Strict singularity of a Volterra-type integral operator on H p

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1 Strict singularity of a Volterra-type integral operator on H p Santeri Miihkinen, University of Helsinki IWOTA St. Louis, July 2016 Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

2 Background and motivation Background and motivation Pommerenke (1977): A novel proof of the deep John-Nirenberg inequality using: An integral operator of the type T g f (z) = z 0 f (ζ)g (ζ)dζ, z D, where f and g are analytic (holomorphic) in the unit disc D = {z C : z < 1} of the complex plane (f, g H(D)). f is a variable here and g is fixed, the symbol of T g. Example 1: g(z) = z T g f (z) = z 0 f (ζ)dζ (the classical Volterra operator) Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

3 Background and motivation z 0 Example 2: g(z) = log 1 1 z 1 z T g f (z) = 1 z ( 1 ) k k=0 k+1 n=0 a n z k (the Cesàro operator). f (ζ) 1 ζ dζ = Characterize the properties of T g in terms of the function-theoretic properties of the symbol g? Aleman and Siskakis (1995): systematic research on T g Hardy spaces H p = { ( 1 2π 1/p } f H(D) : f p = sup f (re it ) dt) p <, 0 r<1 2π 0 where 0 < p <. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

4 Background and motivation T g : H p H p, 1 p <, bounded (compact) iff g BMOA (g VMOA), where { } BMOA = h H(D) : h = sup h σ a h(a) 2 < a D and VMOA = { h BMOA : lim sup h σ a h(a) 2 = 0 a 1 }, where σ a (z) = a z 1 āz, a, z D. Aleman and Cima (2001): scale 0 < p < 1. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

5 Strict singularity Strict singularity A bounded operator L between Banach spaces X and Y is strictly singular, if L restricted to any infinite-dimensional closed subspace of X is not a linear isomorphism onto its range. A notion introduced by T. Kato in 58 (in connection to perturbation theory of Fredholm operators). Compact operators are strictly singular. Denote by S(X ) the strictly singular operators on X. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

6 Strict singularity S(X ) (norm-closed) ideal of the bounded operators B(X ): L S(X ), U B(X ) LU, UL S(X ). Example. For p < q, the inclusion mapping i : l p l q, i(x) = x, is a non-compact strictly singular operator. The strict singularity of T g acting on H p? Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

7 Main result Main result Theorem Let g BMOA \ VMOA and 1 p <. Then there exists a subspace M H p isomorphic to l p s.t. the restriction T g M : M T g (M) is bounded from below. (i.e. A non-compact T g : H p H p fixes a copy of l p.) In particular, T g is not strictly singular. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

8 Main result Strategy of the proof Figure: Operators U, V and T g l p V H p T g U H p Strategy: Construct bounded operators U and V : U bounded from below when T g is non-compact (i.e. g BMOA \ VMOA). The diagram commutes: U = T g V T g M : M T g (M) bounded from below on M = V (l p ) l p. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

9 Main result How to define U and V? We utilize suitably chosen standard normalized test functions f a H p defined by [ ] 1 a 2 1/p f a (z) = (1 āz) 2, z D, for each a D. For p = 2, the functions f a are the normalized reproducing kernels of H 2. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

10 Main result Define and V : l p H p, V (α) = U : l p H p, U(α) = α n f an n=1 α n T g f an, n=1 where the sequence (a n ) D, a n 1 is suitably chosen and α = (α n ) l p. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

11 Tools Tools Theorem V is bounded, if a n 1 fast enough. How to show that U is bounded from below? A result by Aleman and Cima (2001): Let 0 < p < and t (0, p/2). Then there exists a constant C = C(p, t) > 0 s.t. for all a D. T g f a p C g σ a g(a) t Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

12 Tools Recall: g BMOA \ VMOA lim sup a 1 g σ a g(a) p > 0 for any 0 < p <. Thus if T g is non-compact, then there exists a sequence (a n ) D, a n ω T = D s.t. lim n T g f an p > 0. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

13 Tools Lemma A localization result for T g : Let g BMOA, 1 p <, and (a k ) D be a sequence such that a k ω T. Define A ε = {e iθ : e iθ ω < ε} for each ε > 0.Then (i) lim T g f ak p dm = 0 for every ε > 0. k T\A ε (ii) lim T g f ak p dm = 0 for each k. ε 0 A ε Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

14 Operator U is bounded from below Sketch of the proof of U(α) p C α l p for all α l p. Using conditions (i), (ii) and the previous result by Aleman and Cima, we can define a sequence (ε n ), ε 1 > ε 2 >... > 0, and a subsequence (b n ) of (a n ) s.t. (i) (ii) (iii) for every n 1, where and c = lim n T g f an p > 0. ( ) 1/p T g f bj p dm < 4 n c, j = 1,..., n 1; A n ( ) 1/p T g f bn p dm < 4 n c; T\A n ( ) c 1/p 2 T g f bn p dm 2c A n A n = A εn = {e iθ : e iθ ω < ε n } Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

15 Operator U is bounded from below Functions T g f bn resemble disjointly supported peaks in L p (T) near some boundary point ω T. This ensures that U(α) p = n α nt g f bn p C α l p for all α = (α n ) l p. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

16 Operator U is bounded from below p Uα p p = α j T g f bj dm T j=1 n=1 ( α n T g f bn p dm A n\a n+1 n=1... C α p l p. A n\a n+1 ) 1/p α j T g f bj j=1 1/p α j T g f bj p dm A n\a n+1 j n p p dm Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

17 Operator U is bounded from below Figure: Operators U, V and T g l p V H p T g U H p f M = V (l p ) = span{f bn } T g f p = U(α) p C α l p V (α) p = f p. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

18 Strict singularity of T g on Bergman spaces and Bloch space Strict singularity of T g on Bergman spaces and Bloch space Standard Bergman spaces A p α = L p (D, da α ) H(D), where da α (z) = (1 z 2 ) α da(z), α > 1 and 0 < p <, are isomorphic to l p. S(l p ) = K(l p ) the strict singularity and compactness are equivalent for T g acting on A p α. Let B be the Bloch space. Then T g : B B is strictly singular T B 0 is strictly singular. Since the little Bloch space B 0 is isomorphic to c 0 and S(c 0 ) = K(c 0 ), the restriction T B 0 is compact. Finally, the biadjoint (T g B 0 ) can be identified with T g : B B and consequently T g acting on B is compact. Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

19 For further reading For further reading A. Aleman, A class of integral operators on spaces of analytic functions, Topics in complex analysis and operator theory, 3-30, Univ. Málaga, Málaga, A.G. Siskakis, Volterra operators on spaces of analytic functions-a survey, Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp A. Aleman and A.G. Siskakis, An integral operator on H p, Complex Variables Theory Appl. 28 (1995), no. 2, A. Aleman and J.A. Cima, An integral operator on H p and Hardy s inequality, J. Analyse Math. 85 (2001), S. Miihkinen, Strict singularity of a Volterra-type integral operator on H p, Proceedings of the AMS, DOI: (to appear in print). Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20

20 For further reading THANK YOU! Santeri Miihkinen, University of Helsinki Volterra-type integral operator IWOTA July / 20