Some Holomorphic Semi-Groups*

In this note we state some basic spectral properties of a class of operators associated with a representation of a Lie group. We merely indicate the proofs since we intend to give details in another context.

Let T (.) be a strongly continuous representation of the Lie group 𝔊 in the Banach space 𝔛. For a in 𝔄, the Lie algebra of 𝔊, let A(a) be the infinitesimal generator of the one-parameter group, T (exp(ta)), −∞ < t < ∞, and let A^{∗}(a) be the adjoint of A(a). It is known that

𝔚_{k} = {xϵ𝔛|xϵDomain(A(a_{1}) … A(a_{k})) for all a_{1}, …, a_{k}ϵ𝔄}

is dense in X and that

𝔚_{k}^{∗} = {x^{∗}ϵ𝔛^{∗}|x^{∗}ϵ Domain (A^{∗}(a_{1}) … A^{∗}(a_{k})) for all a_{1}, …, a_{k}ϵ𝔄}

is dense in 𝔛^{∗} in the weak* topology. Let {e_{1}, …, e_{n}} be a basis for 𝔄 and set A_{i} = A(e_{i}), A_{i}^{∗} = A^{∗}(e_{i}). If α = (α_{1}, …, α_{k}), 1 ≤ α_{i} ≤ n, is a sequence of integers we denote a product such as A_{α_{1}} … A_{α_{k}} by A_{α}; |α| = k is the length of α. The operator B^{∘} = Σ_{|α|≤m}a_{α}A_{α} (the coefficients are complex numbers) is defined on 𝔚_{m} and denoting α in reversed order by α^{∗}, the operator B̃^{∘} = Σ_{|α|≤m}a_{α}A_{α^{∗}}^{∗} is defined on 𝔚_{m}^{∗}. The closure B of B^{∘} and the weak* closure B̃ of B̃^{∘} are well-defined operators. If for any real n-vector, ξ = (ξ_{1}, …, ξ_{n}), |Σ_{|α|≤m}a_{α}ξ_{α}| ≥ ρ∥ξ∥^{m} (ρ > 0 is fixed and ∥ξ∥ is the ordinary Euclidean norm) then Theorem 1. The adjoint B^{∗} of B is B̃.

To prove this we observe that if B^{∗}x_{1}^{∗} = x_{2}^{∗} then^{1} for x = 𝔛, x_{1} ∗ (T (g)x) is a weak solution of the equation

Σ_{|α|≤m}a_{α}R_{α^{∗}}{x_{1}^{∗}(T (g)x)} = x_{2}^{∗}(T (g)x);

R_{i} is the infinitesimal transformation

R_{i}f(g) = lim_{t→0} t^{−1}{f(exp(te_{i})g) − f(g)}.

We use the properties of weak solutions of elliptic equations to show that x_{1}^{∗} is in 𝔚_{m^{−}1}∗ and then apply an adaptation^{2} of the mollifier technique.

If the stronger condition

(−1)^{j−1}Re{Σ_{|α|≤m}a_{α}ξ_{α}} ≥ ρ∥ξ∥^{m},

with 2j = m, is fulfilled then * Appeared in PNAS, vol. 46 (1960).

Theorem 2. The operator B is the infinitesimal generator of a semi-group, S(t), of class H(ϕ_{1}, ϕ_{2}), for some ϕ_{1} and ϕ_{2}.^{3}

The theorem is equivalent to an inequality ∥R(λ, B)∥ ≤ M/[ρ(λ, S)] for the norm of the resolvent of B; M is a constant and ρ(λ, S) is the distance of λ from a sector

S = \{ζ|ψ_{2} ≤ arg (ζ − ζ_{0}) ≤ ψ_{1}\}, \frac{π}{2} < ψ_{2} ≤ π ≤ ψ_{1} < 3π/2.

To obtain the inequality we observe that

x^{∗}(T (g)(B − λ)x) = Σ_{|α|≤m}a_{α}L_{α}x^{∗}(T (g)x) − λx^{∗}(T (g)x),

with

L_{i}f(g) = lim_{t→0} t^{−1}{f(g exp(te_{i})) − f(g)};

partially invert this equation in a neighborhood of the identity using a parametrix; and then establish appropriate estimates^{4} for the parametrix. It is of interest to note that if a_{α} is real when |α| = m, S(t) is holomorphic in a half-plane.

The semi-groups, S(t), have a canonical representation. Let μ be a left-invariant Haar measure on 𝔊, then Theorem 3. There is a unique function h(t, g) (defined for t in an open sector containing (0, ∞)) in L_{1}(μ) for each t and such that for all representations T (·) of 𝔊

S(t)x = ∫_{𝔊} h(t, g)T (g)xμ(dg).

The integral is a Bochner integral and h(t, g) is analytic in t and g. We remark finally that, when t is in the interior of the domain of S(.) and x is in X, S(t)x is an analytic vector.

The above results generalize theorems of Nelson^{5} and Nelson and Stinespring.^{1}

1Nelson, E., and W.F. Stinespring, Amer.J.Math., 81, 547-560 (1959). 2Langlands, R.P., On Lie Semi-Groups (submitted to Canad.J.Math.). 3For terminology see Hille, E., and R.S. Phillips, Functional Analysis and Semi-Groups. 4These are essentially estimates for the resolvents of strongly elliptic differential operators

with constant coefficients and are obtained from estimates for the fundamental solution of the associated parabolic equation similar to those of Silov G.E., (Usp.Mat.Nauk., 10, 89-100 (1955)).

5Ann.Math., 70, 572-615 (1959).