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${\sL}$-resolvents and pseudo-spectral functions of symmetric linear relations in Hilbert spaces

Volodymyr Derkach

TL;DR

The paper develops a unified framework connecting boundary-triple/Weyl-function methods with Kreĭn–Saakyan ${\mathfrak L}$-resolvent theory for symmetric linear relations having improper gauges. It extends the ${\mathfrak L}$-resolvent matrix to improper gauges, derives explicit right/left ${\mathfrak L}$-resolvent matrices via boundary-data, and characterizes spectral and pseudo-spectral functions through LT-type representations and admissible parameters ${\tau}$. These results are then specialized to canonical systems of differential equations, providing explicit descriptions of spectral data for regular and limit-point cases and yielding generalized Fourier transforms tied to boundary-triple data. The framework thus offers a comprehensive toolkit for spectral analysis of symmetric relations with improper gauges and their canonical-system realizations, with potential applications in indefinite/infinite-dimensional settings. Overall, the work bridges classical extension theory with modern operator-function approaches to deliver concrete resolvent parametrizations and spectral characterizations.

Abstract

Let $A$ be a closed symmetric operator with the deficiency index $(p,p)$, $p<\infty$, acting in a Hilbert space $\sH$ and let $\sL$ be a subspace of $\sH$. The set of $\sL$-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge $\sL(\subset\sH)$ was described by Kreĭn and Saakyan. The Kreĭn--Saakyan theory of $\sL$-resolvent matrices was extended by Shmul'yan and Tsekanovskii to the case of improper gauge $\sL(\not\subset\sH)$ and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Kreĭn--Saakyan theory of $\sL$-resolvent matrices for symmetric linear relations with improper gauges in Hilbert spaces and extend the known formula for the $\sL$-resolvent matrix in terms of boundary operators to this class of relations. Descriptions of spectral and pseudo-spectral functions of symmetric linear relations with improper gauges are given. The results are applied to linear relations generated by a canonical system.

${\sL}$-resolvents and pseudo-spectral functions of symmetric linear relations in Hilbert spaces

TL;DR

The paper develops a unified framework connecting boundary-triple/Weyl-function methods with Kreĭn–Saakyan -resolvent theory for symmetric linear relations having improper gauges. It extends the -resolvent matrix to improper gauges, derives explicit right/left -resolvent matrices via boundary-data, and characterizes spectral and pseudo-spectral functions through LT-type representations and admissible parameters . These results are then specialized to canonical systems of differential equations, providing explicit descriptions of spectral data for regular and limit-point cases and yielding generalized Fourier transforms tied to boundary-triple data. The framework thus offers a comprehensive toolkit for spectral analysis of symmetric relations with improper gauges and their canonical-system realizations, with potential applications in indefinite/infinite-dimensional settings. Overall, the work bridges classical extension theory with modern operator-function approaches to deliver concrete resolvent parametrizations and spectral characterizations.

Abstract

Let be a closed symmetric operator with the deficiency index , , acting in a Hilbert space and let be a subspace of . The set of -resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge was described by Kreĭn and Saakyan. The Kreĭn--Saakyan theory of -resolvent matrices was extended by Shmul'yan and Tsekanovskii to the case of improper gauge and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Kreĭn--Saakyan theory of -resolvent matrices for symmetric linear relations with improper gauges in Hilbert spaces and extend the known formula for the -resolvent matrix in terms of boundary operators to this class of relations. Descriptions of spectral and pseudo-spectral functions of symmetric linear relations with improper gauges are given. The results are applied to linear relations generated by a canonical system.

Paper Structure

This paper contains 21 sections, 41 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear relations in Hilbert spaces
  4. ${\mathcal{R}}$-families
  5. Boundary triples and Weyl functions
  6. Generalized resolvents of a symmetric linear relation $A$
  7. Rigged Hilbert spaces and generalized resolvents
  8. Rigged Hilbert spaces.
  9. Extended generalized resolvents
  10. Extended spectral functions
  11. Extended boundary triples
  12. ${{\mathfrak L}}$-resolvents
  13. Gauge ${{\mathfrak L}}$ and ovf's ${\mathcal{P}}({z})$, ${\mathcal{Q}}({z})$.
  14. The ${{\mathfrak L}}$-preresolvent matrix
  15. Right ${{\mathfrak L}}$-resolvent matrix
  16. ...and 6 more sections

Key Result

Lemma 2.4

There is a one to one correspondence between ${\mathcal{R}}^{p\times p}$-families $\tau({z})=\mathop{\mathrm{ran}}\nolimits$ and ${\mathcal{R}}^{p\times p}$-pairs $$ established by the formulas The ${\mathcal{R}}^{p\times p}$-family $\tau$ admits the following representation

Theorems & Definitions (92)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: RB69
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 82 more