Invariance theorems for Nevanlinna families

2015/03/18 by Derkach, Vladimir, Hassi, Seppo, Malamud, Mark
#30C80 #47A07 #47A10 #47A56 #47B10 #47B44 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 30E20 #Secondary 30C40

paper · doi:10.48550/arxiv.1503.05606

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 z₀∈ \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.

Related

• Nevanlinna classes associated to a closed set on ∂D
• Inverse of generalized Nevanlinna function that is holomorphic at infinity
• Non-invariance of the Nevanlinna class under the action of the Cesaro and Volterra operators
• A general realization theorem for matrix-valued Herglotz-Nevanlinna functions
• An analytic characterization of the symmetric extension of a Herglotz-Nevanlinna function in several variables