On a Question of A. E. Nussbaum on Measurability of Families of Closed Linear Operators in a Hilbert Space

TL;DR

This paper investigates the measurability of families of closed operators in a Hilbert space, clarifying the distinction between different notions of measurability and providing explicit criteria.

Contribution

It answers a longstanding question by showing the non-equivalence of weak measurability and projection matrix measurability, with explicit criteria for the latter.

Findings

Weak measurability and projection matrix measurability are not equivalent.

Explicit criteria for the measurability of projection matrices are provided.

Distinction between direct integral and pointwise operator application is demonstrated.

Abstract

The purpose of this note is to answer a question A. E. Nussbaum formulated in 1964 about the possible equivalence between weak measurability of a family of densely defined, closed operators T(t), t real, in a separable complex Hilbert space H on one hand, and the notion of measurability of the 2 \times 2 operator-valued matrix of projections onto the graph Gamma(T(t)) of T(t) on the other, in the negative. Our results demonstrate an interesting distinction between the direct integral over the family of operators T(t) with respect to Lebesgue measure and the naturally maximally defined operator associated with pointwise application of T(t) in the vector-valued Hilbert space L^2(\mathbb R; dt; H). We also provide explicit criteria for the measurability of the matrix of projections onto the graph Gamma(T(t)) of T(t).