Linear-fractionally-induced Composition. Operators

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1 by Linear-fractionally-induced Spurrier James Madison University March 19, 2011 by

2 Notation Let D denote the open unit disk and T the unit circle. H 2 (D) is the space of all functions f = n=0 a nz n that are analytic on D and satisfy f 2 H 2 = a n 2 <. n=0 For ϕ : D D analytic, C ϕ f = f ϕ for all f H 2 (D). by U ϕ = partial isometry in the polar decomposition of C ϕ T z = the forward shift on H 2 (D). K = the ideal of compact operators on H 2 (D)

3 More Notation We re interested in composition operators induced by linear-fractional maps ϕ(z) = az + b cz + d that map D into but not onto D and fix a point on T. Without loss of generality, take the fixed point to be 1. by Let F := {C ϕ : ϕ : D D non-auto, LFT, ϕ(1) = 1}.

4 Main Questions for Today s Talk Let F := {C ϕ ϕ : D D non-auto, LFT, ϕ(1) = 1}. by 1 What is the structure of C (F), modulo the ideal of compact operators? 2 If C ϕ F, what is the structure of C (C ϕ, K), modulo the ideal of compact operators? 3 What spectral information about algebraic combinations of composition operators and their adjoints can we obtain from these structure results?

5 Induced by Parabolic Non-automorphisms A linear-fractional, non-automorphism self-map of D that fixes the point 1 is parabolic if ϕ (1) = 1. by In this case, ϕ = ρ a = (2 a)z+a az+(2+a), where Re a > 0. Let P = {C ρa : Re a > 0}. Theorem (Kriete, MacCluer, and Moorhouse) If C ϕ P, then C (C ϕ, K) = C (C ϕ ) = C (P) and there exists a unique -isomorphism Γ : C([0, 1]) C (P)/K such that, for all a C with Re a > 0, Γ(x a ) = [C ρa ].

6 Rewriting in F For t > 0, define the map Ψ t (z) = an automorphism of D. (t + 1)z + (1 t), which is (1 t)z + (1 + t) by If C ϕ F, set t = ϕ (1) and a = ϕ (1) t 2 +t t. Then C ϕ = C ρa C Ψt. Applying results of Bourdon and MacCluer, Jury, and Kriete, MacCluer, and Moorhouse, we can then rewrite C ϕ as C ϕ = 1 C ρa U Ψt + K, t where K K and U Ψt is the unitary operator appearing in the polar decomposition of C Ψt.

7 Induced by Automorphisms Let G be a collection of automorphisms of D that form an abelian group under composition. by Theorem (Jury, 2007) C (T z, {C γ : γ G})/K = C (T z, {U γ : γ G})/K = C(T) α G d α γ (f ) = f γ for all f C(T) and γ G G d denotes the locally compact group obtained from G by applying the discrete topology.

8 Crossed Product (Discrete Version) Let G be a discrete group, and let A be a C -algebra. An action α of G on A is a homomorphism α : G Aut(A), g α g. The crossed product A α G is the completion of { } A C c (G, A) = A s χ s : s A, A s = 0 for all but finitely many s s G in a norm that is built from a set of representations of C c (G, A) that come from covariant representations of (A, G, α). by

9 1. What is the structure of C (F)/K? C (F)/K C (P, {U Ψt : t R + })/K by C (P)/K = C([0, 1]) Kriete, MacCluer, and Moorhouse C (T z, {U Ψt : t R + })/K = C(T) R + d Jury We want to show that C (P, {U Ψt : t R + })/K is also a crossed product.

10 1. What is the structure of C (F)/K? C (P, {U Ψt : t R + })/K = C (C (P)/K, {[U Ψt ] : t R + }) by {[U Ψt ] : t R + } is an abelian group of cosets of unitary operators. (Jury 2007) [U Ψt ]Γ(g)[U Ψ t ] = Γ(β t (g)) for all g C([0, 1]) and t R +, where β t (g)(x) = g(x t ) and Γ is the -isomorphism from C([0, 1]) onto C (P)/K (Obtained by applying results of Bourdon, MacCluer 2007; Jury 2007: Kriete, MacCluer, Moorhouse 2007, 2009)

11 We can show that the action β is topologically free and then apply the machinery of Karlovich or Lebedev. Theorem (Q) C (P, {U Ψt : t R + })/K = C([0, 1]) β R + d. The -isomorphism F : C([0, 1]) β R + d C (P, {U Ψt : t R + })/K satisfies F ( finite g t χ t ) = finite Γ(g t )[U Ψt ] for all finite g tχ t C c (R + d, C([0, 1])). by Corollary C (F)/K is isomorphic to a subalgebra of C([0, 1]) β R + d.

12 Identifying the Full All cosets of words in F F look like [bc ρa U Ψt ]. Let C 0 ([0, 1]) := {g C([0, 1]) : g(0) = 0} and set { } N = Γ(g t )[U Ψt ] : g t C 0 ([0, 1]). finite Then C[I ] + N is dense in C (F)/K. Under the iso, N maps onto C c (R + d, C 0([0, 1])). Theorem (Q) Define β : R + d Aut(C 0([0, 1])) by β t (g)(x) := g (x t ) for all t R + d, g C 0([0, 1]), and x [0, 1]. Then C (F)/K is isometrically -isomorphic to the unitization of C 0 ([0, 1]) β R + d. by

13 2. If C ϕ F and ϕ (1) 1, what is the structure of C (C ϕ, K)/K? by Theorem (Q) Let ϕ be a linear-fractional, non-automorphism self-map of D with ϕ(1) = 1 and ϕ (1) = t 1. Define β t : Z Aut(C 0 ([0, 1])) by β t ng(x) := g(x tn ) for all n Z, g C 0 ([0, 1]), and x [0, 1]. Then C (C ϕ, K)/K is isometrically -isomorphic to the unitization of C 0 ([0, 1]) β t Z.

14 Essential Spectra Consider cosets in C (C ϕ, K)/K of the form [A] = N n= N Γ(g n ) [ U Ψ t n for t = ϕ (1), N N, g n C([0, 1]), and g n (0) = 0 for n 0. ] For x [0, 1], define π x ([A]) B(l 2 (Z)) by π x ([A]) = g 1 (x t 2 ) g 0 (x t 2 ) g 1 (x t 2 ) g 1 (x t 1 ) g 0 (x t 1 ) g 1 (x t 1 ) g 1 (x) g 0 (x) g 1 (x) g 1 (x t ) g 0 (x t ) g 1 (x t ) g 1 (x t2 ) g 0 (x t2 ) g 1 (x t2 ) Trajectorial Approach: [A] is invertible the discrete operator π x ([A]) is invertible for all x [0, 1].

15 When is π x ([A]) invertible? If x (0, 1) and t > 1, then π x ([A]) = g 1 (1) g 0 (1) g 1 (1) g 1 (1) g 0 (1) 0 0 g 0 (0) g 1 (0) g 1 (0) g 0 (0) g 1 (0) g 1 (0) g 0 (0) g 1 (0) K By results of Gohberg and Fel dman, π x ([A]) is Fredholm with index zero if and only if p A,0 (z) := N n= N g n (0)z n and p A,1 (z) := N n= N g n (1)z n do not vanish on T and have the same winding number.

16 When is π x ([A]) invertible? If x (0, 1) and t > 1, then π x ([A]) = g 1 (1) g 0 (1) g 1 (1) g 1 (1) g 0 (1) 0 0 g 0 (0) 0 0 g 0 (0) 0 0 g 0 (0) K By results of Gohberg and Fel dman, π x ([A]) is Fredholm with index zero if and only if p A,0 (z) := g 0 (0) and p A,1 (z) := N n= N g n (1)z n do not vanish on T, and p A,1 (z) has winding number 0.

17 When is π x ([A]) = [g i j ( x tj )] i,j= invertible? If g 0 (0) 0, p A,1 does not vanish on T, and p A,1 has winding number 0, define π x ([A]) ν = [g i j (x tj )] ν 1 i,j= ν+1 π x ([A]) ν µ = [g i j (x tj )] i,j { ν+1,..., µ,µ,...,ν 1} Theorem (Karlovich and Kravchenko, 1984) π x ([A]) is invertible on l 2 (Z) if and only if the conditions above hold and there exists µ 0 > 0 such that for all µ µ 0, det π x ([A]) ν lim ν det π x ([A]) ν µ 0. (1) by If π x ([A]) is lower-triangular, then (1) is equivalent to the condition that g 0 (x tj ) 0 for all j.

18 Theorem Let t = ϕ (1) 1, and suppose A C (C ϕ, K) satisfies N [A] = Γ(g n ) [ UΨ ] t n n=0 for some N N, g 0 C([0, 1]), and g 1,..., g n C 0 ([0, 1]). Then σ e (A) = g 0 ([0, 1]) p A,1 (D). by

19 Example Let b 1,..., b n C and suppose that ϕ 1,..., ϕ n are linear fractional, non-automorphism self-maps of D that fix the point 1 and satisfy ϕ 1 (1) =... = ϕ n(1) = s 1. If A = so n b j C ϕj, then j=1 σ e p A,1 (z) = 1 s n j=1 b j n b j C ϕj = λ C : λ j=1 z and g 0 0, 1 n s j=1 b j. by

20 We can apply the Pimsner-Voiculescu exact sequence for crossed products by Z and the six-term exact sequence to determine the of C (C ϕ, K). Theorem (Q, 2009) If ϕ is a linear-fractional, non-automorphism self-map of D with ϕ(1) = 1 and ϕ (1) 1, then K 0 (C (C ϕ, K)) = Z Z and K 1 (C (C ϕ, K)) = 0. by

21 A. Antonevich and A. Lebedev, Functional differential equations: I. C*-theory, Longman Scientific & Technical, Harlow P. Bourdon and B. MacCluer. Selfcommutators of automorphic composition operators. Complex Var. Elliptic Equ., 52(2007), I. Gohberg and I. Fel dman, Convolution equations and projection methods for their solution, American Mathematical Society, Providence, R.I M. Jury, generated by groups of composition operators, Indiana Univ. Math. J., 56(2007), M. Jury, The Fredholm index for elements of Toeplitz-composition, Integral Equations Operator Theory, 58(2007), Y. Karlovich, A local-trajectory method and isomorphism theorems for nonlocal, Modern operator theory and applications, , Oper. Theory Adv. Appl., 170, Birkhäuser, Basel by

22 Y. Karlovich and V. Kravchenko, An algebra of singular integral operators with piecewise-continuous coefficients and a piecewise-smooth shift on a composite contour, Math. USSR Izvestiya, 23(1984) T. Kriete, B. MacCluer, and J. Moorhouse, Toeplitz-composition, J. Operator Theory, 58(2007), T. Kriete, B. MacCluer, and J. Moorhouse, theory for algebraic combinations of Toeplitz and composition operators, J. Funct. Anal., 257(2009), T. Kriete, B. MacCluer, and J. Moorhouse, operators within singly generated composition, Israel J. Math., 179(2010), M. Pimsner and D. Voiculescu, Exact sequences for K-groups and Ext-groups of certain cross-product, J. Operator Theory, 4(1980), by

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24 Example of an Essential Spectrum Calculation Let a, c 1, c 2 C with Re a > 0. Suppose that ϕ is a linear-fractional, non-automorphism self-map of D with ϕ(1) = 1 and ϕ (1) = s 1. Then, there exists b C with Re b > 0 such that [ ] ( ) [c 1 C ρa + c 2 C ϕ ] = Γ(c 1 x a ) UΨ c2 [ ] + Γ x b U (1/s) 0 s Ψ. (1/s) 1 Thus, g 0 (x) = c 1 x a, and p (c1 C ρa +c 2 C ϕ),1(z) = c 1 + c 2 s z. Hence by σ e (c 1 C ρa + c 2 C ϕ ) = {c 1 x a : x [0, 1]} { λ C : λ c 1 c 2 s }.