18. Spectra of generalized Poisson integral operators on Lp(ℝ+)
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Abstract
For α, β, μ > 0, the following integral operators, that generalize the Poisson transform, 
are studied in detail on Lebesgue spaces Lp(ℝ+) for 1 ≤ p < ∞. If 0 < β−1/p < αμ, then these operators Pα,β,μ are bounded (and we compute their operator norms which depend on p); and commute on their range. We calculate and represent explicitly their spectra σ(Pα,β,μ). The main technique is to subordinate these operators in terms of 0 groups of isometries Tt,pf (s) := e-t/p f (e-ts) for f ∈ Lp(ℝ+) (which is isometrically isomorphic to the C0 group of translations on Lp(ℝ)) and transfer properties from some special functions. As consequences, we show that these integral operators are noncompact, have thin spectrum and nontrivial invariant subspaces on Lp(ℝ+) for 1 ≤ p < ∞.
Abstract
For α, β, μ > 0, the following integral operators, that generalize the Poisson transform, are studied in detail on Lebesgue spaces Lp(ℝ+) for 1 ≤ p < ∞. If 0 < β−1/p < αμ, then these operators Pα,β,μ are bounded (and we compute their operator norms which depend on p); and commute on their range. We calculate and represent explicitly their spectra σ(Pα,β,μ). The main technique is to subordinate these operators in terms of 0 groups of isometries Tt,pf (s) := e-t/p f (e-ts) for f ∈ Lp(ℝ+) (which is isometrically isomorphic to the C0 group of translations on Lp(ℝ)) and transfer properties from some special functions. As consequences, we show that these integral operators are noncompact, have thin spectrum and nontrivial invariant subspaces on Lp(ℝ+) for 1 ≤ p < ∞.
Chapters in this book
- Frontmatter I
- Editors’ Introduction V
- Per Enflo’s personal thoughts about Victor Lomonosov VII
- Contents XI
- 1. Bishop–Phelps–Bollobás property for positive operators between classical Banach spaces 1
- 2. Isometric embeddings of finite metric trees into (ℝn, d1) and (ℝn, d∞) 15
- 3. Iterates of the spherical Aluthge transform of 2-variable weighted shifts 25
- 4. The freewheeling twisting of Hilbert spaces 43
- 5. A survey of ball-covering property of Banach spaces 67
- 6. A note on the quantitative local version of the log-Brunn–Minkowski inequality 85
- 7. Spectra of “fattened” open book structures 99
- 8. On some local Bishop–Phelps–Bollobás properties 109
- 9. Bounded point derivations of fractional orders 123
- 10. Invariant subspaces: some minimal proofs 131
- 11. On the Hypercyclicity Criterion for operators of Read’s type 139
- 12. Three-space problem for strictly convex renormings 149
- 13. Norm attaining operators of finite rank 157
- 14. Isometric copies of ℓn∞ and ℓn1 in transportation cost spaces on finite metric spaces 189
- 15. From Lomonosov lemma to radical approach in joint spectral radius theory 205
- 16. Pontryagin–Krein theorem: Lomonosov’s proof and related results 231
- 17. Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube 251
- 18. Spectra of generalized Poisson integral operators on Lp(ℝ+) 281
- 19. Order extreme points and solid convex hulls 297
- 20. Universal block tridiagonalization in ℬ(ℋ) and beyond 317
- 21. Rademacher-type independence in Boolean algebras 327
- Index 349
Chapters in this book
- Frontmatter I
- Editors’ Introduction V
- Per Enflo’s personal thoughts about Victor Lomonosov VII
- Contents XI
- 1. Bishop–Phelps–Bollobás property for positive operators between classical Banach spaces 1
- 2. Isometric embeddings of finite metric trees into (ℝn, d1) and (ℝn, d∞) 15
- 3. Iterates of the spherical Aluthge transform of 2-variable weighted shifts 25
- 4. The freewheeling twisting of Hilbert spaces 43
- 5. A survey of ball-covering property of Banach spaces 67
- 6. A note on the quantitative local version of the log-Brunn–Minkowski inequality 85
- 7. Spectra of “fattened” open book structures 99
- 8. On some local Bishop–Phelps–Bollobás properties 109
- 9. Bounded point derivations of fractional orders 123
- 10. Invariant subspaces: some minimal proofs 131
- 11. On the Hypercyclicity Criterion for operators of Read’s type 139
- 12. Three-space problem for strictly convex renormings 149
- 13. Norm attaining operators of finite rank 157
- 14. Isometric copies of ℓn∞ and ℓn1 in transportation cost spaces on finite metric spaces 189
- 15. From Lomonosov lemma to radical approach in joint spectral radius theory 205
- 16. Pontryagin–Krein theorem: Lomonosov’s proof and related results 231
- 17. Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube 251
- 18. Spectra of generalized Poisson integral operators on Lp(ℝ+) 281
- 19. Order extreme points and solid convex hulls 297
- 20. Universal block tridiagonalization in ℬ(ℋ) and beyond 317
- 21. Rademacher-type independence in Boolean algebras 327
- Index 349
