A fundamental principle for invariant subspaces in convex domains

Krivosheev, A. S.

doi:10.1070/im2004v068n02abeh000476

Cited by 29 publications

(62 citation statements)

References 27 publications

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“…Under this condition, the groups become relatively small. We also observe that in work [4], this result was obtained via solving a rather complicated interpolation problem in the space of entire functions of exponential type.…”

Section: Necessary Conditionsmentioning

confidence: 91%

“…1. Earlier the result of Theorem 3.3 was obtained in Theorem 5.1 in work [4]. In this theorem there was considered a case, when = { } ∞ =1 is the trivial partition; here are immediately groups of relatively small diameter.…”

Section: Necessary Conditionsmentioning

confidence: 97%

“…, . In the case of the trivial partition , the quantity Λ ( ) coincides with the quantity Λ introduced in work [4].…”

Section: =1mentioning

confidence: 99%

“…The studies of both problems were made first independently and they have a rich story. The main milestones were reflected in works [3] and [4]. In [4] the fundamental principle problem was solved for invariant subspaces admitting the spectral synthesis as well as the interpolation problem for an arbitrary convex domain ⊂ C under the only restriction ( (Λ) = 0):…”

Section: Introductionmentioning

confidence: 99%

“…The study of such problems is a rather complicated process. At that, the duality of the representation problem and interpolation problem was established in [4] only under an additional restriction for the multiplicities of the points :…”

Section: Introductionmentioning

confidence: 99%

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Basis in invariant subspace of analytical functions

Krivosheeva¹

2018

Ufimskii Matematicheskii Zhurnal

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Abstract. In this work we study the problem on representing the functions in an invariant subspace of analytic functions on a convex domain in the complex plane. We obtain a sufficient condition for the existence of a basis in the invariant subspace consisting of linear combinations of eigenfunctions and associated functions of differentiation operator in this subspace. The linear combinations are constructed by the system of exponential monomials, whose exponents are partitioned into relatively small groups. We apply the method employing the Leontiev interpolating function. At that, we provide a complete description of the space of the coefficients of the series representing the functions in the invariant subspace. We also find necessary conditions for representing functions in an arbitrary invariant subspace admitting the spectral synthesis in an arbitrary convex domain. We employ the method of constructing special series of exponential polynomials developed by the author.

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Section: Necessary Conditionsmentioning

confidence: 91%

Section: Necessary Conditionsmentioning

confidence: 97%

“…, . In the case of the trivial partition , the quantity Λ ( ) coincides with the quantity Λ introduced in work [4].…”

Section: =1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Basis in invariant subspace of analytical functions

Krivosheeva¹

2018

Ufimskii Matematicheskii Zhurnal

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A criterion for the fundamental principle to hold for invariant subspaces on bounded convex domains in the complex plane

Krivosheeva

Krivosheev

2012

Funct Anal Its Appl

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Let D be a bounded convex domain of the complex plane. We study the problem of whether the fundamental principle holds for analytic function spaces on D invariant with respect to the differentiation operator and admitting spectral synthesis. Earlier this problem was solved under a restriction on the multiplicities of the eigenvalues of the differentiation operator. In the present paper, we lift this restriction. Thus, we present a complete solution of the fundamental principle problem for arbitrary nontrivial closed invariant subspaces admitting spectral synthesis on arbitrary bounded convex domains.

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