subspaces in F which are properly contained in M. The chain F is complete if it contains both ∅ and H, and is closed under span and intersection of any subfamily. It is maximal if the only chain F0 of subspaces with F0 ⊆F is F0 = F. D D j 2 D ∞ Example 8. If ϕ : → is analytic and ϕ fixes zero, then the family {z H ( )}j=0 forms a maximal chain of invariant subspaces for Cϕ. We saw in Example 6 that for n = 1, 2,..., the subspaces znH2(D) are invariant for Cϕ. Note that H2(D) ⊃ zH2(D) ⊃ z2H2 ⊃···⊃ znH2(D) ⊃···⊃∅ A result [6] of D. Hadwin, E. Nordgren, M. Radjabalipour, H. Radjavi and P.Rosenthal applies and we see that the family of family of composition operators whose symbol fixes 0 ∈ D is simultaneously triangularizable. Theorem 9. A collection of operators is simultaneously triangularizable if there is a maximal chain of subspaces all of which are invariant under the collection. 4 REBECCA G. WAHL 3. On a Paper of Mahvidi: Invariant Subspaces of Composition Operators We discuss the work of A. Mahvidi, [7], concerning the invariant subspaces of com- position operators on H2(D). Mahvidi gives two types of results for each of the cases: fixed point a ∈ ∂D or a ∈ D. First, assuming that LatCϕ ⊆ LatCψ he determines that the two self maps (ϕ and ψ) of D must have the same iteration type from the model. The other type of result relates the Denjoy-Wolff point of ϕ to that of ψ under the assumption that Cϕ and Cψ have a common invariant subspace. 3.1. The Model for Iteration. Any analytic self-map of the unit disk ϕ with D-W point a, ϕ0(a) =6 0 can be modeled after a linear fractional transform Φ in the following way Φ ◦ σ = σ ◦ ϕ where σ : D → Ω is analytic, Φ is an automorphism of Ω, and Ω is either the plane or a half-plane. This classifies the self-maps of the disk as: (1) Plane/Dilation: Ω = C, σ(a) = 0, Φ(z)= sz, 0 <s< 1. (2) Plane/Translation: Ω = C, σ(a)= ∞, Φ(z)= z + 1 (3) Half-Plane/Dilation: Ω = RHP , σ(a) = 0, Φ(z)= sz, 0 < |s| < 1. (4) Half-Plane/Translation:Ω = RHP , σ(a)= ∞, Φ(z)= z ± 1. The map ϕ is in the Plane/Dilation case iff ϕ has D-W point in the disk. If ϕ has D-W point a ∈ ∂D and ϕ0(a) < 1, then it is in the Half-Plane/Dilation case. Proposition 10. (Prop. 1.4 p. 455) (1) H1: If ϕ has a fixed point a ∈ D, and Cϕ and Cψ have a non-constant eigen- function, then C1: Cϕ and Cψ commute. (2) H2: If ϕ (not an automorphism) has a fixed point a ∈ D. If ϕ0(a) =6 0, then 0 j C2: Cϕf = λf had non-zero solution iff λ = ϕ (a) , some non-negative integer 0 j j. Moreover, f is a non-zero solution of Cϕf = ϕ (a) f for some non-negative integer j iff f(z)= cσ(z)j where σ is the map from the model and c is a constant. ( [3], p.78) H3: C3: 0 n ∞ (3) If Cϕ is compact, then σ(Cϕ)= {ϕ (a) }n=1 ∪{0, 1}. Recall that the function Kw is called the reproducing kernel function. In the Hardy space H2(D), the reproducing kernel is 1 K (z)= w 1 − wz and in this space has norm given by 1 kK k2 = hK , K i = w w w 1 −|w|2 NOTES ON INVARIANT SUBSPACES OF OPERATORS ON A HILBERT SPACE 5 3.2. Case: a ∈ ∂D. 3.3. Lattice Containment. H: D ∞ ∗ Theorem 11. (2.1, p.456) a ∈ ∂ , n=0{Kϕn(zα)} ∈ LatCψ uncountable collection ∞ C: {zα} with Blaschke orbits under ϕ, n=0W(1 −|ϕn(zα)|) < ∞, then ψ = ϕ(m),m ∈ N ∪{0}.