Extended eigenvalues for Cesàro operators

Lacruz, Miguel; León-Saavedra, Fernando; Petrović, Srdjan; Zabeti, Omid

doi:10.1016/j.jmaa.2015.04.028

Cited by 14 publications

(8 citation statements)

References 18 publications

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“…The fact, σ(T 1 ) = C(1; 1), as well as the resolvent formula (6.78) for T 1 are wellknown, we refer, for instance, to [9], and [8] (see also [1], [19], [29], [30], and the references cited therein). What appears to be less well-known is the a.c. nature of the spectrum of T 1 and the spectral representation in terms of the Mellin transform.…”

Section: )mentioning

confidence: 99%

On Birman's sequence of Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

Michael

et al. 2018

Journal of Differential Equations

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Dedicated with great pleasure to Eduard Tsekanovskii on the occasion of his 80th birthday.Abstract. In 1961, Birman proved a sequence of inequalities {In}, for n ∈ N, valid for functions in C n 0 ((0, ∞)) ⊂ L 2 ((0, ∞)). In particular, I 1 is the classical (integral) Hardy inequality and I 2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite b > 0, these inequalities hold on the standard Sobolev space H n 0 ((0, b)). Furthermore, in all cases, the Birman constants [(2n − 1)!!] 2 /2 2n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L 2 ((0, ∞)) (resp., L 2 ((0, b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.

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Section: )mentioning

confidence: 99%

On Birman's sequence of Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

Michael

et al. 2018

Journal of Differential Equations

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“…There has been renewed interest in the classical Cesàro operator and its generalizations as of late [1,17,28,32,38,44] so perhaps it is a good time to put together an extended survey of what is currently known about this operator. Most of us in analysis know the name Cesàro from his summability method for infinite series and the important role this plays in summing the Fourier series of an integrable function.…”

Section: Introductionmentioning

confidence: 99%

The Cesaro operator

Ross¹

2022

Preprint

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This paper surveys the various aspacts of the Cesàro operator with a special emphasis on the Hilbert space setting of ℓ 2 . We include a discussion of summability methods of Fourier series, the Cesàro matrix and integral operator, and the Cesàro operator on the Hilbert space ℓ 2 . In this setting of ℓ 2 we cover the spectral properties of the Cesàro operator as well as a treatment of its hyponormality (via posinormal operators) and subnormality (via a multiplication operator of Kriete and Trutt). Using the results of Kriete and Trutt, we describe the commutant of the Cesàro operator as well as its bounded square roots. The Cesàro operator has a rich lattice of invariant subspaces whose description remains unknown and we suspect might never be fully understood. We survey some results which display the complexity of these invariant subspaces and provide the reader with several paths forward for further discussion. Though the Cesàro operator was originally explored in the ℓ 2 and Hardy space settings, it has been explored in various other settings such as the general Hardy spaces as well as the Bergman spaces. Finally, we explore some of the so-called generalized Cesàro operator which are classes of integral operators on the Hardy spaces that connect to many areas of classical function theory.

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“…C, the Cesàro operator Cf is de…ned by In classical analysis, the Cesàro operator was investigated from various aspects and a large number of results have appeared recently [1][2][3][4][5]. Titchmarsh [6] also used the operator as a convergence method for divergent integrals and introduced the Cesàro summability of integrals [7, p.11].…”

Section: Introductionmentioning

confidence: 99%