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L-resolvents of symmetric linear relations in Pontryagin spaces

Volodymyr Derkach

TL;DR

This work extends Krein–Saakyan L-resolvent theory to symmetric linear relations in Pontryagin spaces with improper gauges by marrying boundary-triple methods with rigged space techniques. It derives an explicit ${\mathfrak L}$-resolvent matrix ${W}_{\Pi{\mathfrak L}}(\lambda)$ and shows its membership in the class ${\mathcal{W}}_{\kappa_1}(J_p)$, enabling a one-to-one correspondence between ${\mathfrak L}$-regular ${\mathfrak L}$-resolvents and generalized Nevanlinna parameter families via linear-fractional transforms. The framework unifies boundary-data parametrizations with Krein–Saakyan representations for symmetric relations in indefinite spaces and applies it to canonical systems, offering new parametrizations of generalized resolvents in the indefinite setting. This provides a robust toolkit for analyzing minimal linear relations generated by canonical systems and other indefinite problems, with potential implications for spectral theory in Pontryagin spaces. Overall, the paper broadens the scope of L-resolvent techniques to indefinite inner product spaces and delivers concrete formulas to parametrize and compute generalized resolvents in this broader context.

Abstract

Let A be a closed symmetric operator with the deficiency index (p,p), $p<\infty$, acting in a Hilbert space H and let L be a subspace of H. The set of L-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge L was described by Krein and Saakyan. The Krein--Saakyan theory of L-resolvent matrix was extended by Shmul'yan and Tsekanovskii to the case of improper gauge L 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 Krein--Saakyan theory of L-resolvent matrices for symmetric linear relations with improper gauges in Pontryagin spaces. We extend the known formula for the L-resolvent matrix in terms of boundary operators to this class of relations. The results are applied to the minimal linear relation generated by a canonical system.

L-resolvents of symmetric linear relations in Pontryagin spaces

TL;DR

This work extends Krein–Saakyan L-resolvent theory to symmetric linear relations in Pontryagin spaces with improper gauges by marrying boundary-triple methods with rigged space techniques. It derives an explicit -resolvent matrix and shows its membership in the class , enabling a one-to-one correspondence between -regular -resolvents and generalized Nevanlinna parameter families via linear-fractional transforms. The framework unifies boundary-data parametrizations with Krein–Saakyan representations for symmetric relations in indefinite spaces and applies it to canonical systems, offering new parametrizations of generalized resolvents in the indefinite setting. This provides a robust toolkit for analyzing minimal linear relations generated by canonical systems and other indefinite problems, with potential implications for spectral theory in Pontryagin spaces. Overall, the paper broadens the scope of L-resolvent techniques to indefinite inner product spaces and delivers concrete formulas to parametrize and compute generalized resolvents in this broader context.

Abstract

Let A be a closed symmetric operator with the deficiency index (p,p), , acting in a Hilbert space H and let L be a subspace of H. The set of L-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge L was described by Krein and Saakyan. The Krein--Saakyan theory of L-resolvent matrix was extended by Shmul'yan and Tsekanovskii to the case of improper gauge L 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 Krein--Saakyan theory of L-resolvent matrices for symmetric linear relations with improper gauges in Pontryagin spaces. We extend the known formula for the L-resolvent matrix in terms of boundary operators to this class of relations. The results are applied to the minimal linear relation generated by a canonical system.

Paper Structure

This paper contains 16 sections, 30 theorems, 200 equations.

Table of Contents

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

Key Result

Lemma 2.4

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

Theorems & Definitions (64)

  • 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
  • Theorem 2.9
  • ...and 54 more