Difference of two weighted composition operators on Bergman spaces

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1 Difference of two weighted comosition oerators on Bergman saces S. Acharyya, Z. Wu Deartment of Math, Physical, and, Life Sciences Embry - Riddle Aeronautical University Worldwide, Deartment of Mathematical Sciences University of Nevada, Las Vegas Southeastern Analysis Meeting 2017 Southeastern Analysis Meeting / 22

2 Definition H (D) : All Analytic functions on D Weighted { Bergman sace } A α = f H (D) : f (z) da α (z) <, where ( D da α (z) = (α + 1) 1 z 2) α da (z), α > 1 A 2 0 = A2 (Bergman Sace) Southeastern Analysis Meeting / 22

3 Definition H (D) : All Analytic functions on D Weighted { Bergman sace } A α = f H (D) : f (z) da α (z) <, where ( D da α (z) = (α + 1) 1 z 2) α da (z), α > 1 A 2 0 = A2 (Bergman Sace) Southeastern Analysis Meeting / 22

4 Definition H (D) : All Analytic functions on D Weighted { Bergman sace } A α = f H (D) : f (z) da α (z) <, where ( D da α (z) = (α + 1) 1 z 2) α da (z), α > 1 A 2 0 = A2 (Bergman Sace) Southeastern Analysis Meeting / 22

5 Definitions ϕ : Analytic from D D Definition The comosition oerator with symbol ϕ: C ϕ : H (D) H (D), C ϕ (f ) = f ϕ u : Measurable from D C Definition The weighted comosition oerator with weight u and symbol ϕ: uc ϕ : H (D) All measurable functions on D, uc ϕ (f ) = u (f ϕ) Southeastern Analysis Meeting / 22

6 Definitions ϕ : Analytic from D D Definition The comosition oerator with symbol ϕ: C ϕ : H (D) H (D), C ϕ (f ) = f ϕ u : Measurable from D C Definition The weighted comosition oerator with weight u and symbol ϕ: uc ϕ : H (D) All measurable functions on D, uc ϕ (f ) = u (f ϕ) Southeastern Analysis Meeting / 22

7 Introduction Question When is uc ϕ vc ψ comact? (Assume u and v are analytic) It is known that: Theorem (Z.Čučković and R. Zhao, 2007) 1 < < Then uc ϕ is comact from A α into A β if and only if ( ) (2+α) 1 z 2 lim z 1 D 1 z ϕ(w) 2 u(w) da β (w) = 0. Southeastern Analysis Meeting / 22

8 Introduction Question When is uc ϕ vc ψ comact? (Assume u and v are analytic) It is known that: Theorem (Z.Čučković and R. Zhao, 2007) 1 < < Then uc ϕ is comact from A α into A β if and only if ( ) (2+α) 1 z 2 lim z 1 D 1 z ϕ(w) 2 u(w) da β (w) = 0. Southeastern Analysis Meeting / 22

9 A theorem of Moorhouse Question When is C ϕ C ψ comact? Theorem (J. Moorhouse, 2005) C ϕ C ψ is comact on A 2 α if and only if both 1 z 2 1 z 2 lim σ(z) = 0, lim σ(z) z 1 1 ϕ(z) 2 z 1 1 ψ(z) 2 = 0. Here σ(z) = ϕ(z) ψ(z) 1 ϕ(z)ψ(z), z D Note: σ is often referred to as the Cancellation Factor. Southeastern Analysis Meeting / 22

10 A theorem of Moorhouse Question When is C ϕ C ψ comact? Theorem (J. Moorhouse, 2005) C ϕ C ψ is comact on A 2 α if and only if both 1 z 2 1 z 2 lim σ(z) = 0, lim σ(z) z 1 1 ϕ(z) 2 z 1 1 ψ(z) 2 = 0. Here σ(z) = ϕ(z) ψ(z) 1 ϕ(z)ψ(z), z D Note: σ is often referred to as the Cancellation Factor. Southeastern Analysis Meeting / 22

11 Connection between the Difference oerator and Weighted Comosition oerators The next theorem links the two tyes of oerators. Theorem (E. Saukko, 2011) 1 < < Then C ϕ C ψ is comact from A α into A β if and only if σc ϕ and σc ψ are both comact from A α into L (A β ). Southeastern Analysis Meeting / 22

12 Another Version of Saukko s theorem Theorem (Another Version) C ϕ C ψ is comact from A α into A β if and only if each of the following holds: (a) (b) ( ) (2+α) 1 z 2 lim z 1 D 1 z ϕ(w) 2 σ(w) da β (w) = 0, ( ) (2+α) 1 z 2 lim z 1 D 1 z ψ(w) 2 σ(w) da β (w) = 0 Southeastern Analysis Meeting / 22

13 Answering the Original Question Question When is uc ϕ vc ψ comact? (Assume u and v are analytic) Definition For γ R, M(γ) is defined as follows: M(γ) = {f : f (z)(1 z 2 ) γ L < } Southeastern Analysis Meeting / 22

14 Answering the Original Question Question When is uc ϕ vc ψ comact? (Assume u and v are analytic) Definition For γ R, M(γ) is defined as follows: M(γ) = {f : f (z)(1 z 2 ) γ L < } Southeastern Analysis Meeting / 22

15 Comactness of uc ϕ vc ψ 0 < < 2+α 2+β u, v M( 2+β 2+α ) Theorem (Acharyya and Wu, 2017) uc ϕ vc ψ : A α A β is comact if and only if each of the following holds: (a) (b) lim σ(z) z 1 ( u(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α lim (1 σ(z) 2 ) 2+α u(z) v(z) z 1 ( + v(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α (1 z 2 ) 2+β (1 ψ(z) 2 ) 2+α ) = 0, ) + (1 z 2 ) 2+β = 0 (1 ψ(z) 2 ) 2+α Southeastern Analysis Meeting / 22

16 Comactness of uc ϕ vc ψ 0 < < 2+α 2+β u, v M( 2+β 2+α ) Theorem (Acharyya and Wu, 2017) uc ϕ vc ψ : A α A β is comact if and only if each of the following holds: (a) (b) lim σ(z) z 1 ( u(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α lim (1 σ(z) 2 ) 2+α u(z) v(z) z 1 ( + v(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α (1 z 2 ) 2+β (1 ψ(z) 2 ) 2+α ) = 0, ) + (1 z 2 ) 2+β = 0 (1 ψ(z) 2 ) 2+α Southeastern Analysis Meeting / 22

17 Comactness of uc ϕ vc ψ Theorem (Acharyya and Wu, 2017) uc ϕ vc ψ : A α A β is comact if and only if each of the following holds: (a) (b) lim σ(z) z 1 ( u(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α lim (1 σ(z) 2 ) 2+α u(z) v(z) z 1 ( + v(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α (1 z 2 ) 2+β (1 ψ(z) 2 ) 2+α Proof: "= " Suose uc ϕ vc ψ : A α A β is comact. Let ϕ a (z) = z a 1 az. Note that k a, ϕ a k a 0 weakly. Thus ) = 0, ) + (1 z 2 ) 2+β = 0 (1 ψ(z) 2 ) 2+α lim uc ϕ (k a ) vc ψ (k a ) a 1,β = 0, lim uc ϕ (ϕ a k a ) vc ψ (ϕ a k a ) a 1,β = 0. Southeastern Analysis Meeting

18 Comactness of uc ϕ vc ψ Aly the lemma: Lemma Suose 0 < < and 0 < r < 1. There is a constant C > 0 such that for any z D and f A α f (z) C (1 z 2 ) 2+α f (w) da α (w). (z,r) Also, use the elementary facts that C ϕ (ϕ ϕ(z) )(z) = 0 and C ψ (ϕ ϕ(z) )(z) = σ(z), and a chain of ineualities to obtain σ(z) u(z) (1 z 2 ) lim z 1 (1 ϕ(z) 2 ) 2+α u(z) v(z) (1 σ(z) 2 ) lim z 1 (1 ψ(z) 2 ) 2+α 2+α 2+β = 0, (1 z 2 ) 2+β = 0. Southeastern Analysis Meeting

19 Comactness of uc ϕ vc ψ Aly the lemma: Lemma Suose 0 < < and 0 < r < 1. There is a constant C > 0 such that for any z D and f A α f (z) C (1 z 2 ) 2+α f (w) da α (w). (z,r) Also, use the elementary facts that C ϕ (ϕ ϕ(z) )(z) = 0 and C ψ (ϕ ϕ(z) )(z) = σ(z), and a chain of ineualities to obtain σ(z) u(z) (1 z 2 ) lim z 1 (1 ϕ(z) 2 ) 2+α u(z) v(z) (1 σ(z) 2 ) lim z 1 (1 ψ(z) 2 ) 2+α 2+α 2+β = 0, (1 z 2 ) 2+β = 0. Southeastern Analysis Meeting

20 Comactness of uc ϕ vc ψ Similarly σ(z) v(z) (1 z 2 ) lim z 1 (1 ψ(z) 2 ) 2+α 2+β = 0, u(z) v(z) (1 σ(z) 2 ) lim z 1 (1 ϕ(z) 2 ) 2+α 2+α = (has root in Moorhouse and Saukko s work:) (1 z 2 ) 2+β = 0. It is sufficient to show that for any seuence {f n } in A α with f n,α 1 and f n (z) 0 as n uniformly on any comact set of D, we have (uc ϕ vc ψ )(f n ),β 0 as n. Partition the disk into E and E, with E = {z D : σ(z) < } Southeastern Analysis Meeting

21 Comactness of uc ϕ vc ψ Similarly σ(z) v(z) (1 z 2 ) lim z 1 (1 ψ(z) 2 ) 2+α 2+β = 0, u(z) v(z) (1 σ(z) 2 ) lim z 1 (1 ϕ(z) 2 ) 2+α 2+α = (has root in Moorhouse and Saukko s work:) (1 z 2 ) 2+β = 0. It is sufficient to show that for any seuence {f n } in A α with f n,α 1 and f n (z) 0 as n uniformly on any comact set of D, we have (uc ϕ vc ψ )(f n ),β 0 as n. Partition the disk into E and E, with E = {z D : σ(z) < } Southeastern Analysis Meeting

22 Comactness of uc ϕ vc ψ We can write (uc ϕ vc ψ )(f n ) = (uc ϕ vc ψ )(f n )χ E +(u v)c ψ (f n )χ E +u(c ϕ C ψ )(f n )χ E. Therefore we need to establish the following three statements. lim (uc ϕ vc ψ )(f n )χ E n,β = 0, lim (u v)c ψ(f n )χ E n,β = 0, lim u(c ϕ C ψ )(f n )χ E n,β = 0. The first two statements are true, due to the following lemma. Southeastern Analysis Meeting

23 Comactness of uc ϕ vc ψ We can write (uc ϕ vc ψ )(f n ) = (uc ϕ vc ψ )(f n )χ E +(u v)c ψ (f n )χ E +u(c ϕ C ψ )(f n )χ E. Therefore we need to establish the following three statements. lim (uc ϕ vc ψ )(f n )χ E n,β = 0, lim (u v)c ψ(f n )χ E n,β = 0, lim u(c ϕ C ψ )(f n )χ E n,β = 0. The first two statements are true, due to the following lemma. Southeastern Analysis Meeting

24 Comactness of uc ϕ vc ψ We can write (uc ϕ vc ψ )(f n ) = (uc ϕ vc ψ )(f n )χ E +(u v)c ψ (f n )χ E +u(c ϕ C ψ )(f n )χ E. Therefore we need to establish the following three statements. lim (uc ϕ vc ψ )(f n )χ E n,β = 0, lim (u v)c ψ(f n )χ E n,β = 0, lim u(c ϕ C ψ )(f n )χ E n,β = 0. The first two statements are true, due to the following lemma. Southeastern Analysis Meeting

25 Comactness of uc ϕ vc ψ Lemma Suose s, t > 0, ω is a nonnegative locally bounded measurable function on D, ϕ is a holomorhic self ma of D, and lim z 1 ω(z) (1 z 2 ) s (1 ϕ(z) 2 ) t = 0. (a) If β > s 1, then the measure ϕ (ω, A β ) is a comact (2 + β + t s)-carleson measure. (b) If β > 1 and ω M(γ) with γ < 1 + β, then the measure ϕ (ω, A β ) is a comact (2 + β γ + ɛ(γ + t s))-carleson measure for any ɛ (0, min{ 1+β γ s γ, 1}) if γ < s, or ɛ (0, 1) if γ s. Southeastern Analysis Meeting

26 Comactness of uc ϕ vc ψ To rove the third statement lim u(c ϕ C ψ )(f n )χ E n,β = 0, we aly Fubini, the revious lemma, and the following lemma: Lemma Let 0 < <. There exists a constant C > 0, such that for all a D, z (a, ), and f A α with f,α 1 f (z) f (a) ϕ a (z) C (1 a 2 ) (2+α)/ f (w) da α. (a, 1 2 ) Southeastern Analysis Meeting

27 The theorem of Moorhouse Theorem (Acharyya and Wu, 2017) uc ϕ vc ψ : A α A β is comact if and only if each of the following holds: (a) (b) lim σ(z) z 1 ( u(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α lim (1 σ(z) 2 ) 2+α u(z) v(z) z 1 Corollary (J. Moorhouse, 2005) ( + v(z) (1 z 2 ) 2+β (1 ϕ(z) 2 ) 2+α C ϕ C ψ is comact on A 2 α if and only if both (1 z 2 ) 2+β (1 ψ(z) 2 ) 2+α ) = 0, ) + (1 z 2 ) 2+β = 0 (1 ψ(z) 2 ) 2+α 1 z 2 1 z 2 lim σ(z) = 0, lim σ(z) z 1 1 ϕ(z) 2 z 1 1 ψ(z) = 0. 2 Southeastern Analysis Meeting

28 Hilbert-Schmidt oerator (definition) X : Searable Hilbert sace {e j } : Orthonormal basis Definition T is Hilbert-Schmidt if T HS(X) = Te j 2 j=0 1 2 < Notational Simlicity: T HS(X) = T HS Southeastern Analysis Meeting

29 Hilbert-Schmidt oerator (definition) X : Searable Hilbert sace {e j } : Orthonormal basis Definition T is Hilbert-Schmidt if T HS(X) = Te j 2 j=0 1 2 < Notational Simlicity: T HS(X) = T HS Southeastern Analysis Meeting

30 Hilbert - Schmidtness of C ϕ C ψ Theorem (B.R. Choe, T. Hosokawa and H. Koo, 2010) Let α 1. Consider C ϕ C ψ acting on A 2 α. Then C ϕ C ψ 2 HS D σ 2 (z) da α (z) (1 ϕ (z) 2 ) 2 + α + σ 2 (z) da α (z) D (1 ψ (z) 2 ) 2 + α Here α = 1 corresonds to H 2, and the comarability constants deend only on α. Southeastern Analysis Meeting

31 Hilbert - Schmidtness of uc ϕ vc ψ Theorem (Acharyya and Wu, 2017) E : D or T u, v : Measurable uc ϕ vc ψ acting from A 2 α L 2 (µ) Then ( uc ϕ vc ψ 2 HS σ 2 u 2 E (1 ϕ 2 ) 2+α + v 2 ) (1 ψ 2 ) 2+α dµ ( ) + (1 σ 2 ) 2+α u v 2 1 E (1 ϕ 2 ) 2+α + 1 (1 ψ 2 ) 2+α dµ Southeastern Analysis Meeting

32 Hilbert - Schmidtness of uc ϕ vc ψ Theorem (Acharyya and Wu, 2017) E : D or T u, v : Measurable uc ϕ vc ψ acting from A 2 α L 2 (µ) Then ( uc ϕ vc ψ 2 HS σ 2 u 2 E (1 ϕ 2 ) 2+α + v 2 ) (1 ψ 2 ) 2+α dµ ( ) + (1 σ 2 ) 2+α u v 2 1 E (1 ϕ 2 ) 2+α + 1 (1 ψ 2 ) 2+α dµ Southeastern Analysis Meeting

33 Hilbert - Schmidtness of uc ϕ vc ψ A Key Lemma: Lemma (Acharyya and Wu, 2017) For z, w D, define ρ = z w 1 zw α > 2 Then A 2 K (α) z (z) + B 2 K w (α) ( (w) + 2R ρ 2 ( A 2 K (α) z ABK w (α) ) (z) + B 2 K (α) (w) +(1 ρ 2 ) 2+α A + B 2 ( K (α) z w ) (z) (z) + K w (α) (w) ). Southeastern Analysis Meeting

34 Hilbert - Schmidtness of uc ϕ vc ψ A Key Lemma: Lemma (Acharyya and Wu, 2017) For z, w D, define ρ = z w 1 zw α > 2 Then A 2 K (α) z (z) + B 2 K w (α) ( (w) + 2R ρ 2 ( A 2 K (α) z ABK w (α) ) (z) + B 2 K (α) (w) +(1 ρ 2 ) 2+α A + B 2 ( K (α) z w ) (z) (z) + K w (α) (w) ). Southeastern Analysis Meeting

35 Corollary Theorem (Acharyya and Wu, 2017) ( uc ϕ vc ψ 2 HS σ 2 u 2 E (1 ϕ 2 ) 2+α + v 2 ) (1 ψ 2 ) 2+α dµ ( ) + (1 σ 2 ) 2+α u v 2 1 E (1 ϕ 2 ) 2+α + 1 (1 ψ 2 ) 2+α dµ Corollary Consider the following oerators σuc ϕ, σvc ψ, (1 σ 2 ) 1+ α 2 (u v)cϕ and (1 σ 2 ) 1+ α 2 (u v)cψ from A 2 α or H 2 to L 2 (µ). Then uc ϕ vc ψ is Hilbert-Schmidt if and only if all of the four oerators are Hilbert-Schmidt. Southeastern Analysis Meeting

36 Corollary Theorem (Acharyya and Wu, 2017) ( uc ϕ vc ψ 2 HS σ 2 u 2 E (1 ϕ 2 ) 2+α + v 2 ) (1 ψ 2 ) 2+α dµ ( ) + (1 σ 2 ) 2+α u v 2 1 E (1 ϕ 2 ) 2+α + 1 (1 ψ 2 ) 2+α dµ Corollary Consider the following oerators σuc ϕ, σvc ψ, (1 σ 2 ) 1+ α 2 (u v)cϕ and (1 σ 2 ) 1+ α 2 (u v)cψ from A 2 α or H 2 to L 2 (µ). Then uc ϕ vc ψ is Hilbert-Schmidt if and only if all of the four oerators are Hilbert-Schmidt. Southeastern Analysis Meeting

37 Questions? Thank You! Southeastern Analysis Meeting

LINEAR FRACTIONAL COMPOSITION OPERATORS OVER THE HALF-PLANE

LINEAR FRACTIONAL COMPOSITION OPERATORS OVER THE HALF-PLANE BOO RIM CHOE, HYUNGWOON KOO, AND WAYNE SMITH Abstract. In the setting of the Hardy saces or the standard weighted Bergman saces over the unit