Papers by Nikolaos Chalmoukis
We construct a family of Schur multipliers for lower triangular matrices on ℓ^p, 1<p<∞ rela... more We construct a family of Schur multipliers for lower triangular matrices on ℓ^p, 1<p<∞ related to θ-summability kernels, a class of kernels including the classical Fejer, Riesz and Bochner kernels. From this simple fact we derive diverse applications. Firstly we find a new class of Schur multipliers for Hankel operators on ℓ^2, generalizing a result of E. Ricard. Secondly we prove that any space of analytic functions in the unit disc which can be identified with a weighted ℓ^p space, has the property that the space of its multipliers is contained in the space of symbols g that induce a bounded generalized Cesáro operator T_g.
We study the Riesz (a, p)-capacity of the so called Dobiński set. We characterize the values of t... more We study the Riesz (a, p)-capacity of the so called Dobiński set. We characterize the values of the parameters a and p for which the (a, p)-Riesz capacity of the Dobiński set is positive. In particular we show that the Dobiński set has positive logarithmic capacity, thus answering a question of Dayan, Fernandéz and González. We approach the problem by considering the dyadic analogues of the Riesz (a, p)-capacities which seem to be better adapted to the problem.
Onto Interpolation for the Dirichlet Space and for $W^{1,2}(\mathbb{D})$
arXiv: Complex Variables, 2018
We give a characterization of onto interpolating sequences with finite associated measure for the... more We give a characterization of onto interpolating sequences with finite associated measure for the Dirichlet space in terms of capacity of some condensers. The same condition in fact characterizes all onto interpolating sequences for $ W^{1,2}(\mathbb{D}) $ even if the associated measure is infinite.
We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most... more We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most of the results are already known but some results are new. Among the new results that we prove there is the characterization of the compactness of the Hardy operator, a reverse Hölder inequality for trace measures and a simple proof of the characterization of trace measures based on a monotonicity argument. Furthermore we give a probabilistic proof of an inequality due to Wolff. We also provide a list of open problems and suggest some possible lines of future research. 2010 Mathematics Subject Classification. 05C05, 05C63, 31E05, 42B25, 42B35, 31A15, 30C85.
Abstract. In this work we study what we call Siegel–dissipative vector of commuting operators pA1... more Abstract. In this work we study what we call Siegel–dissipative vector of commuting operators pA1, . . . , Ad`1q on a Hilbert space H and we obtain a von Neumann type inequality which involves the Drury–Arveson space DA on the Siegel upper half-space U. The operator Ad`1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup teuτă0. We then study the operator ed`1A where A “ A1 1 ̈ ̈ ̈Ad d for α P Nd0 and prove that can be studied by means of model operators on a weighted L space. To prove our results we obtain a Paley–Wiener type theorem for DA and we investigate some multiplier operators on DA as well.
It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA)... more It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA), is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (φt) that can replace the semigroup of rotations in Sarason’s Theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (φt). In addition we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch space and surprisingly enough we find that the semigroups for which the Bloch version of Sarason’s Theorem holds are exactly the same as in the BMOA case.
Interpolation problems in Dirichlet type spaces
This thesis deals with interpolation problems in spaces of analytic functions of Dirichlet type, ... more This thesis deals with interpolation problems in spaces of analytic functions of Dirichlet type, that is spaces in which the corresponding norm is defined in terms of some integral involving (partial) derivatives of the function. These problems can be categorized roughly into two categories. The first one is problems of interpolation by sequences where one tries to find a function in a given space such that on prescribed points assumes prescribed values. The second is boundary interpolation where one prescribes the values of the function on a set that sits in the boundary of the natural domain of definition. In the first part we explore simply interpolating sequences for the Dirichlet space. We manage to give a partial answer to a problem first raised by Bishop. We prove that for sequences which satisfy the so called Shapiro–Shields condition there exists a potential theoretic characterization of when they are simply interpolating. In the second part we study random interpolating se...
In the context of Dirichlet type spaces on the unit ball of $\mathbb{C}^d$, also known as Hardy-S... more In the context of Dirichlet type spaces on the unit ball of $\mathbb{C}^d$, also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthenings of boundary interpolation theorems of Peller and Khrushchev and of Cohn and Verbitsky.
A family of Schur multipliers for lower triangular matrices with applications
We construct a family of Schur multipliers for lower triangular matrices on $\ell^p, 1<p<\i... more We construct a family of Schur multipliers for lower triangular matrices on $\ell^p, 1<p<\infty$ related to $\theta$-summability kernels, a class of kernels including the classical Fejer, Riesz and Bochner kernels. From this simple fact we derive diverse applications. Firstly we find a new class of Schur multipliers for Hankel operators on $\ell^2 $, generalizing a result of E. Ricard. Secondly we prove that any space of analytic functions in the unit disc which can be identified with a weighted $\ell^p$ space, has the property that the space of its multipliers is contained in the space of symbols $g$ that induce a bounded generalized Cesaro operator $T_g$.
We discuss random interpolation in weighted Dirichlet spaces $\mathcal{D}_\alpha$, $0\leq \alpha\... more We discuss random interpolation in weighted Dirichlet spaces $\mathcal{D}_\alpha$, $0\leq \alpha\leq 1$. While conditions for deterministic interpolation in these spaces depend on capacities which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $\alpha=1/2$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for $\mathcal{D}_\alpha$ are exactly the almost sure separated sequences when $0\le \alpha<1/2$ (which includes the Hardy space $H^2=\mathcal{D}_0$), and they are exactly the almost sure zero sequences for $\mathcal{D}_\alpha$ when $1/2 \leq \alpha\le 1$ (which includes the classical Dirichlet space $\mathcal{D}=\mathcal{D}_1$).
Onto interpolation for the Dirichlet space and for H1(D)
Advances in Mathematics
Proceedings of the American Mathematical Society
Inom komplex analys och operatorteori studerar man vanligtvis begränsade linjära operatorer mella... more Inom komplex analys och operatorteori studerar man vanligtvis begränsade linjära operatorer mellan Banachrum bestående av analytiska funktioner. Detta görs för att kunna erhålla information om själva Banachrummets struktur. Ett klassiskt exempelär Cesaros medelvärdes operator påH p , ett Hardyrum bestående av analytiska funktioner. En generalisering av denna operatorär det såkallade Cesaros generaliserad operator, T g , som kan spåras tillbaka till arbetet av Ch. Pommerenke, 1970. Operatorns egenskaper har varit ett aktivt forskningsområde i de senaste 20åren. I detta arbete, som ar inspirerad av studien av T g , försöker vi ge svar till några frågor angående operatorns variation. Acknowledgments I would like to express my deep gratitude to my supervisor Alexandru Aleman for his invaluable help and guidance during the preparation of this master's thesis and for his willingness to give his time whenever I needed. Also, I would like to thank "Alexander S. Onassis Public Benefit Foundation" for supporting me financially during my studies at Lund University. Finally, I wish to thank my family and friends for their presence and encouragement throughout my study.
Concrete Operators
We consider the Dirichlet problem on in_nite and locally _nite rooted trees, andwe prove that the... more We consider the Dirichlet problem on in_nite and locally _nite rooted trees, andwe prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev W1,p of the tree.
Results in Mathematics
We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Ta... more We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $$\ell ^p$$ ℓ p spaces $$1
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Papers by Nikolaos Chalmoukis