Intertwining property and generator domain

Ichiro Shigekawa (Kyoto University)

1. Intertwining property In this talk, we discuss the intertwining property of semigroups. Suppose we are given

two semigroups {T_{t}} and {T̂_{t}} on Banach spaces B and B̂, respectively. In addition, we are given a closed operator D: B → B̂. The intertwining property between {T_{t}} and {T̂_{t}} is characterized as follows:

DT_{t} = T̂_{t}D.

This property can be rewite in terms of the generators. Let A and Â be the generators of {T_{t}} and {T̂_{t}}, respectively. Then the above identity is equivalent to

DA = ÂD.  (1)

We relax the identity (1) as follows:

DA = ÂD + R.  (2)

We call this the defective intertwining property. Here, R is a bounded operator on Dom(D). The equation (2) is rather unviguous. We have to give the presice domain where (2). To answer this question, we have the following theorem. We denote the resolvent of A and Â by G_{λ} and Ĝ_{λ}. G_{λ} is defined for λ ∈ ρ(A) where ρ(A) is the set of resolvent of A. We regard Dom(D) as a Banach space with the graph norm ∥x∥_{D} := ∥Dx∥B̂ +∥x∥_{B}.

Theorem 1 Suppose R: Dom(D) → B̂ is a bounded operator. Then the following three conditions are equivalent to each other.

(1) There exists a subspace D ⊆ B satisfying the following conditions i) D ⊆ Dom(A) ∩ Dom(D) ii) AD ⊆ Dom(D), DD ⊆ Dom(Â), iii) there exists Λ ∈ R such that if λ ≥ Λ, then λ ∈ ρ(A) and (λ − A)D is dense in

Dom(D),

iv) it holds that

DAx = ÂDx + Rx, for any x ∈ D.  (3)

(2) There exists Λ ∈ R such that if λ ≥ Λ, then λ ∈ ρ(A) ∩ ρ(Â), G_{λ}Dom(D) ⊆ Dom(D) and

DG_{λ}x = Ĝ_{λ}Dx + Ĝ_{λ}RG_{λ}, for any x ∈ Dom(D).  (4)

(3) For any t ≥ 0, T_{t} is not only a semitoup on B but also a semigroup on Dom(D) and the following holds:

DT_{t}x = T̂_{t}Dx + ∫_{0}^{t} T̂_{t−s}RT_{s}xds, for any x ∈ Dom(D).  (5)

2. Generator domanin for symmetric bilinear form Using the above intertwining prperty, we give an application to an issue of generator

domain.

We suppose, in addition, that semigrouups are defined on a Hilbert spaces. We denote H and Ĥ in place of B and B̂. We further assume that A and Â are symmetric operators and bounded from below. Associated bilinear forms are denoted by E and Ê, respectively. Finally we assume that

E(u, v) = (Du, Dv)Ĥ.  (6)

Theorem 2 Assume the intertwining property (2). Then, u ∈ Dom(A) if and only if u ∈ Dom(E) = Dom(D) and Du ∈ Dom(Ê). Morover, under the condition, the following identity holds:

(Au, Au) = Ê(Du, Du) − (Ru, u).  (7)

We give concrete two examples. Example 1.

Let us take H = L^{2}(R^{d}; μ). Here μ is a measure defied by μ(dx) = e^{−U(x)}dx. We assume that the Hessian ∇^{2}U is bounded from below uniformly, i.e., there exists m ∈ R such that

∇^{2}U(x)[X, X] ≥ m|X|^{2}, ∀x ∈ R^{d}, ∀X ∈ R^{d}.

We consider the following Dirichlet form E:

E(u, v) = ∫_{R^{d}}(∇u, ∇v)μ(dx).  (8)

Then the domanin A (the genrator associated with E) is the set of all u such that u, ∇u, ∇^{2}u ∈ L^{2}(μ) and ∇u ∈ Dom(∇^{2}U), i.e.

∫_{R^{d}} ∇^{2}U(x)[∇u(x), ∇u(x)]μ(dx) < ∞.

Example 2.

Let us take H = L^{2}(R^{d}; dx) and consider a symmetric operator A = Δ − V. Here Δ is the laplacian and V is a potential function. We assume that V ≥ 1 and there exists c > 0 such that |∇V |^{2} ≤ cV. Then u ∈ Dom(A) if and only if u, ∇u, ∇^{2}u, V u ∈ L^{2}.

References

[1] I. Shigekawa, The domain of a generator and the intertwining property, preprint

[2] N. Yoshida, Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto Univ., 32 (1992), 621–654.

e-mail: ichiro@kusm.kyoto-u.ac.jp URL: http://www.kusm.kyoto-u.ac.jp/˜ ichiro/