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Theory of sectorial operators and its application in Fractional calculus

Maksim V. Kukushkin

Abstract

In this monograph, we formulated the sufficient conditions of the Abel-Lidskii basis property for a sectorial operator. Having studied such an operator class, we strengthened the conditions regarding the semi-angle of the sector and weakened a great deal conditions regarding the involved parameters. Thus, we clarified the results by Lidskii V.B. devoted to the decomposition on the root vector system of the non-selfadjoint operator. We used a technique of the entire function theory and introduced the so-called Schatten-von Neumann class of the convergence exponent. Having considered strictly accretive operators satisfying special conditions formulated in terms of the norm and used a sequence of contours of the power type, we invented a peculiar method how to calculate a contour integral involved in the problem. Finally, we consider evolution equations in the abstract Hilbert space.

Theory of sectorial operators and its application in Fractional calculus

Abstract

In this monograph, we formulated the sufficient conditions of the Abel-Lidskii basis property for a sectorial operator. Having studied such an operator class, we strengthened the conditions regarding the semi-angle of the sector and weakened a great deal conditions regarding the involved parameters. Thus, we clarified the results by Lidskii V.B. devoted to the decomposition on the root vector system of the non-selfadjoint operator. We used a technique of the entire function theory and introduced the so-called Schatten-von Neumann class of the convergence exponent. Having considered strictly accretive operators satisfying special conditions formulated in terms of the norm and used a sequence of contours of the power type, we invented a peculiar method how to calculate a contour integral involved in the problem. Finally, we consider evolution equations in the abstract Hilbert space.
Paper Structure (80 sections, 79 theorems, 1466 equations)

This paper contains 80 sections, 79 theorems, 1466 equations.

Table of Contents

  1. Preliminaries
  2. Properties of fractional differential operators
  3. Multidimensional integro-differential operators
  4. Mapping and representation theorems
  5. Strictly accretive property
  6. Sectorial property
  7. Compactness of the resolvent
  8. Existence theorems via the Lax-Milgram method
  9. Remarks
  10. Spectral properties of the sectorial operators
  11. Historical review
  12. Special operator class
  13. Asymptotic equivalence
  14. Singular numbers and completeness of the root vectors
  15. Comparison analysis with the subordination concept
  16. ...and 65 more sections

Key Result

Theorem 1

The directional fractional integral operators are bounded in $L_{p}(\Omega),$$1\leq p<\infty,$ the following estimates holds

Theorems & Definitions (163)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 1
  • proof
  • ...and 153 more