# Schatten Index of the Sectorial Operator via the Real Component of Its Inverse

^{1}

^{2}

## Abstract

**:**

**non-self-adjoint**operators with the discrete spectrum. The main challenge is to represent a complete description of belonging to the Schatten class through the properties of the Hermitian real component. The method of estimating the singular values is elaborated by virtue of the established asymptotic formulas. The latter fundamental result is advantageous since, of many theoretical statements based upon it, one of them is a concept on the root vectors series expansion, which leads to a wide spectrum of applications in the theory of evolution equations. In this regard, the evolution equations of fractional order with the sectorial operator in the term not containing the time variable are involved. The concrete well-known operators are considered and the advantage of the represented method is convexly shown.

## 1. Introduction

## 2. Preliminaries

#### 2.1. Sectorial Sesquilinear Forms and the Hermitian Components

**H1:**

**H2:**

#### 2.2. Previously Obtained Results

- (A)
- The following Schatten classification holds:

- (B)
- In the case $\nu \left({R}_{W}\right)=\infty ,\phantom{\rule{0.166667em}{0ex}}\mu \ne 0,$ the following relation holds:

## 3. Main Results

#### The Main Refinement of the Result A

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

## 4. Mathematical Applications

#### 4.1. The Low Bound for the Schatten Index of the Perturbed Differential Operator

- 2.
- Let us show the application of Lemma 2; firstly, consider the following reasonings:

#### 4.2. Existence and Uniqueness Theorems for Evolution Equations via Obtained Results

**S1:**

**Theorem**

**2.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Kukushkin, M.V.
Schatten Index of the Sectorial Operator via the Real Component of Its Inverse. *Mathematics* **2024**, *12*, 540.
https://doi.org/10.3390/math12040540

**AMA Style**

Kukushkin MV.
Schatten Index of the Sectorial Operator via the Real Component of Its Inverse. *Mathematics*. 2024; 12(4):540.
https://doi.org/10.3390/math12040540

**Chicago/Turabian Style**

Kukushkin, Maksim V.
2024. "Schatten Index of the Sectorial Operator via the Real Component of Its Inverse" *Mathematics* 12, no. 4: 540.
https://doi.org/10.3390/math12040540