Pusz--Woronowicz's functional calculus revisited

TL;DR

This paper revisits Pusz--Woronowicz's functional calculus for positive forms, providing a Hilbert space operator perspective that unifies operator connection operations and perspectives.

Contribution

It offers an elementary, self-contained operator-theoretic explanation of their functional calculus, demonstrating its capacity to encompass all operator connection operations.

Findings

Operator connection operations are fully captured by the functional calculus.

The paper provides a Hilbert space operator perspective on the functional calculus.

It unifies various operator connection concepts under a single framework.

Abstract

This note is a complement to Pusz--Woronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, self-contained and purely Hilbert space operator explanation of their functional calculus, we show that any operator connection type operations (including any operator perspectives) are captured by their functional calculus.