Cartan Isometries and Toeplitz Operators on Cartan domains

TL;DR

This paper characterizes Cartan isometries and Toeplitz operators on Cartan domains, revealing their invariance properties, boundary structure, and conditions for compactness and reflexivity, advancing understanding of operator theory in complex analysis.

Contribution

It provides an intrinsic characterization of Cartan isometries and establishes invariance and boundary properties of Toeplitz operators on Cartan domains.

Findings

Cartan isometries are invariant under biholomorphic automorphisms.

The zero operator is the only compact Toeplitz operator.

A Brown-Halmos type condition for Toeplitz operators is established.

Abstract

We provide a description of the Shilov boundary of the classical Cartan domain in terms of Jordan triple determinant. As a consequence, we obtained an intrinsic characterization of Cartan isometries. Further, we obtain (i) invariance of Cartan isometries under the action of the biholomorphic automorphism group, and (ii) a Brown-Halmos type condition for Toeplitz operators on the Cartan domain. Also, we show that the zero operator is the only compact Toeplitz operator. Finally, we study the $\boldsymbol T$-Toeplitz operators and reflexivity of a Cartan isometry $\boldsymbol T.$