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Local form subordination without a power decay and a criterion of Riesz basesness

Boris Mityagin, Petr Siegl

Abstract

We revisit the local form subordination condition on the perturbation of a self-adjoint operators with compact resolvent, which is used to show the Riesz basis property of the eigensystem of the perturbed operator. Our new assumptions and new proof allow for establishing the Riesz basis property also in the case of slow and non-monotone decay in this subordination condition.

Local form subordination without a power decay and a criterion of Riesz basesness

Abstract

We revisit the local form subordination condition on the perturbation of a self-adjoint operators with compact resolvent, which is used to show the Riesz basis property of the eigensystem of the perturbed operator. Our new assumptions and new proof allow for establishing the Riesz basis property also in the case of slow and non-monotone decay in this subordination condition.

Paper Structure

This paper contains 16 sections, 18 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Perturbed operator $T$ and completeness of its eigensystem
  5. Main result (Riesz basis property)
  6. Proofs
  7. Perturbed operator $T$ and completeness of its eigensystem
  8. Localization of the eigenvalues of the perturbed operator
  9. Proof of Theorem \ref{['thm:RB.slow']}
  10. Simpler proof if $\{\omega_j\}_{j \in \mathbb{N}}$ decays faster
  11. Sufficient conditions for \ref{['asm:omega.G']} if $\mu_n=n$, $n \in \mathbb{N}$, and examples in this case
  12. Sufficiently fast decay of $\{\omega_j\}_{j\in \mathbb{N}}$
  13. An arbitrarily slow decay and sufficiently large gaps in the support of $\{\omega_j\}_{j\in \mathbb{N}}$
  14. Application in finite-band matrices
  15. Sufficiently fast decay of $\{\omega_j\}_{j\in \mathbb{N}}$ for \ref{['asm:omega.fast']}
  16. ...and 1 more sections

Key Result

Lemma 2.1

If $\Lambda_j$, $j=1,2$, are bounded, closed, isolated and disjoint subsets of $\sigma(T)$, then the corresponding Riesz projections $P_{\Lambda_j}$$, j=1,2$, are disjoint.

Theorems & Definitions (41)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Remark 3.5
  • proof : Proof of Proposition \ref{['prop:T.basic.def']}
  • proof : Proof of Proposition \ref{['prop:T.complete']}
  • ...and 31 more