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On the essential decreasing of the summation order in the Abel-Lidskii sense

Maksim V. Kukushkin

TL;DR

This work analyzes the Abel-Lidskii summation for root-vector series of sectorial operators in the trace class, showing that the summation order $s$ can be made arbitrarily small under mild growth assumptions on algebraic multiplicities. The authors develop a qualitative theory based on characteristic determinants, canonical products, and Abel-Lidskii series to decompose the Hilbert space into invariant subspaces and control convergence, deriving an infinitesimalness result that links multiplicity growth to the summation order. Key contributions include invariant-subspace splitting, counting-function splitting with explicit asymptotics, sharper canonical-product bounds, and a rigorous framework for applying Abel-Lidskii summation to fractional evolution problems and spectral asymptotics. The results open a pathway to extend the approach to general Schatten classes and to solve abstract Cauchy problems for fractional and nonlocal evolution equations arising in applications.

Abstract

In this paper, we consider a problem of decreasing the summation order in the Abel-Lidskii sense. The problem has a significant prehistory since 1962 created by such mathematicians as Lidskii V.B., Katsnelson V.E., Matsaev V.I., Agranovich M.S. As a main result, we will show that the summation order can be decreased from the values more than a convergence exponent, in accordance with the Lidskii V.B. result, to an arbitrary small positive number. Additionally, we construct a qualitative theory of summation in the Abel-Lidkii sense and produce a number of fundamental propositions representing the interest themselves.

On the essential decreasing of the summation order in the Abel-Lidskii sense

TL;DR

This work analyzes the Abel-Lidskii summation for root-vector series of sectorial operators in the trace class, showing that the summation order can be made arbitrarily small under mild growth assumptions on algebraic multiplicities. The authors develop a qualitative theory based on characteristic determinants, canonical products, and Abel-Lidskii series to decompose the Hilbert space into invariant subspaces and control convergence, deriving an infinitesimalness result that links multiplicity growth to the summation order. Key contributions include invariant-subspace splitting, counting-function splitting with explicit asymptotics, sharper canonical-product bounds, and a rigorous framework for applying Abel-Lidskii summation to fractional evolution problems and spectral asymptotics. The results open a pathway to extend the approach to general Schatten classes and to solve abstract Cauchy problems for fractional and nonlocal evolution equations arising in applications.

Abstract

In this paper, we consider a problem of decreasing the summation order in the Abel-Lidskii sense. The problem has a significant prehistory since 1962 created by such mathematicians as Lidskii V.B., Katsnelson V.E., Matsaev V.I., Agranovich M.S. As a main result, we will show that the summation order can be decreased from the values more than a convergence exponent, in accordance with the Lidskii V.B. result, to an arbitrary small positive number. Additionally, we construct a qualitative theory of summation in the Abel-Lidkii sense and produce a number of fundamental propositions representing the interest themselves.
Paper Structure (13 sections, 11 theorems, 233 equations)

This paper contains 13 sections, 11 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Historical review
  3. Preliminaries
  4. Overview of the supplementary results
  5. Characteristic determinant
  6. Abel-Lidskii Series expansion
  7. Main results
  8. Splitting to the infinite set of the invariant subspaces
  9. Splitting of the counting function
  10. Sharper estimate for the canonical product
  11. Infinitesimalness of the summation order
  12. Supplementary remarks and mathematical applications
  13. Conclusions

Key Result

Lemma 1

If the series 2 converges, then the corresponding infinite product 3 converges uniformly on every compact subset and satisfies the estimate where $k_{p}=3e(p+1)(2+\ln p),\,p>0,\,k_{0}=1.$

Theorems & Definitions (21)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • Definition 1
  • Lemma 6
  • proof
  • ...and 11 more