Eigenvalues of Toeplitz operators on the Annulus and Neil Algebra

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1 Eigenvalues of Toeplitz operators on the Annulus and Neil Algebra Adam Broschinski University of Florida March 15, 2013

2 Outline On the Annulus Definitions Toeplitz Operators on the Annulus Existence of Eigenvectors for S-A Toeplitz Operators Proofs The Neil Algebra Definitions Toeplitz Operators Relative to the Neil Algebra Existence of Eigenvectors for S-A Toeplitz Operators Why the spaces of Davidson, Paulsen, Raghupathi, Singh [DPRS]? Definitions Characterization of Representations

3 On the Annulus Some basic definitions: Unlike A(D) A {z C : 0 < q < z < 1} B {z C : z = 1 or z = q} A(A) span {z n : z Z} { L 2 (B) f measurable : f 2 H 2 (A) A(A) L2 B } f 2 ds < L 2 (B) (A(A) + A(A) ) = span { log ( q 1 2 z )} 3 / 20

4 Sarason s modulus automorphic Hardy Spaces For each α [0, 1) let Note: H 2 α(a) { z α f(z) L 2 (B) : f H 2 (A) } e i2πα f(re iθ ) = f(re i(θ+2π) ), f H 2 α(a). H 2 0 (A) = H2 (A) H 2 α(a) = H 2 α+1 (A) H 2 β (A) = { z β α f : f H 2 α(a) } If φ H (A) and f H 2 α(a), then φf H 2 α(a). 4 / 20

5 Toeplitz Operators on the Annulus Let P α : L 2 (B) H 2 α(a) denote projection onto H 2 α(a). Given a real valued φ L (A) and α [0, 1) define the Toeplitz Operator, T α φ : H2 α(a) H 2 α(a) by T α φ f = P αφf. Since φ is real valued, Tφ α is self adjoint. ( Since Tφ α λ) f = Tφ λ α f, if T φ α has an eigenvalue, the eigenvalue can be assumed to be zero. 5 / 20

6 Inner-Outer Factorization of Functions in H 2 α(a) A function F H 2 α(a) is outer if A(A)F = H 2 α(a). A function ψ H 2 α(a) is inner if ψ = 1 almost everywhere on B. Sarason showed each function g H 2 α(a) has an inner-outer factorization; i.e., there exists a β [0, 1), an inner ψ H 2 β, and an outer F H 2 α β (A) such that g = ψf. 6 / 20

7 Eigenvalues in the Gap Abrahamse showed that if 1 on B 1 φ = 1 on B q, then T 0 φ has sequences of eigenvalues approaching 1 and 1. Theorem (Aryana-Clancy) For φ L (A) if essinf { } { } φ(z) : z B 1 = M > 0 > m = esssup φ(z) : z Bq or vice-versa, and log φ M = (respectively B log φ m = ), then there exists a sequence of eigenvalues B for Tφ 0 approaching M (respectively m). 7 / 20

8 Eigenvalues for Self-Adjoint Toeplitz Operators Theorem (B) Fix a real valued φ L (B) and let a α [0, 1) and a nonzero g H 2 α be given. (i) If Tφ α g = 0, then g is outer and moreover there is a nonzero c R such that, on B, φ g 2 = c log ( q 1 2 z ). (1) (ii) Conversely, if there is a c R, an α [0, 1), and an outer function g H 2 α such that (1) holds, then T α φ g = 0. (iii) In particular, there exists at most one α such that T α φ has eigenvalue 0 and the dimension of this eigenspace is at most one. 8 / 20

9 Existence of Multiple Eigenvalues We will say λ is an eigenvalue of φ relative to A(A) if there exists a nonzero α [0, 1) and a nontrivial solution to T α φ g = λg. Corollary Suppose essinf { φ(z) : z B 1 } = M > 0 > m = esssup { φ(z) : z Bq }. (i) Each λ (m, M) is an eigenvalue of φ relative to A(A). (ii) M is an eigenvalue if and only if c log(q 1/2 z ) φ M L 1. (iii) If log φ M =, then for any α [0, 1) there exists a B sequence of eigenvalues approaching M. The space is determined by B 1 log φ λ B q log φ λ. 9 / 20

10 Proof of Existence of Eigenvalues Fix a nonzero real valued φ L (B). Assume that there exists a g H 2 α(a) such that T α φ g = 0. If a A(A), then 0 = Tφ α g, ag = P v φg, ag = Thus for all n Z. B φ g 2 z n = 0 and B B φ g 2 a. φ g 2 z n = 0 So since L 2 (B) (A(A) + A(A) ) = span { log ( q 1 2 z )} there exists a c R such that φ g 2 = c log ( q 1 2 z ). The condition that g is outer allows us to reverse this proof. 10 / 20

11 Proof of Existence of Eigenvalues continued To show that g is outer let g = ψf be the inner outer factorization of g and let β be the index of ψ. Let C z β ψ H 2 0. For n N, let Ĉ 1 (n) Ĉ q (n) T T C(z)z n and C(qz)z n. By Cauchy s Theorem, Ĉ1(n) = q n Ĉ q (n). 11 / 20

12 Proof of Existence of Eigenvalues continued Recall, g = ψf. Hence z β F H 2 α(a) and 0 = Tφ α g, zn z β F = φ g 2 ψz β z 2β z n B = ( ) c log q 1 2 z C z 2β z n B = ( ) ( c log q 1 2 q n+2β Ĉ q (n) Ĉ1(n) ) = ( ) c log q 1 2 Ĉq (n) ( q n+2β q n). So Ĉq(n) = 0 for all n 0 and β = 0. Thus ψ is a constant function, and g is outer. 12 / 20

13 Under the Neil Algebra The Neil algebra is A { f A(D) : f (0) = 0 }. Following (DPRS) to a nonzero v = α + βz C + Cz we associate H 2 v(d) H 2 (D) v. The subspace H 2 v is invariant for A: If f H2 v(d) and a A, then af H 2 v(d). Unlike A(D) L 2 (T) (A + A ) = span {z, z}. 13 / 20

14 Toeplitz Operators Relative to the Neil Algebra Let P v : L 2 (T) H 2 v(d) denote the projection onto H 2 v(d). Given a real valued φ L (T) and v C + Cz define the Toeplitz Operator, T v φ : H2 v(d) H 2 v(d) by T v φ f = P vφf. Since φ is real-valued, Tφ v is self-adjoint. Since (Tφ v λ)f = T φ λ v f, if T φ v has an eigenvalue, the eigenvalue can be assumed to be zero. 14 / 20

15 Inner-Outer Factorization of Functions in H 2 (D) A function F H 2 (D) is outer if A(D)F = H 2 (D). A function ψ H (D) is inner if ψ = 1 almost everywhere on T. Each function g H 2 (D) has an inner-outer factorization; i.e., there exists an inner ψ H (D) and an outer F H 2 (D) such that g = ψf. If F H 2 (D) is outer and 0 v C + Cz satisfies F, v = 0, then F H 2 v and AF = H 2 v(d). 15 / 20

16 Existence of Eigenvalues for Self-Adjoint Toeplitz Operators Theorem (B) Fix a real valued φ L (T) and let a nonzero v C + Cz and a nonzero g H 2 v be given. (i) If Tφ v g = 0, then g is outer and moreover there is a nonzero c C such that, on T, φ g 2 = R(cz). (2) (ii) Conversely, if there is a c C and an outer function g H 2 such that (2) holds, then Tφ v g = 0, where v 0 is orthogonal to g(0) + g (0)z. (iii) In particular, there exists at most one v such that T v φ has eigenvalue 0 and the dimension of this eigenspace is at most one. 16 / 20

17 Existence of Multiple Eigenvalues We will say λ is an eigenvalue of φ relative to A if there exists a nonzero v C + Cz and a nontrivial solution to T v φ g = λg. Corollary Suppose essinf { φ : R(cz) > 0 } = M > 0 > m = esssup { φ : R(cz) < 0 }, (i) Each λ (m, M) is an eigenvalue of φ relative to A. (ii) M is an eigenvalue if and only if R(cz) φ M is in L1. For most v C + Cz, the operator T v φ has no eigenvalues. 17 / 20

18 Why the H 2 v spaces of [DPRS]? For a Hilbert space N H 2 N (D) = f = f i z i : f i N, f i 2 <. i=0 i=0 Following Raghupathi given V N + Nz H 2 N H 2 V = H2 N V Define π V : A B ( H 2 V) by πv (a) M a. 18 / 20

19 Characterization of Representations Theorem (B) If π is a unital pure extremal completely contractive representation of A, then there exists a Hilbert space N and V N + Nz such that π is unitarily equivalent to π V. We will say a representation π of A on B(H) has rank 1 if there do not exist a pair of orthogonal invariant subspaces. Corollary If a unital pure extremal completely contractive representation has rank 1, then it is unitarily equivalent to π V where N = C. 19 / 20

20 Fin Thanks for listening. For more details see Eigenvalues of Toeplitz operators on the Annulus and Neil Algebra on arxiv.org. 20 / 20

TOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :

TOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C : TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted