[Paper Review] Invariance theorems for Nevanlinna families

Vladimir Derkach, Seppo Hassi+1 more|arXiv (Cornell University)|Mar 18, 2015

Spectral Theory in Mathematical Physics25 references3 citations

TL;DR

This paper establishes invariance theorems for the point and continuous spectra of operator-valued Herglotz-Nevanlinna functions in Hilbert spaces, proving that spectral properties such as the presence of zero in the imaginary part or real eigenvalues remain invariant across the upper half-plane. The results rely on Harnack’s inequality and maximum principles, extending classical spectral invariance from scalar to operator-valued settings.

ABSTRACT

A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real value $a$ in a single point $z_0\in {\mathbb C}_+$ should be identically equal to $a$. In the present note we prove similar invariance results both for the point and the continuous spectra of an operator-valued Herglotz-Nevanlinna function with values in the set of bounded or unbounded linear operators (or relations) in a Hilbert space. The proof of this invariance result for continuous spectrum is based on Harnack's inequality. This inequality is systematically used to characterize operator-valued Herglotz-Nevanlinna functions with form-domain invariance property for their imaginary parts or Herglotz-Nevanlinna functions with values in the Schatten-von Neumann classes.

Motivation & Objective

• To establish invariance theorems for the point and continuous spectra of operator-valued Herglotz-Nevanlinna functions in Hilbert spaces.
• To characterize spectral invariance of the imaginary part and form-domain properties using Harnack’s inequality.
• To extend classical maximum principle results from scalar to operator-valued Herglotz-Nevanlinna functions.
• To provide spectral characterizations of subclasses $ R^s[h] $ and $ R^u[h] $ via single-point conditions at $ z = i $.
• To link spectral invariance to geometric and boundary triplet structures in symmetric operator theory.

Proposed method

• The proof of spectral invariance for the continuous spectrum relies on Harnack’s inequality applied to the imaginary part of the operator-valued function.
• The authors use integral representations of Herglotz-Nevanlinna functions with operator-valued measures to analyze spectral behavior.
• Key techniques include the maximum principle and nonnegative kernel theory to analyze the kernel and range of the imaginary part.
• The study employs sesquilinear forms and their closures to define the imaginary part of the operator-valued function in a form-domain invariant way.
• The framework incorporates Schatten-von Neumann classes to characterize functions with trace-class imaginary parts.
• Boundary triplet and Weyl function realizations are used to interpret spectral invariance geometrically.

Experimental results

Research questions

• RQ1 Does the presence of zero in the point spectrum of the imaginary part of a Herglotz-Nevanlinna function remain invariant across the upper half-plane?
• RQ2 Can the continuous spectrum of the imaginary part of an operator-valued Herglotz-Nevanlinna function be invariant under analytic continuation in the upper half-plane?
• RQ3 To what extent do spectral invariance properties of $ F(z) $ depend on the form-domain of its imaginary part?
• RQ4 How can the subclasses $ R^s[h] $ and $ R^u[h] $ be characterized by single-point conditions at $ z = i $?
• RQ5 What is the role of Harnack’s inequality in proving invariance of the continuous spectrum for operator-valued Nevanlinna families?

Key findings

• The point spectrum of the imaginary part of $ F(z) $ is invariant across $ bC_+ $: $ 0 otin ho( ext{Im} hinspace F(z_0)) $ if and only if $ 0 otin ho( ext{Im} hinspace F(z)) $ for all $ z eq bR $.
• The continuous spectrum of the imaginary part is invariant: $ 0 otin ho( ext{Im} hinspace F(z_0)) $ implies $ 0 otin ho( ext{Im} hinspace F(z)) $ for all $ z eq bR $.
• The real spectrum of $ F(z) $ is invariant: $ a otin ho(F(z_0)) $ if and only if $ a otin ho(F(z)) $ for all $ z eq bR $.
• The class $ R^s[h] $ is characterized by $ ext{ker} hinspace ext{Im} hinspace F(i) = \{0\} $, and $ R^u[h] $ by $ 0 otin ho( ext{Im} hinspace F(i)) $.
• For $ G_(z) $, the continuous spectrum of the imaginary part contains $ [0, ) $, and $ 0 otin ho(G_(z)) $ for all $ z eq bR $.
• In the case of compact resolvent, $ s_j((G_(z))^{-1}) = O(j^{-2}) $, indicating trace-class behavior.