Hyponormality and quasinormality of unbounded Toeplitz operators with non harmonic symbol on the Fock Sobolev space

TL;DR

The paper investigates hyponormality and quasinormality of unbounded Toeplitz operators with non-harmonic symbols on the Fock-Sobolev space $F^{2,m}(\mathbb{C})$. It derives necessary conditions for hyponormality and quasinormality for symbols of the form $\varphi(z)=a z^{p}\overline{z}^{n} + b z^{s}\overline{z}^{t}$, expressed as intricate inequalities involving factorial terms. The authors show these conditions are not sufficient via explicit counterexamples and demonstrate that quasinormality does not imply hyponormality in this unbounded setting, highlighting a divergence from the bounded-case theory. The work extends operator-theoretic analysis of Toeplitz operators to the Fock-Sobolev framework, with implications for spectral properties and potential applications in signal processing and related areas.

Abstract

In this paper, we establish the essential criteria for the hyponormality and quasinormality of the unbounded Toeplitz operator $T_{\varphi}$ with non-harmonic symbol, acting on the Fock-Sobolev space $F^{2, m}(\mathbb{C})$. The study shows that quasinormality does not inherently imply hyponormality of unbounded Toeplitz operator with non-harmonic symbols.

Theorems & Definitions (18)

• Proposition 1
• Lemma 2
• Remark 3
• theorem 4
• proof
• Example 5
• Example 6
• Example 7
• Remark 8
• Remark 9