## Defective Intertwining Property And Generator Domain

Citations: | 4 - 4 self |

### BibTeX

@MISC{Shigekawa_defectiveintertwining,

author = {Ichiro Shigekawa},

title = {Defective Intertwining Property And Generator Domain},

year = {}

}

### OpenURL

### Abstract

. We discuss the defective intertwining property of generators of semigroups. We give an equivalent conditions in terms of generator, resolvent and semigroup. As an application, using this property, we give a example that we can determine the exact generator domain of a Schrodinger operator. 1. Introduction The intertw#rt]L property plays an important role to deal w#al semigroups. The intertw #A0(D property takes the follow#ll form: DA= AD w#1]( A and A are generators of semigroups and D is a closed operator. In the privious paper [8],w e discuss the intertw#rt]4 property and apply it to the issue of the domain of a generator. But there are many issuesw#su h are notw#t](F a scope of intertw#rt]) property. In this paper,w e extend it to the follow#ll defective intertw#rt]( property: DA = AD + R. Here, an additional term R appears. We discuss the equivalent conditions in terms of resolvents and semigroups. We formulate the issue in theframew ork of Banach space. In the case o...

### Citations

835 |
Semigroups of Linear Operators and Applications to Partial Differential Equations
- Pazy
- 1983
(Show Context)
Citation Context ...≥ ˜ω, ‖R(λ; Aμ) n ‖ L(Dom(D)) ≤ On the other hand, it holds that (see e.g., [9, §1.7 (7.7)]) R(λ; Aμ) =(λ+ μ) −1 ( μλ (μ − A)R μ + λ = μ2 ( μλ R ; A (μ + λ) 2 μ + λ Set κ = μλ μκ μ+λ . Then λ = μ−κ . =-=(9)-=- implies {( μ ∥ μ + λ ˜M . (9) (λ − ˜ω) n ) ) ; A − 1 μ + λ . ) 2 R(κ; A) − 1 } n∥ ∥∥∥L(Dom(D)) ≤ μ + λ We fix κ and let μ →∞. Then λ → κ and we have ‖R(κ; A) n ‖ L(Dom(D)) ≤ ˜M . (κ − ˜ω) n ˜M . (λ −... |

201 |
M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms
- Ma, Röckner
- 1992
(Show Context)
Citation Context ... Then Aμ is a bounded operator not only on B but also on Dom(D). We also set Âμ = μ Â ˆ Gμ. Then (4) yields DAμx − ÂμDx = D(μ 2 Gμ − μx) − (μ 2 ˆ GμDx − μDx) = μ 2 (DGμ − ˆ GμDx) = μ 2 GμRGμx ˆ = Rμx =-=(7)-=- Here Rμ = μ 2 ˆ GμRGμ. It is easy to see that Rμ ∈L(Dom(D), ˆ B) and the operator norm of Rμ is uniformly bounded for large μ. Now we claim the following identity. De tAµ x − e tÂµ ∫ t Dx = e 0 (t−s)... |

114 |
One-Parameter Semigroups
- Davies
- 1980
(Show Context)
Citation Context ...fying the following conditions: i) D⊆Dom(A) ∩ Dom(D). ii) AD ⊆Dom(D), DD ⊆Dom( Â). iii) For sufficiently large λ, (λ − A)D is dense in Dom(D). iv) The following equality holds. DAx = ÂDx + Rx, ∀x ∈D. =-=(3)-=- (b) For sufficiently large λ, Gλ Dom(D) ⊆ Dom(D) and DGλx = ˆ GλDx + ˆ GλRGλx, ∀x ∈ Dom(D). (4) (c) For any t ≥ 0, {Tt} is a (C0)-semigroup not only on B but also on Dom(D) and the following holds: D... |

89 |
Equations of Evolution
- Tanabe
- 1979
(Show Context)
Citation Context ...E and Ê, e.g., E(x, y) =−(Ax, y)H ∀x ∈ Dom(A), ∀y ∈ Dom(E). We fix δ>ωand set Eδ(x, y) =E(x, y)+δ(x, y)H. Then F = Dom(E) is a Hilbert space with the inner product (x, y)F = 1 { } Eδ(x, y)+Eδ(y, x) . =-=(14)-=- 2 Here denotes the complex conjugation. By the weak sector condition, E is a bounded sesqui-linear form on F×F, i.e., there exists a constant C>0 such that Similarly we define |E(x, y)| ≤C(x, x) 1/2 ... |

28 |
L p estimates for Schrödinger operators with certain potentials
- Shen
- 1995
(Show Context)
Citation Context ...ppose that f,∇ 2 f,V f ∈ L 2 . Then, clearly (Δ − V )f ∈ L 2 which implies f ∈ Dom(Δ − V ). This completes the proof. Remark 4. 1. This result is known when V is a polynomial (see, Guibourg [4], Shen =-=[10]-=-.) 1 (28) is equivalent to Dom(Δ − V ) = Dom(Δ) ∩ Dom(V ). Under this condition, Ichinose-Tamura [5] proved the norm convergence of TrotterKato product formula. Our case include, e.g., the case V (x) ... |

9 | Estimates in L p for magnetic Schrödinger operators - Shen - 1996 |

9 |
The norm convergence of the Trotter-Kato product formula with error bound
- Ichinose, Tamura
(Show Context)
Citation Context ...letes the proof. Remark 4. 1. This result is known when V is a polynomial (see, Guibourg [4], Shen [10].) 1 (28) is equivalent to Dom(Δ − V ) = Dom(Δ) ∩ Dom(V ). Under this condition, Ichinose-Tamura =-=[5]-=- proved the norm convergence of TrotterKato product formula. Our case include, e.g., the case V (x) =e x as a special case. We can discuss similar problem on a Riemannian manifold. In this case, the s... |

8 |
Inégalités maximales pour l’opérateur de Schrödinger
- Guibourg
- 1993
(Show Context)
Citation Context ...ersely, suppose that f,∇ 2 f,V f ∈ L 2 . Then, clearly (Δ − V )f ∈ L 2 which implies f ∈ Dom(Δ − V ). This completes the proof. Remark 4. 1. This result is known when V is a polynomial (see, Guibourg =-=[4]-=-, Shen [10].) 1 (28) is equivalent to Dom(Δ − V ) = Dom(Δ) ∩ Dom(V ). Under this condition, Ichinose-Tamura [5] proved the norm convergence of TrotterKato product formula. Our case include, e.g., the ... |

6 |
Note on the paper “The norm convergence of the Trotter–Kato product formula with error bound” by Ichinose and
- Ichinose, Tamura, et al.
(Show Context)
Citation Context ...ˆ Gλ to both sides of the preceding equality, we have ˆGλD(A − λ)x = −Dx + ˆ GλRx. We recall that (A − λ)x = −y and hence which yields ˆGλDy = DGλy − ˆ GλRGλy DGλy = ˆ GλDy + ˆ GλRGλy, ∀y ∈ (λ − A)D. =-=(6)-=- We have to show that the identity above holds for all y ∈ Dom(D). We recall that there exist M>0 and ω ≥ 0 such that and hence, for λ>ω, ‖Tt‖L(B) ≤ Me ωt , ‖ ˆ Tt‖L( B) ˆ ≤ Me ωt ‖λGλ‖ L(B) ≤ M/(λ − ... |

5 |
A remark on coercive forms and associated semigroups. “Partial differential operators and mathematical physics,” (Holzhau
- Albeverio, Ru-Zong, et al.
- 1994
(Show Context)
Citation Context ... B into ˆ B with the domain Dom(D). We always denote by Dom the domain of an operator or, later, the domain of quadratic form. The following property is called the intertwining property: DTt = ˆ TtD. =-=(1)-=- We denote the generator of {Tt} and { ˆ Tt} by A and Â, respectively. Then the intertwining property above is (at least formally) equivalent to DA = ÂD. For the moment, we use this notation formally.... |

5 |
Estimates in Lp for magnetic Schrödinger operators
- Shen
- 1996
(Show Context)
Citation Context ... B ∗ + ˆ B 〈 ˆ GλRGλx, θ〉 ˆ B ∗ = ˆ B 〈Dx, ˆ G ∗ λθ〉 ˆ B ∗ + B〈x, V ∗ ˆ G ∗ λθ〉B ∗. Setting θ =(λ − Â∗ )ξ for ξ ∈ Dom( Â∗ ), we get ˆB 〈Dx, ξ〉 Bˆ ∗ = 〈x, S ∗ (λ − Â∗ )ξ〉B∗ − B〈x, V ∗ ξ〉B∗ ∀x ∈ Dom(D) =-=(11)-=- which implies ξ ∈ Dom(D ∗ ) and D ∗ ξ = S ∗ (λ − Â∗ )ξ − V ∗ ξ. Further, by putting x =(λ − A)y, y ∈ Dom(A) in (11), B〈(λ − A)y, D ∗ ξ〉B ∗ = ˆ B 〈S(λ − A)y, (λ − Â∗ )ξ〉 ˆ B ∗ − ˆ B 〈V (λ − A)y, ξ〉 ˆ ... |

3 |
The domain of a generator and the intertwining property
- Shigekawa
(Show Context)
Citation Context ... an important role in dealing with semigroups. The intertwining property takes the following form: DA = ÂD where A and Â are generators of semigroups and D is a closed operator. In the previous paper =-=[12]-=-, we discussed the intertwining property and applied it to the issue of the domain of a generator. The intertwining property was used e.g., in Bakry’s paper [2] to discuss the Riesz transformation. Bu... |

2 |
Sobolev spaces on a Riemannian manifold and their equivalence
- Yoshida
- 1992
(Show Context)
Citation Context ..., 2000 12 ICHIRO SHIGEKAWA results (see e.g., [9, Chapter 3]). And so we can say that this work is a generalization of perturbation theory to some extent. Such a relation appeared in Yoshida’s paper =-=[15]-=- in connection with the Littlewood-Paley theory. Yoshida noticed the importance of this relation but he treated only bounded R. One of our motivations is to remove this restriction. We discuss equival... |

2 |
Etude des transfomations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Prob
- Bakry
- 1987
(Show Context)
Citation Context ...property above is (at least formally) equivalent to DA = ÂD. For the moment, we use this notation formally. This property is sometimes too restrictive and so we will relax it as follows. DA = ÂD + R. =-=(2)-=-DEFECTIVE INTERTWINING PROPERTY 3 Here R is an appropriate operator. If A and Â satisfy this identity, we say that the defective intertwining property holds. We have to precisely give the subspace wh... |

1 | The domain of a generator and the intertwining property, preprint - Shigekawa |

1 |
On equivalence of Lp-norms related to Schrödinger type operator on Riemannian manifolds, Probab. Theory Related Fields
- Miyokawa, Shigekawa
(Show Context)
Citation Context ... we discuss the Schrödinger operator of the form Δ − V on Rd where V is a scalar potential. We give a characterization of the domain of this operator. Further applications are discussed in the papers =-=[13, 8]-=- where the Littlewood-Paley theory is developed for the Schrödinger operators on a Riemannian manifold. In this case, the defective term R is unbounded and our extension in this paper is crucial. The ... |

1 | Lp multiplier theorem for the Hodge-Kodaira operator, “Séminaire de Probabilités XXXVIII
- Shigekawa
- 2005
(Show Context)
Citation Context ... we discuss the Schrödinger operator of the form Δ − V on Rd where V is a scalar potential. We give a characterization of the domain of this operator. Further applications are discussed in the papers =-=[13, 8]-=- where the Littlewood-Paley theory is developed for the Schrödinger operators on a Riemannian manifold. In this case, the defective term R is unbounded and our extension in this paper is crucial. The ... |