]> 2.6 Semigroup generation

2.6 Semigroup generation

Observe, using Definition 2.3.3 that
𝒜v = v,D(𝒜) = {v H2(0, 1) : v(0) = 0,v(1) = Kv(0)} .

Solving the eigenproblem for 𝒜 we determine

vn(θ) = 2 s¯2 1 sinh μn¯θ,0 θ 1,n (2.41)

Lemma 2.6.1. If K1, then the operator 𝒜 defined by (2.4) is the infinitesimal generator of a linear analytic semigroup {S(t)}t0 on H. It has the representation

S(t)x = n=eλntx,v nHxn = n=eλnt Res λ=λn(λI 𝒜)1xt 0,x H ,

where {xn}n is the system of eigenfunctions given by (2.34) and {vn}n is its biorthogonal system (2.41). Moreover, the condition

cosh π < K < 1 (2.42)

is necessary and sufficient for EXS of the semigroup {S(t)}t0.

Proof. From Proposition 2.5.1 and Corollary 2.3.1 we know that the operator 𝒜 is similar to a normal discrete operator 𝒩 with a spectrum located on the negative real semiaxis or on the parabola having the branches directed to the left. Thus by Theorem 2.3.4 the operator 𝒜 generates an analytic semigroup. Using the boundedness of a semigroup and the basis property we get

S(t)x =Tet𝒩T1x = T lim l n=lletλn T1x,e nHen = n=etλn T1x,e nHTen = = n=etλn x, (T)1e nHTen = n=etλn x,vnHxn ,

where {et𝒩} t0 denotes an analytic semigroup generated by 𝒩, {en}n is a system of its eigenvectors being an orthonormal basis of H, T stands for the similarity transformation, {vn}n is the (unique) systems biorthogonal with respect to {xn}n, being the system of eigenvectors of 𝒜. The condition (2.42) follows easily from (2.33). □

Lemma 2.6.1 permits us to strengthen the results concerning the regularity of the functional 𝒞 defined in (2.7).

Lemma 2.6.2. For K1 and α > 1 4, the functional 𝒞 defined by (2.7) is linear and bounded on the Banach space Hα, 0 α 1, where Hα = D(𝒜α), is the domain of a fractional power of 𝒜, equipped with a topology induced by the norm xHα = 𝒜αxH.

Proof. It follows from Lemma 2.6.1 that, for K1, the fractional powers 𝒜α (0 α 1) of 𝒜 and the corresponding spaces Hα are well defined [48, pp. 24 - 30]. Since for α > 1 4 we have Hα C(0, 1) [48, Theorem 1.6.1, p. 39], then there exists Mα > 0 such that 𝒞x = x(1) xC(0,1) Mα xHα for all x Hα. □

Remark 2.6.1. There is another way to obtain the above result. By (2.8), (2.9), we have

𝒞x = 𝒜1αd,𝒜αx Hx Hα ,

where α (0, 1) is such that d D[(𝒜)1α]. Now

𝒜v = n=λ¯ nv,xnHvnv D(𝒜) ,

and d D[(𝒜)β] iff n=d,x nH 2 λ n 2β < . Hence, taking (2.8), (2.37), and (2.41) into account, we get β < 3 4.