31 research outputs found
Compactness properties of Volterra-type integral operators on analytic function spaces
No full textThe topic of this dissertation lies at the intersection of analytic function theory and operator theory. In the thesis, compactness and structural properties of a class of Volterra-type (integral) operators acting on analytic function spaces are investigated. The Volterra-type operator is obtained by integrating a product of two analytic functions, where one of these functions, the so-called symbol of the operator, is fixed and the other one is considered to be a variable. This integral operator was introduced by C. Pommerenke in 1977 in connection to exponential integrability of BMOA-functions. A systematic research of Volterra-type operators was initiated by Aleman and Siskakis in the mid-1990s when they characterized the boundedness and compactness of these operators on the Hardy spaces and weighted Bergman spaces.
In the first article of the thesis, we derive estimates for the essential and weak essential norms of a Volterra-type operator in terms of its symbol when the operator is acting on the Hardy spaces, BMOA and VMOA. The essential and weak essential norms of a linear operator are its distances from compact and weakly compact operators respectively. In particular, it follows from our estimates that the compactness and weak compactness of Volterra-type operator coincide when its domain is the non-reflexive Hardy space, BMOA, or VMOA.
In the second article, a notion of strict singularity of a linear operator is investigated in the case of the Volterra-type operator acting on the Hardy spaces. An operator between Banach spaces is strictly singular if its restriction to any closed infinite-dimensional subspace is not a linear isomorphism onto its range. We construct an isomorphic copy M of the sequence space of p-summable sequences and show that a non-compact Volterra-type operator restricted to M is a linear isomorphism onto its range. This implies that the strict singularity and compactness of this operator coincide in the Hardy space case.
In the third article, we provide estimates for the operator norms and essential norms of the Volterra-type operator acting between weighted Bergman spaces, where the weight function satisfies a doubling condition.Väitöskirja kuuluu operaattoriteorian ja kompleksianalyysin alaan. Työssä tarkastellaan Volterra-tyyppisiä lineaarisia integraalioperaattoreita ja niiden kompaktisuusominaisuuksia. Kyseisen operaattorin esitteli Pommerenke vuonna 1977 ns. BMOA-funktioiden eksponentiaalisen integroituvuuden yhteydessä. Volterra-tyyppinen operaattori esiintyy usealla matemaattisen analyysin alalla kuten kompositio-operaattoreiden puoliryhmien teorian yhteydessä. Operaattorin systemaattisen tutkimuksen aloitti Aleman ja Siskakis yhteistyökumppaneineen 1990-luvun puolivälissä karakterisoimalle sen rajoittuneisuuden ja kompaktisuuden Hardy-ja Bergman-avaruuksilla.
Väitöskirjassa johdetaan estimaatteja Volterra-tyyppisen operaattorin etäisyydelle kompaktien ja heikosti kompaktien operaattoreiden ideaaleista sen symbolin avulla ilmaistuna. Näistä arvioista seuraa esimerkiksi kyseisen operaattorin kompaktisuuden ja heikon kompaktisuuden yhtäpitävyys tiettyjen funktio-avaruuksien tapauksissa. Lisäksi työssä osoitetaan, että Hardy-avaruuksilla määritellyn Volterra-tyyppisen operaattorin rajoittuma eräälle p-summautuvien jonojen avaruuden kanssa isomorfiselle aliavaruudelle on alhaalta rajoitettu. Tästä seuraa erityisesti, että tämän operaattorin aito singulaarisuus ja kompaktisuus ovat yhtäpitäviä Hardy-avaruuden tapauksessa.ei saavutettav
Strict singularity of a Volterra-type integral operator on H^p
No full textWe consider a Volterra-type integral operator acting on the Hardy spaces H^p of the unit disc. The operator T_g was introduced by Ch. Pommerenke and it has been studied systematically by several people including A. Aleman, A.G. Siskakis and R. Zhao among others. From a functional analytic point of view, one interesting notion is the strict singularity of a linear operator between Banach spaces. An operator is strictly singular if its restriction to any infinite-dimensional subspace is not an isomorphism onto its range. We discuss our recent result, which states that a non-compact T_g fixes an isomorphic copy of the sequence space l^p. In particular, the strict singularity of T_g coincides with its compactness on spaces H^p
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On the exponential integrability of conjugate functions
No full textWe relate the exponential integrability of the conjugate function ˜f to the size of the
gap in the essential range of f . Our main result complements a related theorem of
Zygmund
Structural rigidity of generalised Volterra operators on H-P
No full textWe show that the non-compact generalised analytic Volterra operators T-g, where g is an element of BMOA, have the following structural rigidity property on the Hardy spaces H-P for 1 H-p is l(2)-singular for p not equal 2. (C) 2018 Elsevier Masson SAS. All rights reserved.Peer reviewe
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Toeplitz operators with piecewise continuous symbols on the Hardy space H 1
No full textThe geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces with . In the Hardy space , the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra . It is natural to ask whether the theory for piecewise continuous symbols can also be extended to . We answer this question in negative and show in particular that the Toeplitz operator is never bounded on if its symbol has a jump discontinuity
Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces
No full textWe compute the exact value of the essential norm of ageneralized Hilbert matrix operator acting on weightedBergman spaces Apv and weighted Banach spaces H∞v ofanalytic functions, where v is a general radial weight. Inparticular, we obtain the exact value of the essential normof the classical Hilbert matrix operator on standard weightedBergman spaces Apα for p > 2 + α, α ≥ 0, and on Korenblumspaces H∞α for 0 < α < 1. We also cover the Hardy spaceHp, 1 < p < ∞, case. In the weighted Bergman space case, theessential norm of the Hilbert matrix is equal to the conjecturedvalue of its operator norm and similarly in the Hardy spacecase the essential norm and the operator norm coincide. Wealso compute the exact value of the norm of the Hilbert matrixon H∞wα with weights wα(z) = (1 − |z|)α for all 0 < α < 1. Also in this case, the values of the norm and essential normcoincide
On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces
No full textIn this article, the open problem of finding the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces is adressed. The norm was conjectured to be by Karapetrovic. We obtain a complete solution to the conjecture for \alpha > 0 and 2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3} \le p < 2(2+\alpha) and a partial solution for 2+2\alpha < p < 2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3}. Moreover, we also show that the conjecture is valid for small values of when 2+2\alpha < p \le 3+2\alpha. Finally, the case is considered
