SPECTRUM OF THE INFINITE-DIMENSIONAL LAPLACIAN

GIORGIO METAFUNE, ABDELAZIZ RHANDI, AND ROLAND SCHNAUBELT

Abstract. We prove that the spectrum of the infinite-dimensional Laplacian is the left half plane {λ ∈ ℂ : Re λ ≤ 0}. As a consequence, we obtain a simple proof of the norm discontinuity of the generated semigroup.

Let H be a separable, infinite dimensional, real Hilbert space and let (e_{k}) be an orthonormal basis. We denote by BUC(H) the space of all bounded, uniformly continuous functions from H to ℂ. We shall regard BUC(ℝ^{n}) as a subspace of BUC(H) via the isometric embedding

J_{n} : BUC(ℝ^{n}) → BUC(H), (J_{n}φ)(x) := φ(x_{1}, …, x_{n}),

for φ ∈ BUC(ℝ^{n}), x ∈ H, and x_{k} := (x, e_{k}). By D_{k} we denote the partial derivative in the direction e_{k}. Let λ_{k} > 0 with ∑_{k=1}^{∞} λ_{k} < ∞ be given. The infinite dimensional heat equation

\frac{∂u}{∂t} (t, x) = ∑_{k=1}^{∞} λ_{k} D_{k}^{2} u(t, x),  t > 0, x ∈ H,
u(0, x) = φ(x),  x ∈ H,

on BUC(H) is solved by the strongly continuous contraction semigroup

P_{t}φ = lim_{n→∞} P_{t}^{n}φ, φ ∈ BUC(H).

The above limit exists in BUC(H) uniformly in t on bounded subsets of [0, ∞) and, for φ ∈ BUC(H), x ∈ H and t > 0, we set

P_{t}^{n}φ(x) := (4πt)^{− \frac{n}{2}} (λ_{1} ⋯ λ_{n})^{− \frac{1}{2}} ∫_{ℝ^{n}} e^{− ∑_{k=1}^{n} \frac{y^{2}_{k}}{4tλ_{k}}} φ(x − ∑_{k=1}^{n} y_{k}e_{k})dy. (1)

We denote by A the generator of (P_{t}). Let BUC^{1,1}(H) be the space of Fréchet differentiable φ ∈ BUC(H) with a bounded Lipschitz continuous derivative. It is known that

D_{0}(H) := {φ ∈ BUC^{1,1}(H) : D_{k}D_{l}φ ∈ BUC(H), sup_{k,l∈ℕ}∥D_{k}D_{l}φ∥_{∞} < ∞}

is a P_{t}–invariant core of A and

Aφ = ∑_{k=1}^{∞} λ_{k} D_{k}^{2} φ for φ ∈ D_{0}(H).

The reader is referred to [2] and [1] for the proofs of the above results.

1991 Mathematics Subject Classification. 35R15.

This work was done during a visit of the second and third author at the Università di Lecce. A. Rhandi was supported by CNR (Italy) and CNCPRST (Morocco) and R Schnaubelt by the Vigoni project of DAAD.

Theorem 1. The spectrum of A is the left half plane {λ ∈ ℂ : Re λ ≤ 0} and the spectrum of P_{t} is the unit disc {λ ∈ ℂ : |λ| ≤ 1}. Moreover, every λ ∈ σ(A) is an approximate eigenvalue.

Proof. Note that the restriction of P_{t} to BUC(ℝ^{n}) coincides with the semigroup generated by A_{n} := ∑_{k=1}^{n} λ_{k} D_{k}^{2}. In particular, A_{n} is the part of A in BUC(ℝ^{n}) and, hence, R(λ, A_{n}) = R(λ, A)_{|BUC(ℝ^{n})} for λ ∈ ρ(A) ∩ ρ(A_{n}). Therefore, for these values of λ, the sequence ∥R(λ, A_{n})∥_{∞} is bounded. Let V : BUC(ℝ^{n}) → BUC(ℝ^{n}) be the isometry defined by

(V φ)(x) := φ(\sqrt{λ_{1}}x_{1}, …, \sqrt{λ_{n}}x_{n}), φ ∈ BUC(ℝ^{n}), x ∈ ℝ^{n}.

A simple change of variables in (1) shows that e^{tA_{n}} = V^{−1}e^{tΔ_{n}}V for t ≥ 0, n ∈ ℕ, where Δ_{n} denotes the Laplacian on BUC(ℝ^{n}). This implies that R(λ, A_{n}) = V^{−1}R(λ, Δ_{n})V for λ ∈ Σ_{π} := {0 ≠ λ ∈ ℂ : | arg λ| < π}, and thus

∥R(λ, Δ_{n})∥_{∞} = ∥R(λ, A_{n})∥_{∞} ≤ C for λ ∈ Σ_{π} ∩ ρ(A), n ∈ ℕ.

Fix λ ∈ Σ_{π} with Re λ < 0. For n ∈ ℕ, the function g_{λ,n}(x) := e^{\frac{λ}{2n} |x|^{2}} belongs to BUC(ℝ^{n}) and ∥g_{λ,n}∥_{∞} = 1. Setting

f_{λ,n}(x) := (λ − Δ_{n})g_{λ,n}(x) = −\frac{λ^{2}}{n^{2}} |x|^{2}e^{\frac{λ}{2n} |x|^{2}},

we compute

∥f_{λ,n}∥_{∞} = \frac{2|λ|^{2}}{ne|Re λ|}.

So we derive

∥R(λ, A_{n})∥_{∞} = ∥R(λ, Δ_{n})∥_{∞} ≥ \frac{∥R(λ, Δ_{n})f_{λ,n}∥_{∞}}{∥f_{λ,n}∥_{∞}} = \frac{ne|Re λ|}{2|λ|^{2}}.

As a result, λ must belong to the spectrum of A. Since A generates a contraction semigroup, the first and second assertion readily follow from standard spectral theory (cf. [4, Chap. IV]).

To prove the last assertion, we observe that iℝ is contained in the approximate point spectrum of A. Let λ = −a^{2} +ib with a > 0 and b ∈ ℝ. The first part of the proof applies to the operator Ã on BUC(H̃) corresponding to the sequence (λ_{2}, λ_{3}, ⋯), where H̃ is the closed subpace of H spanned by the vectors e_{k}, k ≥ 2. Thus there exist g̃_{n} ∈ D_{0}(H̃) such that ∥g̃_{n}∥_{∞} = 1 and ∥Ãg̃_{n} − ibg̃_{n}∥_{∞} → 0 as n → ∞. We now define

f_{n}(x) := exp(iaλ_{1}^{− \frac{1}{2}} x_{1}) g̃_{n}(x_{2}, x_{3}, ⋯), x ∈ H.

Clearly, f_{n} ∈ D_{0}(H), ∥f_{n}∥_{∞} = 1, and

Af_{n}(x) = ∑_{k=1}^{∞} λ_{k} D_{k}^{2} f_{n}(x) = −a^{2}f_{n}(x) + exp(iaλ_{1}^{− \frac{1}{2}} x_{1}) (Ãg̃_{n})(x_{2}, x_{3}, ⋯), x ∈ H.

Hence λ is an approximate eigenvalue of A.

As a consequence of [4, Thm.II.4.18] we immediately obtain the following result from [3], see also [1], [5], [7].

Corollary 2. The semigroup (P_{t}) is not eventually norm continuous on BUC(H).

In fact we have proved slightly more. A semigroup T (·) with growth bound 0 is called asymptotically norm continuous if

lim_{t→∞} lim sup_{h→0} ∥T (t + h) − T (t)∥ = 0.

In view of [6, Thm.1.9] our theorem yields

Corollary 3. The semigroup (P_{t}) is not asymptotically norm continuous on BUC(H).

References

[1] W. Arendt, A. Driouich and O. El-Mennaoui, On the infinite product of C_{0}-semigroups. J. Funct. Anal. 160, 524–542 (1998).

[2] P. Cannarsa and G. Da Prato, Infinite-dimensional elliptic equations with Hölder-continuous coefficients. Adv. Differ. Equ. 1, 425-452 (1996).

[3] W. Desch and A. Rhandi, On the norm continuity of transition semigroups in Hilbert spaces. Archiv Math. 70, 52-56 (1998).

[4] K.J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations. To appear in Graduate Texts in Mathematics, Springer-Verlag.

[5] P. Guiotto, Non-differentiability of heat semigroups in infinite-dimensional Hilbert spaces. Semigroup Forum 55, 232–236 (1997).

[6] J. Martinez and J.M. Mazon, C_{0}–semigroups norm continuous at infinity. Semigroup Forum 52, 213–224 (1996).

[7] J.M.A.M. van Neerven and J. Zabczyk, Norm discontinuity of Ornstein-Uhlenbeck semigroups. Semigroup Forum, to appear.

Giorgio Metafune, Dipartimento di Matematica, Università di Lecce, C.P. 193, 73100 Lecce, Italy.

  E-mail address: metafune@le.infn.it

Abdelaziz Rhandi, Department of Mathematics, University of Marrakesh, B.P.:S. 15, 40000-Marrakesh, Morocco.

  E-mail address: rhandi@ucam.ac.ma

Roland Schnaubelt, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

  E-mail address: rosc@michelangelo.mathematik.uni-tuebingen.de