Strict singularity of a Volterra-type integral operator on H p Santeri Miihkinen, University of Eastern Finland IWOTA Chemnitz, 14-18 August 2017 Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 1 / 22
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)). g is xed, the symbol of T g. Example 1: g(z) = z T g f (z) = z f (ζ)dζ (the classical Volterra 0 operator) Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 2 / 22
Background and motivation Example 2: g(z) = log 1 1 z 1 z T g f (z) = 1 z ( ) 1 k a k=0 k+1 n=0 n z k (the Cesàro operator). z 0 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, UEF Volterra-type integral operator 14-18 August 2017 3 / 22
Background and motivation T g : H p H p, 1 p <, bounded (compact) i 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, UEF Volterra-type integral operator 14-18 August 2017 4 / 22
Background and motivation Boundedness, compactness and spectral properties of T g known in many analytic function spaces. Structural properties of T g have not been widely studied. Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 5 / 22
Strict singularity and l p -singularity Strict singularity and l p -singularity A bounded operator L between Banach spaces X and Y is strictly singular, if L restricted to any innite-dimensional closed subspace of X is not a linear isomorphism onto its range. In other words, it is not bounded below on any innite dimensional closed subspace M X. 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, UEF Volterra-type integral operator 14-18 August 2017 6 / 22
Strict singularity and l p -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. A bounded operator L: X Y is l p singular if it is not bounded below on any subspace M isomorphic to l p, i.e. it does not x a copy of l p. The class of l p singular operators is denoted by S p (H p ). Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 7 / 22
Strict singularity and l p -singularity Example. The projection P : H p H p, P( k=0 a kz k ) = k=0 a 2 k z 2k is bounded and span{z 2k } is isomorphic to l 2 by Paley's theorem. Hence P S p (H p ) \ S 2 (H p ). The strict singularity of T g acting on H p? Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 8 / 22
Main result Main result Theorem 1 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 below. (i.e. A non-compact T g : H p H p xes a copy of l p.) In particular, T g is not strictly singular. T g is rigid in the sense that T g S(H p ) T g K(H p ) T g S p (H p ). Not true for an arbitrary operator. Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 9 / 22
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 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 below on M = V (l p ) l p. Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 10 / 22
Main result How to dene U and V? We utilize suitably chosen standard normalized test functions f a H p dened by [ ] 1 a 2 1/p f a (z) =, z D, (1 āz) 2 for each a D. For p = 2, the functions f a are the normalized reproducing kernels of H 2. Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 11 / 22
Main result Dene 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, UEF Volterra-type integral operator 14-18 August 2017 12 / 22
Tools Tools V is bounded, if a n 1 fast enough. How to show that U is bounded below? A result by Aleman and Cima (2001): Theorem 2 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, UEF Volterra-type integral operator 14-18 August 2017 13 / 22
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, UEF Volterra-type integral operator 14-18 August 2017 14 / 22
Tools Lemma 3 A localization result for T g : Let g BMOA, 1 p <, and (a k ) D be a sequence such that a k ω T. Dene 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, UEF Volterra-type integral operator 14-18 August 2017 15 / 22
Tools Using a result by Aleman and Cima (Theorem 2) and conditions (i) and (ii) in Lemma 3, it is possible to choose a subsequence (b n ) of (a n ) (where lim n T g f an p > 0) s.t. Functions T g f bn resemble disjointly supported peaks in L p (T) near some boundary point ω T, i.e. (T g f bn ) is equivalent to the natural basis of l p. 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, UEF Volterra-type integral operator 14-18 August 2017 16 / 22
Tools 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, UEF Volterra-type integral operator 14-18 August 2017 17 / 22
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 g 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 g B 0 is compact. Finally, the biadjoint (T g B 0 ) can be identied with T g : B B and consequently T g acting on B is compact. Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 18 / 22
Strict singularity of T g on Bergman spaces and Bloch space Spaces BMOA and VMOA A result of Leibov ensures that there exists isomorphic copies of space c 0 of null-sequences inside VMOA : If (h n ) VMOA is a sequence of VMOA-functions s.t. h n 1 and h n 2 0, then there exists a subsequence (h nk ) which is equivalent to the standard basis of c 0. If T g : BMOA BMOA is non-compact, then it turns out that (T g h nk ) satises conditions in Leibov's result and we can apply it again. Consequently, T g xes a copy of c 0 inside VMOA. Thus strict singularity and compactness coincide for T g acting on BMOA (or VMOA). Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 19 / 22
Further results and questions Further results and questions T g : H p H p, 1 p <, is always l 2 singular (joint work with Nieminen, Saksman, and Tylli) An example of an analytic function space X H(D) and a symbol g H(D) s.t. T g S(X ) \ K(X )? What about the case T g : H p H q, where 1 p < q <? Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 20 / 22
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, 2007. 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. 51-68. A. Aleman and A.G. Siskakis, An integral operator on H p, Complex Variables Theory Appl. 28 (1995), no. 2, 149-158. A. Aleman and J.A. Cima, An integral operator on H p and Hardy's inequality, J. Analyse Math. 85 (2001), 157-176. S. Miihkinen, Strict singularity of a Volterra-type integral operator on H p, Proceedings of the AMS, 145 (2017), no. 1, 165-175. Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 21 / 22
For further reading THANK YOU! Santeri Miihkinen, UEF Volterra-type integral operator 14-18 August 2017 22 / 22