Isometric tuples are hyperreflexive

TL;DR

This paper proves that all isometric tuples of operators on a Hilbert space are hyperreflexive, providing explicit bounds on the hyperreflexivity constants for different tuple sizes.

Contribution

It establishes the hyperreflexivity of isometric operator tuples and determines explicit bounds on their hyperreflexivity constants.

Findings

All isometric n-tuples are hyperreflexive.

Hyperreflexivity constant is at most 95 for n=1.

Hyperreflexivity constant is at most 6 for n≥2.

Abstract

An $n$-tuple of operators $(V_1,...,V_n)$ acting on a Hilbert space $H$ is said to be isometric if the row operator $(V_1,...,V_n) : H^n \to H$ is an isometry. We prove that every isometric $n$-tuple is hyperreflexive, in the sense of Arveson. For $n = 1$, the hyperreflexivity constant is at most 95. For $n \geq 2$, the hyperreflexivity constant is at most 6.