20. Universal block tridiagonalization in ℬ(ℋ) and beyond
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Sasmita Patnaik
Abstract
For ℋ, a separable infinite dimensional complex Hilbert space, we prove that every ℬ(ℋ) operator has a basis with respect to which its matrix representation has a universal block tridiagonal form with block sizes given by a simple exponential formula independent of the operator. From this, such a matrix representation can be further sparsified to slightly sparser forms; it can lead to a direct sum of even sparser forms reflecting in part some of its reducing subspace structure; and in the case of operators without invariant subspaces (if any exists), it gives a plethora of sparser block tridiagonal representations. An extension to unbounded operators occurs for a certain domain of definition condition. Moreover, this process gives rise to many different choices of block sizes.
Abstract
For ℋ, a separable infinite dimensional complex Hilbert space, we prove that every ℬ(ℋ) operator has a basis with respect to which its matrix representation has a universal block tridiagonal form with block sizes given by a simple exponential formula independent of the operator. From this, such a matrix representation can be further sparsified to slightly sparser forms; it can lead to a direct sum of even sparser forms reflecting in part some of its reducing subspace structure; and in the case of operators without invariant subspaces (if any exists), it gives a plethora of sparser block tridiagonal representations. An extension to unbounded operators occurs for a certain domain of definition condition. Moreover, this process gives rise to many different choices of block sizes.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
