Composition operators: from dimension 1 to innity (but not beyond)

Transcription

1 : from dimension 1 to innity (but not beyond) Université d'artois (Lens) Lille 26 May 2016

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3 From works with Frédéric Bayart Pascal Lefèvre Hervé Queélec Luis Rodríguez-Piazza

4 First part: dimension 1 Hardy space H 2 : D = {z C ; z < 1} f : D C analytic The space H 2 = H 2 (D) is that of the analytic functions f on D such that: f 2 H 2 := sup 0<r<1 2π 0 f (r e it ) 2 dt 2π < +

5 First part: dimension 1 Hardy space H 2 : D = {z C ; z < 1} f : D C analytic The space H 2 = H 2 (D) is that of the analytic functions f on D such that: f 2 H 2 := sup 0<r<1 2π 0 f (r e it ) 2 dt 2π < + if f (z) = c n z n then f 2 H 2 n=0 = c n 2. n=0

6 Let ϕ: D D analytic (symbol)

7 Let ϕ: D D analytic (symbol) Littlewood's subordination principle (1925): f H 2 = C ϕ (f ) := f ϕ H 2

8 Let ϕ: D D analytic (symbol) Littlewood's subordination principle (1925): Hence: f H 2 = C ϕ (f ) := f ϕ H 2 C ϕ : H 2 H 2 is a bounded operator, called composition operator with symbol ϕ.

9 Compact composition operators Compactness was characterized in various manners by:

10 Compact composition operators Compactness was characterized in various manners by: B. McCluer (1980) (Carleson measure)

11 Compact composition operators Compactness was characterized in various manners by: B. McCluer (1980) (Carleson measure) for H 2 (B d ); certainly known before for d = 1

12 Compact composition operators Compactness was characterized in various manners by: B. McCluer (1980) (Carleson measure) B. McCluer and J. Shapiro (1986) (angular derivative)

13 Compact composition operators Compactness was characterized in various manners by: B. McCluer (1980) (Carleson measure) B. McCluer and J. Shapiro (1986) (angular derivative) J. Shapiro (1987) (Nevanlinna counting function)

14 Compact composition operators Compactness was characterized in various manners by: B. McCluer (1980) (Carleson measure) B. McCluer and J. Shapiro (1986) (angular derivative) J. Shapiro (1987) (Nevanlinna counting function) All these characterization says that C ϕ is compact i ϕ(d) touches the unit circle (if it happens) only sharply (roughly speaking: non tangentially).

15 Compact composition operators Compactness was characterized in various manners by: B. McCluer (1980) (Carleson measure) B. McCluer and J. Shapiro (1986) (angular derivative) J. Shapiro (1987) (Nevanlinna counting function) All these characterization says that C ϕ is compact i ϕ(d) touches the unit circle (if it happens) only sharply (roughly speaking: non tangentially). But actually, the knowledge of ϕ(d) is not sucient (unless ϕ is univalent).

16 Compact composition operators non compact compact

17 Compact composition operators Schatten classes membership was characterized by:

18 Compact composition operators Schatten classes membership was characterized by: D. Luecking (1987) (Carleson measure)

19 Compact composition operators Schatten classes membership was characterized by: D. Luecking (1987) (Carleson measure) D. Luecking and K. Zhu (1992) (Nevanlinna counting function)

20 Approximation numbers Approximation numbers H Hilbert space; T : H H operator; its n-th approximation number is:

21 Approximation numbers Approximation numbers H Hilbert space; T : H H operator; its n-th approximation number is: a n (T ) = inf T R rk R<n

22 Approximation numbers Approximation numbers H Hilbert space; T : H H operator; its n-th approximation number is: Then a n (T ) = inf T R rk R<n a 1 (T ) = T a 2 (T ) a n (T ) a n+1 (T )

23 Approximation numbers Approximation numbers H Hilbert space; T : H H operator; its n-th approximation number is: Then a n (T ) = inf T R rk R<n a 1 (T ) = T a 2 (T ) a n (T ) a n+1 (T ) They measure how much an operator is compact:

24 Approximation numbers Approximation numbers H Hilbert space; T : H H operator; its n-th approximation number is: Then a n (T ) = inf T R rk R<n a 1 (T ) = T a 2 (T ) a n (T ) a n+1 (T ) They measure how much an operator is compact: T compact a n (T ) n 0

25 Approximation numbers Approximation numbers H Hilbert space; T : H H operator; its n-th approximation number is: Then a n (T ) = inf T R rk R<n a 1 (T ) = T a 2 (T ) a n (T ) a n+1 (T ) They measure how much an operator is compact: T compact a n (T ) n 0 T S p (H) n [a n (T )] p <

26 Approximation numbers GOAL Give estimates on approximation numbers of composition operators

27 Lower estimates: no fast decay Lower estimates: no fast decay Approximation numbers of composition operators cannot be arbitrary small:

28 Lower estimates: no fast decay Lower estimates: no fast decay Approximation numbers of composition operators cannot be arbitrary small: Proposition For every symbol ϕ, there exists c > 0 such that: a n (C ϕ ) C e cn

29 Lower estimates: no fast decay Lower estimates: no fast decay Approximation numbers of composition operators cannot be arbitrary small: Proposition For every symbol ϕ, there exists c > 0 such that: a n (C ϕ ) C e cn = r n 0 < r < 1

30 Lower estimates: no fast decay Lower estimates: no fast decay Approximation numbers of composition operators cannot be arbitrary small: Proposition For every symbol ϕ, there exists c > 0 such that: a n (C ϕ ) C e cn = r n 0 < r < 1 First stated by Parfenov (1988) under a cryptic form; explicitely by LQR (2012).

31 Lower estimates: no fast decay That use the fact that, if C ϕ is compact, we can assume that ϕ(0) = 0 and ϕ (0) 0, and then the non-zero eigenvalues of C ϕ are [ϕ (0)] n, n = 0, 1,..., and: Weyl Lemma If T : H H is a compact operator on a Hilbert space H and if (λ n ) n 1 is the sequence of the eigenvalues of T, numbered in non-increasing order, then: n a k (T ) k=1 n λ k. k=1

32 Lower estimates: no fast decay Note: When ϕ < 1, it is easy to see:

33 Lower estimates: no fast decay Note: When ϕ < 1, it is easy to see: a n (C ϕ ) ϕ n

34 Lower estimates: no fast decay Note: When ϕ < 1, it is easy to see: a n (C ϕ ) ϕ n It is actually the only case: Theorem LQR 2012 If a n (C ϕ ) r n with r < 1, then ϕ < 1.

35 Lower estimates: no fast decay Two proofs:

36 Lower estimates: no fast decay Two proofs: The rst proof uses Pietsch's factorization theorem

37 Lower estimates: no fast decay Pietsch's factorization theorem is used for getting a measure µ supported by a compact set K ϕ(d) such that, for the restriction operator R µ : H L 2 (µ), one has a n (C ϕ ) a n (R µ ).

38 Lower estimates: no fast decay R µ is more tractable than C ϕ and it can be proved (assuming ϕ(0) = 0 for simplicity) that: ϕ > r = a n (R µ ) sn n with s = s(r) r 1 1.

39 Lower estimates: no fast decay Two proofs: The rst proof uses Pietsch's factorization theorem The second one uses the Green capacity of ϕ(d)

40 Lower estimates: no fast decay Theorem LQR 2014 If ϕ < 1, one has: lim [a n(c ϕ )] 1/n 1/Cap [ϕ(d)] = e n and (assuming ϕ(0) = 0 for simplicity): Cap [ϕ(d)] +. ϕ 1

41 Lower estimates: no fast decay Hence: a n (C ϕ ) r n implies ϕ < 1.

42 Lower estimates: no fast decay The proof is based on a theorem of H. Widom: Theorem H. Widom (1972) Let K be a compact set of D and r n = sup inf f R C(K), f H, f 1 R where the inmum is taken over all rational functions R with at most n poles, all outside of D. Then: lim r n 1/n = e 1/Cap (K). n

43 Lower estimates: no fast decay The proof is based on a theorem of H. Widom: Theorem H. Widom (1972) Let K be a compact set of D and r n = sup inf f R C(K), f H, f 1 R where the inmum is taken over all rational functions R with at most n poles, all outside of D. Then: lim r n 1/n = e 1/Cap (K). n and of the fact (communicated by A. Ancona) that if V is an open connected set such that V D, then Cap (V ) = Cap (V ).

44 Lower estimates: slow decay Lower estimates: slow decay If a n (C ϕ ) cannot decay fast, it can decay arbitrarily slowly: Theorem LQR 2012 For every vanishing non-increasing sequence (ε n ) n 1, there exists a symbol ϕ such that: a n (C ϕ ) ε n for all n 1.

45 Lower estimates: slow decay Lower estimates: slow decay If a n (C ϕ ) cannot decay fast, it can decay arbitrarily slowly: Theorem LQR 2012 For every vanishing non-increasing sequence (ε n ) n 1, there exists a symbol ϕ such that: a n (C ϕ ) ε n for all n 1. That allows to have compact composition operators which are in no Schatten class S p, p <. This question of Sarason (1988) was rst solved by Carroll and Cowen in 1991.

46 Lower estimates: slow decay However, one has a better result: Theorem H. Queélec - K. Seip 2015 For every function u : R + R + such that u(x) 0 as x and u(x 2 )/u(x) bounded below, there exists a compact composition operator such that: a n (C ϕ ) u(n)

47 Lower estimates: slow decay However, one has a better result: Theorem H. Queélec - K. Seip 2015 For every function u : R + R + such that u(x) 0 as x and u(x 2 )/u(x) bounded below, there exists a compact composition operator such that: a n (C ϕ ) u(n) It is actually a corollary of a result which will be stated later.

48 More specic lower etimates More specic lower estimates If u = (u 1,..., u n ) is a nite sequence of complex numbers, its interpolation constant M u is the smallest M > 0 such that: w 1,..., w n with w j 1 f H with f M s.t. f (u j ) = w j, j = 1,..., n.

49 More specic lower estimates Proposition LQR 2013 Let ϕ a symbol, u = (u 1,..., u n ) D n, such that the v j = ϕ(u j )'s are distinct, and v = (v 1,..., v n ). Set: µ 2 1 u j 2 n = inf 1 j n 1 ϕ(u j ) 2

50 More specic lower estimates Proposition LQR 2013 Let ϕ a symbol, u = (u 1,..., u n ) D n, such that the v j = ϕ(u j )'s are distinct, and v = (v 1,..., v n ). Set: µ 2 1 u j 2 n = inf 1 j n 1 ϕ(u j ) = inf K ϕ(uj ) j n K uj 2 where K z is the reproducing kernel at z

51 More specic lower estimates Proposition LQR 2013 Let ϕ a symbol, u = (u 1,..., u n ) D n, such that the v j = ϕ(u j )'s are distinct, and v = (v 1,..., v n ). Set: Then: µ 2 1 u j 2 n = inf 1 j n 1 ϕ(u j ) 2 a n (C ϕ ) µ n M 2 v.

52 More specic lower estimates That follows from the fact that if Kuj = K uj / K uj, then: M 1 u ( n ) 1/2 n c j 2 H c j Kuj 2 j=1 j=1 ( n ) 1/2. M u c j 2 j=1

53 More specic lower estimates By a suitable choice of u 1,..., u n, we get then: Theorem LQR 2013 Assume that ϕ(] 1, 1[) R and that 1 ϕ(r) ω(1 r), 0 r < 1, where ω is continuous, increasing, sub-additive, vanishes at 0, and ω(h)/h h 0.Then: where a = 1 ϕ(0) > 0. ω a n (C ϕ ) sup e 20/(1 s) 1 (as n ) 0<s<1 as n

54 Lower estimates: Examples Examples Lens maps For 0 < θ < 1, the lens map λ θ is the conformal representation (suitably determined) of D onto the lens domain below:

55 Lower estimates: Examples Examples Lens maps For 0 < θ < 1, the lens map λ θ is the conformal representation (suitably determined) of D onto the lens domain below:

56 Lower estimates: Examples Examples Lens maps For 0 < θ < 1, the lens map λ θ is the conformal representation (suitably determined) of D onto the lens domain below: One has ω 1 (h) h 1/θ

57 Lower estimates: Examples Examples Lens maps For 0 < θ < 1, the lens map λ θ is the conformal representation (suitably determined) of D onto the lens domain below: One has ω 1 (h) h 1/θ ; so we get: a n (C λθ ) α e C n where α, C > 0 depends only on θ.

58 Lower estimates: Examples Corollary LLQR 2013 If ϕ is univalent and ϕ(d) contains an angular sector centered on the unit cercle and with opening πθ, 0 < θ < 1, then: a n (C ϕ ) e C n. with C > 0 depending only on θ.

59 Lower estimates: Examples Examples Cusp map The cusp map χ is the conformal representation (suitably determined) of D onto the cusp domain below:

60 Lower estimates: Examples Examples Cusp map The cusp map χ is the conformal representation (suitably determined) of D onto the cusp domain below:

61 Lower estimates: Examples Examples Cusp map The cusp map χ is the conformal representation (suitably determined) of D onto the cusp domain below: One has ω 1 (h) e C 0/h

62 Lower estimates: Examples Examples Cusp map The cusp map χ is the conformal representation (suitably determined) of D onto the cusp domain below: One has ω 1 (h) e C 0/h ; so wet get: C n/ log n a n (C χ ) e

63 Lower estimates: Counter-example Counter-example: the Shapiro-Taylor map We shall see that these lower estimates for the lens maps and the cusp map are sharp.

64 Lower estimates: Counter-example Counter-example: the Shapiro-Taylor map We shall see that these lower estimates for the lens maps and the cusp map are sharp. However, let ϑ > 0. For ε = ε ϑ > 0 small enough, let V ε = {z C ; Re z > 0 and z < ε}, f ϑ (z) = z( log z) ϑ for z V ε and ς ϑ = exp( f ϑ g ϑ ), where g ϑ is a conformal representation of D onto V ε.

65 Lower estimates: Counter-example Counter-example: the Shapiro-Taylor map We shall see that these lower estimates for the lens maps and the cusp map are sharp. However, let ϑ > 0. For ε = ε ϑ > 0 small enough, let V ε = {z C ; Re z > 0 and z < ε}, f ϑ (z) = z( log z) ϑ for z V ε and ς ϑ = exp( f ϑ g ϑ ), where g ϑ is a conformal representation of D onto V ε. Then ω(t) = t(log 1/t) ϑ and a n (C ςϑ ) n ϑ/2 so that gives C ςϑ S p p > 2/ϑ

66 Lower estimates: Counter-example Counter-example: the Shapiro-Taylor map We shall see that these lower estimates for the lens maps and the cusp map are sharp. However, let ϑ > 0. For ε = ε ϑ > 0 small enough, let V ε = {z C ; Re z > 0 and z < ε}, f ϑ (z) = z( log z) ϑ for z V ε and ς ϑ = exp( f ϑ g ϑ ), where g ϑ is a conformal representation of D onto V ε. Then ω(t) = t(log 1/t) ϑ and a n (C ςϑ ) n ϑ/2 so that gives C ςϑ S p p > 2/ϑ though actually C ςϑ S p p > 4/θ.

67 Upper estimates Upper estimates Proposition (Parfenov 1988) For every symbol ϕ: [a n (C ϕ )] 2 1 sup 0<h<1, ξ D h S(ξ,h) B(z) 2 dm ϕ (z)

68 Upper estimates Upper estimates Proposition (Parfenov 1988) For every symbol ϕ: [a n (C ϕ )] 2 1 sup 0<h<1, ξ D h S(ξ,h) B(z) 2 dm ϕ (z) where B is a Blaschke product with less than n zeroes, S(ξ, h) = D D(ξ, h) and m ϕ is the pull-back measure of m (the normalized Lebesgue measure on D) by ϕ.

69 Upper estimates Upper estimates Proposition (Parfenov 1988) For every symbol ϕ: [a n (C ϕ )] 2 1 sup 0<h<1, ξ D h S(ξ,h) B(z) 2 dm ϕ (z) where B is a Blaschke product with less than n zeroes, S(ξ, h) = D D(ξ, h) and m ϕ is the pull-back measure of m (the normalized Lebesgue measure on D) by ϕ. Follows from: 1) the subspace BH 2 has codimension n 1, so c n (C ϕ ) C ϕ BH 2 where c n (C ϕ ) is the Gelfand number; and: 2) a n (C ϕ ) = c n (C ϕ ) (because H 2 is a Hilbert space).

70 Upper estimates Upper estimates Proposition (Parfenov 1988) For every symbol ϕ: [a n (C ϕ )] 2 1 sup 0<h<1, ξ D h S(ξ,h) B(z) 2 dm ϕ (z) where B is a Blaschke product with less than n zeroes, S(ξ, h) = D D(ξ, h) and m ϕ is the pull-back measure of m (the normalized Lebesgue measure on D) by ϕ. Follows from: 1) the subspace BH 2 has codimension n 1, so c n (C ϕ ) C ϕ BH 2 where c n (C ϕ ) is the Gelfand number; and: 2) a n (C ϕ ) = c n (C ϕ ) (because H 2 is a Hilbert space). We deduce from this proposition the following theorem.

71 Upper estimates Theorem LQR 2013 Assume that ϕ is continuous on D and that ϕ(d) is contained in a polygon with vertices ϕ(e it 1 ),..., ϕ(eit N ). Then, if: ϕ(e it ) ϕ(e it j ) ϖ( t tj ), for t t 0 small enough and j = 1,..., N, and ϖ is continuous, increasing, sub-additive, vanishes at 0, and ϖ(h)/h, one has, for some constants κ, σ > 0: h 0 ϖ 1 (κ 2 a n (C ϕ ) kn ), κ 2 kn where k n is the largest integer such that N k d k < n and d k is the integer part of σ log κ 2 n + ϖ(κ 2 n ) 1.

72 Upper estimates: Examples Examples

73 Upper estimates: Examples Examples Lens maps For λ θ, one has N = 2, ϖ 1 (h) h 1/θ, d k k and k n n; hence, for β, c > 0 depending only on θ: a n (C λθ ) β e c n.

74 Upper estimates: Examples Examples Lens maps For λ θ, one has N = 2, ϖ 1 (h) h 1/θ, d k k and k n n; hence, for β, c > 0 depending only on θ: α e C n a n (C λθ ) β e c n.

75 Upper estimates: Examples Examples Lens maps For λ θ, one has N = 2, ϖ 1 (h) h 1/θ, d k k and k n n; hence, for β, c > 0 depending only on θ: α e C n a n (C λθ ) β e c n. Remark. Similarly, H. Queélec and K. Seip (2015) showed that if 1 ϕ(z) = 1 + (1 z) α, 0 < α < 1, then: e π(1 α) (2n)/α a n (C ϕ ) e π(1 α) n/(2α)

76 Upper estimates: Examples Examples Lens maps For λ θ, one has N = 2, ϖ 1 (h) h 1/θ, d k k and k n n; hence, for β, c > 0 depending only on θ: a n (C λθ ) β e c n. Cusp map For the cusp map χ, one has N = 1, ϖ 1 (h) = e 1/h, d k 2 k and 2 kn n/ log n; hence: a n (C χ ) e c n/ log n.

77 Upper estimates: Examples Examples Lens maps For λ θ, one has N = 2, ϖ 1 (h) h 1/θ, d k k and k n n; hence, for β, c > 0 depending only on θ: α e C n a n (C λθ ) β e c n. Cusp map For the cusp map χ, one has N = 1, ϖ 1 (h) = e 1/h, d k 2 k and 2 kn n/ log n; hence: e C n/ log n a n (C χ ) e c n/ log n.

78 Upper estimates: Examples Examples Lens maps For λ θ, one has N = 2, ϖ 1 (h) h 1/θ, d k k and k n n; hence, for β, c > 0 depending only on θ: a n (C λθ ) β e c n. Cusp map For the cusp map χ, one has N = 1, ϖ 1 (h) = e 1/h, d k 2 k and 2 kn n/ log n; hence: a n (C χ ) e c n/ log n.

79 Upper estimates: Examples The example of lens maps can be generalized as follows: Theorem LQR 2013 If ϕ(d) is contained in a polygon P with vertices on the unit circle, then, for constants α, β > 0 depending only of P, one has: a n (C ϕ ) α e β n.

80 Upper estimates: Examples The example of lens maps can be generalized as follows: Theorem LQR 2013 If ϕ(d) is contained in a polygon P with vertices on the unit circle, then, for constants α, β > 0 depending only of P, one has: a n (C ϕ ) α e β n. For the proof, we may assume that ϕ is conformal from D onto P and we use the Schwarz-Christoel formula, which allows to take ϖ(h) = h θ in the previous theorem (where πθ is the greatest angle of the polygon).

81 Upper estimates: Examples The example of lens maps can be generalized as follows: Theorem LQR 2013 If ϕ(d) is contained in a polygon P with vertices on the unit circle, then, for constants α, β > 0 depending only of P, one has: a n (C ϕ ) α e β n. For the proof, we may assume that ϕ is conformal from D onto P and we use the Schwarz-Christoel formula, which allows to take ϖ(h) = h θ in the previous theorem (where πθ is the greatest angle of the polygon). Remark. This should be compared with the previous result return

82 Upper estimates: Examples The example of lens maps can be generalized as follows: Theorem LQR 2013 If ϕ(d) is contained in a polygon P with vertices on the unit circle, then, for constants α, β > 0 depending only of P, one has: a n (C ϕ ) α e β n. For the proof, we may assume that ϕ is conformal from D onto P and we use the Schwarz-Christoel formula, which allows to take ϖ(h) = h θ in the previous theorem (where πθ is the greatest angle of the polygon). Remark. This should be compared with the previous result return

83 Upper estimates: Examples Spread lens maps In the two previous examples, the composition operators are in all the Schatten classes S p, p > 0. For the following example, it is not the case.

84 Upper estimates: Examples Spread lens maps In the two previous examples, the composition operators are in all the Schatten classes S p, p > 0. For the following example, it is not the case. (Theorem LLQR 2013) Let λ θ be a lens map and φ θ (z) = λ θ (z) exp ( 1+z 1 z ). Then: a n (C φθ ) (log n/n) 1/2θ n = 2, 3,...

85 Upper estimates: Examples We get that C φθ S p for p > 2θ.

86 Upper estimates: Examples We get that C φθ S p for p > 2θ. However, with Luecking's characterization, one can show: Theorem LLQR 2013 C φθ S p p > 2θ.

87 Upper estimates: Examples We get that C φθ S p for p > 2θ. However, with Luecking's characterization, one can show: Theorem LLQR 2013 C φθ S p p > 2θ. Open question Find a lower estimate for a n (C φθ ).

88 Remark Our proofs for the lower and upper estimates give in particular the following remark. If B is a Blaschke product, (BH 2 ) is the model space associated to B. One has:

89 Remark For any symbol ϕ: a n (C ϕ ) sup u 1,...,u n (0,1) inf f (BH 2 ) f =1 C ϕf where the supremum is taken over all Blaschke products with n zeros on the real interval (0, 1). a n (C ϕ ) inf B sup f BH 2 f =1 C ϕ f where the inmum is taken over all Blaschke products with less than n zeros.

90 Two sides estimates Two sides estimates To end this part, we state results of H. Queélec and K. Seip with two sides estimates.

91 Two sides estimates Two sides estimates To end this part, we state results of H. Queélec and K. Seip with two sides estimates. Notation. Let u : D R be in the disk algebra (i.e. continuous on D and analytic in D) such that u( z) = u(z); let ũ its harmonic conjugate, and: ϕ u = exp( u iũ).

92 Two sides estimates Two sides estimates To end this part, we state results of H. Queélec and K. Seip with two sides estimates. Notation. Let u : D R be in the disk algebra (i.e. continuous on D and analytic in D) such that u( z) = u(z); let ũ its harmonic conjugate, and: ϕ u = exp( u iũ). One assumes that, if U(t) = u(e it ), U is increasing on[0, π], U(0) = 0, and that U is smooth, except perhaps at t = 0.

93 Two sides estimates Two sides estimates To end this part, we state results of H. Queélec and K. Seip with two sides estimates. Notation. Let u : D R be in the disk algebra (i.e. continuous on D and analytic in D) such that u( z) = u(z); let ũ its harmonic conjugate, and: ϕ u = exp( u iũ). One assumes that, if U(t) = u(e it ), U is increasing on[0, π], U(0) = 0, and that U is smooth, except perhaps at t = 0. Moreover, one assumes that: to ensure that C ϕu h u (t) := π t is compact. U(x) x 2 dx t 0 +

94 Two sides estimates Then: Theorem (smooth case) H. Queélec - K. Seip 2015 Assume that, for some c > 1 and C > 0, one has, for t small enough: t U (t) U(t) 1 + c log t and U(t) t h u (t) C log t (log log t )

95 Two sides estimates Then: Theorem (smooth case) H. Queélec - K. Seip 2015 Assume that, for some c > 1 and C > 0, one has, for t small enough: t U (t) U(t) 1 + c log t and U(t) t h u (t) C log t (log log t ) These assumptions mean that ϕ u is tangentially smooth at 1

96 Two sides estimates Then: Theorem (smooth case) H. Queélec - K. Seip 2015 Assume that, for some c > 1 and C > 0, one has, for t small enough: t U (t) U(t) 1 + c log t and U(t) t h u (t) C log t (log log t ) Then: a n (C ϕu ) 1 hu (e n ) return

97 Two sides estimates Then: Theorem (smooth case) H. Queélec - K. Seip 2015 Assume that, for some c > 1 and C > 0, one has, for t small enough: t U (t) U(t) 1 + c log t and U(t) t h u (t) C log t (log log t ) Then: a n (C ϕu ) 1 hu (e n ) return

98 Two sides estimates For the second result, we need a bit more notation. Writing: U(t) = e η U( log t ) for 0 < t 1 and U(t) 1/e, one denes ω U by the implicit equation: for x 0 such that η U (x) 1. η U ( x/ωu (x) ) = ω U (x)

99 Two sides estimates Then: Theorem (sharp cusp case) H. Queélec - K. Seip 2015 Assume that: η U (x) η U (x) = o (1/x) as x ;

100 Two sides estimates Then: Theorem (sharp cusp case) H. Queélec - K. Seip 2015 Assume that: η U (x) η U (x) = o (1/x) as x ; These assumptions mean that ϕ is sharp cusped at 1

101 Two sides estimates Then: Theorem (sharp cusp case) H. Queélec - K. Seip 2015 Assume that: η U (x) η U (x) = o (1/x) as x ; Then: [ ( ) ] π 2 n a n (C ϕu ) = exp 2 + o (1) ω U (n)

102 Two sides estimates Then: Theorem (sharp cusp case) H. Queélec - K. Seip 2015 Assume that: η U (x) η U (x) = o (1/x) as x ; Then: [ ( ) ] π 2 n a n (C ϕu ) = exp 2 + o (1) ω U (n) The proofs are rather involved.

103 Dimension d 2 Second part: dimension d 2

104 Dimension d 2 Second part: dimension d 2 Two domains are classical: the open ball B d = {z = (z 1,..., z d ) C d ; z z d 2 < 1}

105 Dimension d 2 Second part: dimension d 2 Two domains are classical: the open ball B d = {z = (z 1,..., z d ) C d ; z z d 2 < 1} the polydisc D d

106 Dimension d 2 Second part: dimension d 2 Two domains are classical: the open ball B d = {z = (z 1,..., z d ) C d ; z z d 2 < 1} the polydisc D d The Hardy space H 2 (Ω) (with Ω = B d or D d ) is dened similarly as in dimension 1.

107 Dimension d 2 Second part: dimension d 2 Two domains are classical: the open ball B d = {z = (z 1,..., z d ) C d ; z z d 2 < 1} the polydisc D d The Hardy space H 2 (Ω) (with Ω = B d or D d ) is dened similarly as in dimension 1. Caution Not all symbols ϕ: Ω Ω give a bounded composition operator on H 2 (Ω).

108 Dimension d 2 Second part: dimension d 2 Two domains are classical: the open ball B d = {z = (z 1,..., z d ) C d ; z z d 2 < 1} the polydisc D d The Hardy space H 2 (Ω) (with Ω = B d or D d ) is dened similarly as in dimension 1. Caution Not all symbols ϕ: Ω Ω give a bounded composition operator on H 2 (Ω). In the sequel, we shall assume the symbol ϕ is such that ϕ(ω) has non-void interior.

109 Dimension d 2 Lower estimates One has, as in dimension 1: Proposition BLQR 2015 Let C ϕ : H 2 (Ω) H 2 (Ω) be compact (Ω = B d or D d ). Let u = (u 1,..., u n ) Ω n and v j = ϕ(u j ) be distinct. Let M v be the interpolation constant of v = (v 1,..., v n ). Then, setting: one has: µ 2 n = inf d 1 j n k=1 1 u j,k 2 1 v j,k 2, a n (C ϕ ) µ n M 2 v.

110 Dimension d 2 Lower estimates Then: Theorem BLQR 2015 Let C ϕ : H 2 (Ω) H 2 (Ω) be compact (Ω = B d or D d ). Then, for some constant C > 0, one has: a n (C ϕ ) e Cn1/d.

111 Dimension d 2 Lower estimates Then: Theorem BLQR 2015 Let C ϕ : H 2 (Ω) H 2 (Ω) be compact (Ω = B d or D d ). Then, for some constant C > 0, one has: a n (C ϕ ) e Cn1/d. The interesting point is the dependence with the dimension d.

112 Dimension d 2 Lower estimates It is obtained with a good choice of the sequence (u 1,..., u n ) in the previous proposition, and using estimates on its interpolation constant due to: P. Beurling when Ω = D d ; B. Berndtsson (1985) when Ω = B d.

113 Dimension d 2 Lower estimates Generalization A bounded symmetric domain of C d is a bounded open convex and circled subset Ω of C d such that for every point a Ω, there is an involutive bi-holomorphic map u : Ω Ω such that a is an isolated xed point of u (equivalently, as shown by J.-P. Vigué: u(a) = a and u (a) = id).

114 Dimension d 2 Lower estimates Generalization A bounded symmetric domain of C d is a bounded open convex and circled subset Ω of C d such that for every point a Ω, there is an involutive bi-holomorphic map u : Ω Ω such that a is an isolated xed point of u (equivalently, as shown by J.-P. Vigué: u(a) = a and u (a) = id). The unit ball B d and the polydisk D d are examples of bounded symmetric domains.

115 Dimension d 2 Lower estimates Hardy space The Shilov boundary S Ω of Ω is the smallest closed set F Ω such that sup z Ω f (z) = sup f (z) z Ω for every function f holomorphic in a neighbourhood of Ω. It is also the set of extreme points of Ω.

116 Dimension d 2 Lower estimates Hardy space The Shilov boundary S Ω of Ω is the smallest closed set F Ω such that sup z Ω f (z) = sup f (z) z Ω for every function f holomorphic in a neighbourhood of Ω. It is also the set of extreme points of Ω. The Shilov boundary of B d is its usual boundary S d 1.

117 Dimension d 2 Lower estimates Hardy space The Shilov boundary S Ω of Ω is the smallest closed set F Ω such that sup z Ω f (z) = sup f (z) z Ω for every function f holomorphic in a neighbourhood of Ω. It is also the set of extreme points of Ω. The Shilov boundary of B d is its usual boundary S d 1. But the Shilov boundary of the bidisk D 2 is {(z 1, z 2 ) C 2 z 1 = z 2 = 1}, though D 2 = {(z 1, z 2 ) C 2 ; z 1, z 2 1 and z 1 = 1 or z 2 = 2}.

118 Dimension d 2 Lower estimates There is a unique probability measure σ on S Ω which is invariant by the automorphisms u of Ω such that u(0) = 0. The Hardy space H 2 (Ω) is the space of analytic functions f : Ω C such that: ( ) 1/2 f 2 = sup f (rξ) 2 dσ(ξ) <. 0<r<1 S Ω

119 Dimension d 2 Lower estimates We have: Theorem BLQR 2015 Let Ω be a bounded symmetric domain of C d and C ϕ : H 2 (Ω) H 2 (Ω) compact. Then, for some constant C > 0, one has: a n (C ϕ ) e Cn1/d.

120 Dimension d 2 Lower estimates That use Weyl Lemma and: Theorem (D. Clahane 2005) Let Ω be a bounded symmetric domain of C d and ϕ: Ω Ω be a holomorphic map inducing a compact composition operator C ϕ : H 2 (Ω) H 2 (Ω). Then ϕ has a unique xed point z 0 Ω and the spectrum of C ϕ consists of 0, and all possible products of eigenvalues of the derivative ϕ (z 0 ).

121 Dimension d 2 Lower estimates That use Weyl Lemma and: Theorem (D. Clahane 2005) Let Ω be a bounded symmetric domain of C d and ϕ: Ω Ω be a holomorphic map inducing a compact composition operator C ϕ : H 2 (Ω) H 2 (Ω). Then ϕ has a unique xed point z 0 Ω and the spectrum of C ϕ consists of 0, and all possible products of eigenvalues of the derivative ϕ (z 0 ). However, the rst proof give more information to construct examples.

122 Dimension d 2 Upper estimates Upper estimates Theorem BLQR 2015 Let Ω = B d1 B dn, d d N = d.

123 Dimension d 2 Upper estimates Upper estimates Theorem BLQR 2015 Let Ω = B d1 B dn, d d N = d. N = 1 gives the ball B d N = d and d 1 = = d N = 1 give the polydisk D d.

124 Dimension d 2 Upper estimates Upper estimates Theorem BLQR 2015 Let Ω = B d1 B dn, d d N = d. If ϕ < 1, then C ϕ is compact and a n (C ϕ ) e Cn1/d.

125 Dimension d 2 Upper estimates Upper estimates Theorem BLQR 2015 Let Ω = B d1 B dn, d d N = d. If ϕ < 1, then C ϕ is compact and a n (C ϕ ) e Cn1/d. Open question Does that hold for Ω a general bounded symmetric domain?

126 Dimension d 2 Upper estimates Upper estimates Theorem BLQR 2015 Let Ω = B d1 B dn, d d N = d. If ϕ < 1, then C ϕ is compact and a n (C ϕ ) e Cn1/d. Open question Does that hold for Ω a general bounded symmetric domain? Open question Does the converse hold?

127 Dimension d 2 Upper estimates In the case of the polydisk Ω = D d, and diagonal symbols, one has: Theorem BLQR 2015 Let ϕ 1,..., ϕ D : D D be symbols inducing compact composition operators on H 2 (D), and let: ϕ(z 1,..., z d ) = ( ϕ 1 (z 1 ),..., ϕ d (z d ) ). Then, for C ϕ : H 2 (D d ) H 2 (D d ), one has: ( d a n (C ϕ ) 2 d 1 j=1 ) C ϕj inf n 1 n d n [ an1 (C ϕ1 )+ +a nd (C ϕd ) ].

128 Dimension d 2 Upper estimates To prove that, for xed n 1,..., n d such that n 1 n d n, one consider, for each j = 1,..., d, an operator R j : H 2 (D) H 2 (D) with rank < n j such that C ϕj R j = a nj (C ϕj ).

129 Dimension d 2 Upper estimates To prove that, for xed n 1,..., n d such that n 1 n d n, one consider, for each j = 1,..., d, an operator R j : H 2 (D) H 2 (D) with rank < n j such that C ϕj R j = a nj (C ϕj ). One denes R : H 2 (D d ) H 2 (D d ) by: where α = (α 1,..., α d ). R(z α ) = R 1 (z α 1 1 ) R d(z α d d ),

130 Dimension d 2 Upper estimates To prove that, for xed n 1,..., n d such that n 1 n d n, one consider, for each j = 1,..., d, an operator R j : H 2 (D) H 2 (D) with rank < n j such that C ϕj R j = a nj (C ϕj ). One denes R : H 2 (D d ) H 2 (D d ) by: where α = (α 1,..., α d ). R(z α ) = R 1 (z α 1 1 ) R d(z α d d ), Then R has rank < n 1 n d n and C ϕ R is less than the upper estimate given in the theorem.

131 Dimension d 2 Examples Examples Multi-lens maps Let 0 < θ 1,..., θ d < 1 and λ θ1,..., λ θd maps. Then, if: be the associated lens ϕ(z 1,..., z d ) = ( λ θ1 (z 1 ),..., λ θd (z d ) ), one has: e αn1/(2d) a n (C ϕ ) e βn1/(2d).

132 Dimension d 2 Examples Multi-cusp map Let χ be the above cusp map, and ϕ(z 1,..., z d ) = ( χ(z 1 ),..., χ(z d ) ). Then: e αn1/d / log n a n (C ϕ ) e βn1/d / log n.

133 Dimension d 2 Examples Another type of example Let c 1,..., c d > 0 such that c c d 1 and ϕ(z 1,..., z d ) = (c 1 z c d z d, 0,..., 0). Then: a n (C ϕ ) (c c d ) n n (d 1)/4

134 Dimension d 2 Examples Another type of example Let c 1,..., c d > 0 such that c c d 1 and ϕ(z 1,..., z d ) = (c 1 z c d z d, 0,..., 0). Then: a n (C ϕ ) (c c d ) n n (d 1)/4 In particular, if c c d = 1, then C ϕ is compact, and C ϕ S p p > 4/(d 1).

135 Dimension d 2 Examples This example is called by Hervé toy example

136 Dimension d 2 Examples This example is called by Hervé toy example : this is the Toy Story

137 Dimension d 2 Examples This example is called by Hervé toy example : this is the Toy Story

138 Innite dimension Introduction Third part: innite dimension Introduction We saw that: Theorem For C ϕ : H 2 (D d ) H 2 (D d ), one has: C n1/d always a n (C ϕ ) e c n1/d if ϕ < 1, then a n (C ϕ ) e

139 Innite dimension Introduction Third part: innite dimension Introduction We saw that: Theorem For C ϕ : H 2 (D d ) H 2 (D d ), one has: C n1/d always a n (C ϕ ) e c n1/d if ϕ < 1, then a n (C ϕ ) e As e C n1/d d e C > 0, one might believe that there is no compact composition operator in innite dimension.

140 Innite dimension Introduction Third part: innite dimension Introduction We saw that: Theorem For C ϕ : H 2 (D d ) H 2 (D d ), one has: C n1/d always a n (C ϕ ) e c n1/d if ϕ < 1, then a n (C ϕ ) e As e C n1/d d e C > 0, one might believe that there is no compact composition operator in innite dimension. Actually, it is not the case, as we shall see.

141 Innite dimension Hardy space Hardy space We consider the innite polydisk D. We have to dene the Hardy space H 2.

142 Innite dimension Hardy space Hardy space We consider the innite polydisk D. We have to dene the Hardy space H 2. It is natural to ask that it is the space of all functions f with (1) f (z) = α 0 c α z α and f 2 2 := α 0 c α 2 <, where α = (α j ) j 1, z = (z j ) j 1 and z α = j 1 z α j j. If one asks absolute convergence in (1), we should have α 0 z α 2 <.

143 Innite dimension Hardy space Since one has the Euler type formula: z α 2 = α 0 1 j=1 1 z j 2 we get that: z j 2 <. j=1

144 Innite dimension Hardy space Since one has the Euler type formula: z α 2 = α 0 1 j=1 1 z j 2 we get that: z j 2 <. j=1 Hence it is natural to consider Ω 2 = D l 2 instead of the whole polydisk.

145 Innite dimension Hardy space Actually, we will work with Ω 1 = D l 1 (which is an open subset of l 1 ) because of the following proposition: Proposition LQR 2016 Let ϕ j : D D be analytic, j = 1, 2,... and ϕ(z) = ( ϕ j (z j ) ) j 1. Then C ϕ(f ) 2 < for all f 2 < if and only if: ϕ j (0) <. j=1

146 Innite dimension Hardy space Actually, we will work with Ω 1 = D l 1 (which is an open subset of l 1 ) because of the following proposition: Proposition LQR 2016 Let ϕ j : D D be analytic, j = 1, 2,... and ϕ(z) = ( ϕ j (z j ) ) j 1. Then C ϕ(f ) 2 < for all f 2 < if and only if: ϕ j (0) <. j=1 Hence H 2 = H 2 (Ω 1 ) will be the space of all f : Ω 1 C such that f 2 <.

147 Innite dimension We will say that ϕ is truly innite-dimensional if ϕ (a): l 1 l 1 is one-to-one for some a Ω 1.

148 Innite dimension We will say that ϕ is truly innite-dimensional if ϕ (a): l 1 l 1 is one-to-one for some a Ω 1. First, if ϕ(ω 1 ) remains far from Ω 1, one has: Theorem LQR 2016 Let ϕ: Ω 1 Ω 1 truly innite-dimensional such that ϕ(ω 1 ) Ω 1 is compact. Then: 1) C ϕ : H 2 (Ω 1 ) H 2 (Ω 1 ) is bounded, and even compact; 2) ϕ (0): l 1 l 1 is compact; 3) for all p > 0, one has: 1 [ ( log 1/an (C ϕ ) )] p =. n=1

149 Innite dimension Caution There exist compact composition operators C ϕ : H 2 (Ω 1 ) H 2 (Ω 1 ) such that ϕ(ω 1 ) is unbounded in l 1. One can take a diagonal symbol.

150 Innite dimension Caution There exist compact composition operators C ϕ : H 2 (Ω 1 ) H 2 (Ω 1 ) such that ϕ(ω 1 ) is unbounded in l 1. One can take a diagonal symbol. Remark. Assuming ϕ(ω 1 ) compact in l 1 instead compact in Ω 1 is not sucient. ( 1 + z1 ) Example: ϕ(z) =, 0, 0,.... 2

151 Innite dimension Proposition LQR 2016 Let λ 1, λ 2,... < 1 and ϕ(z) = (λ j z j ) j 1. Then ϕ: Ω 1 Ω 1 and, for every p > 0 : (λ j ) j 1 l p C ϕ S p. In particular, there exist truly innite-dimensional symbols on Ω 1 such that C ϕ is in all Schatten classes S p, p > 0.

152 Innite dimension Theorem LQR 2016 For every 0 < δ < 1, there exist compact composition operators on H 2 (Ω 1 ) with truly innite-dimensional symbol such that: a n (C ϕ ) exp [ c e b(log n)δ ].

153 Th a t' s a l l F o lk s!

154 References D. Li, H. Queélec, L. Rodríguez-Piazza, On approximation numbers of composition operators, J. Approximation Theory 164, (2012) P. Lefèvre, D. Li, H. Queélec, L. Rodríguez-Piazza, Some new properties of composition opertors associated with lens maps, Israel J. Math. 195, (2013) D. Li, H. Queélec, L. Rodríguez-Piazza, Estimates for approximation numbers of some classes of composition operators on the Hardy space, Annales Acad. Sci. Fennicae Math. 38, (2013) D. Li, H. Queélec, L. Rodríguez-Piazza, A spectral radius type formula for approximation numbers of composition operators, J. Funct. Anal. 267, (2014) H. Queélec, K. Seip, Decay rates for approximation numbers of composition operators, J. Anal. Math. 125, (2015) F. Bayart, D. Li, H. Queélec, L. Rodríguez-Piazza, Approximation numbers of composition operators on the Hardy and Bergman spaces of the ball and of the polydisk, submitted D. Li, H. Queélec, L. Rodríguez-Piazza, on the Hardy space of the innite polydisk, in preparation