Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations
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Mark M. Malamud
Abstract
This survey is concerned with the spectral theory of Volterra operators An = ⊕nj bjJαj , αj > 0, which are direct sums of multiples of fractional order Riemann- Liouville operators Jαj . We discuss the lattices of invariant and hyperinvariant subspaces of operators An, as well as their commutants, double commutants, and other operator algebras related to An. We describe the sets of extended eigenvalues and the corresponding eigenvectors of the operators Jα. The Gohberg-Krein conjecture on equivalence of unicellularity and cyclicity properties of a dissipative Volterra operator is also discussed. The problem of the similarity of the Volterra integral operators to the operators Jα is discussed too.
Abstract
This survey is concerned with the spectral theory of Volterra operators An = ⊕nj bjJαj , αj > 0, which are direct sums of multiples of fractional order Riemann- Liouville operators Jαj . We discuss the lattices of invariant and hyperinvariant subspaces of operators An, as well as their commutants, double commutants, and other operator algebras related to An. We describe the sets of extended eigenvalues and the corresponding eigenvectors of the operators Jα. The Gohberg-Krein conjecture on equivalence of unicellularity and cyclicity properties of a dissipative Volterra operator is also discussed. The problem of the similarity of the Volterra integral operators to the operators Jα is discussed too.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Recent history of the fractional calculus: data and statistics 1
- Basic FC operators and their properties 23
- Mathematical and physical interpretations of fractional derivatives and integrals 47
- Generalized fractional calculus operators with special functions 87
- General fractional calculus 111
- Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus 127
- Fractional Laplace operator and its properties 159
- Applications of the Mellin integral transform technique in fractional calculus 195
- Fractional Fourier transform 225
- The Wright function and its applications 241
- Mittag-Leffler function: properties and applications 269
- Asymptotics of the special functions of fractional calculus 297
- Analysis of fractional integro-differential equations of thermistor type 327
- A survey on fractional variational calculus 347
- Variational principles with fractional derivatives 361
- Continuous time random walks and space-time fractional differential equations 385
- Inverse subordinators and time fractional equations 407
- Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations 427
- Fractional differentiation in p-adic analysis 461
- Index 473
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Recent history of the fractional calculus: data and statistics 1
- Basic FC operators and their properties 23
- Mathematical and physical interpretations of fractional derivatives and integrals 47
- Generalized fractional calculus operators with special functions 87
- General fractional calculus 111
- Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus 127
- Fractional Laplace operator and its properties 159
- Applications of the Mellin integral transform technique in fractional calculus 195
- Fractional Fourier transform 225
- The Wright function and its applications 241
- Mittag-Leffler function: properties and applications 269
- Asymptotics of the special functions of fractional calculus 297
- Analysis of fractional integro-differential equations of thermistor type 327
- A survey on fractional variational calculus 347
- Variational principles with fractional derivatives 361
- Continuous time random walks and space-time fractional differential equations 385
- Inverse subordinators and time fractional equations 407
- Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations 427
- Fractional differentiation in p-adic analysis 461
- Index 473
