The Lévy laplacian and differential operators of 2-nd order in Hilbert spaces.

*(English)*Zbl 0945.47037Let \(\mathbb{H}\) be a real separable Hilbert space, \(\mathcal L_s\) be the Banach space of self-adjoint bounded linear operators on \(\mathbb{H}\), \(\mathcal L_{\infty}\) the Banach space of compact linear operators on \(\mathbb{H}\) and \(\mathcal L_1\) the Banach space of nuclear operators on \(\mathbb{H}\). Write \(tr(A)\) for the trace of \(A\in \mathcal L_1\).

Let \(\psi \in \mathcal L_s^*(=\) the dual of \(\mathcal L_s)\). Then \(\psi \) is said to be regular, if there exists an \(A\in \mathcal L_1\cap \mathcal L_s\) such that \(\psi (B)=tr(AB)\) whenever \(B\in \mathcal L_s\). Further, \(\psi \) is said to be irregular, if \(\psi (B)=0\) for every \(B\in \mathcal L_{\infty}\cap \mathcal L_s\). The following result is proved in the paper: For every \(\psi \in \mathcal L_s^*\) there exists a uniquely determined regular \(\psi _1\in \mathcal L_s^*\) and an irregular \(\psi _2\in \mathcal L_s^*\) such that \(\psi =\psi _1+\psi _2\) (Theorem 1).

Identifying the Banach space \(\mathcal N^2_s\) of symmetric bounded bilinear functionals on \(\mathbb{H}\) with \(\mathcal L_s\), a second order differential operator associated to \(\psi \in (\mathcal N^2_s)^*\) is defined as follows: For \(x\in \mathbb{H}\), let \(\mathcal G_x\) stand for real functionals \(f\) on \(\mathbb{H}\) possessing the second order Fréchet derivative \(f''(x)\in \mathcal N_s^2\) at \(x\). For \(\psi \in (\mathcal N^2_s)^*\) and \(x\in \mathbb{H}\), put \(\mathbf D^{\psi}_x{:}f\to \psi (f''(x))\), \(f\in \mathcal G_x\). Then \(\mathbf D^{\psi}=\{\mathbf D^{\psi}_x;\;x\in \mathbb{H}\}\) is called a second order differential operator on \(\mathbb{H}\). Thus, by Theorem 1, every differential operator can be, in an obvious sense, uniquely split into a regular and an irregular part.

Given an orthonormal basis \(E=\{e_j;\;j\in \mathbb{N}\}\) and \(x\in \mathbb{H}\), define \[ L^Ef(x)=\lim _{n\to \infty}\frac 1n\sum _{j=1}^nf''(x)(e_j,e_j) \] for those \(f\in \mathcal G_x\), for which the limit exists. It is shown that the Lévy Laplacian \(L^E\) can be extended to an irregular differential operator on \(\mathbb{H}\).

In the paper, properties of irregular operators are studied. In particular, it turns out that irregular differential operators actually behave as the first order differential operators. Certain invariance properties are shown to characterize irregular operators (Theorem 4). Further, all differential operators enjoying the maximum principle are described (Theorem 5). The last part of the paper deals with the Dirichlet and the Poisson problem for irregular differential operators enjoying the maximum principle. For a class of domains and a class of boundary conditions (or right hand sides for the Poisson problem), uniqueness and existence are established (Theorem 7). Relations to domains and functionals investigated by G. E. Šilov are also studied. The paper under review generalizes and extends results recently obtained for the Lévy Laplacian by A. B. Mingarelli and S. Wang in [Differ. Eqn. Dyn. Syst. 1, 23-34 (1993; Zbl 0885.35142)].

Let \(\psi \in \mathcal L_s^*(=\) the dual of \(\mathcal L_s)\). Then \(\psi \) is said to be regular, if there exists an \(A\in \mathcal L_1\cap \mathcal L_s\) such that \(\psi (B)=tr(AB)\) whenever \(B\in \mathcal L_s\). Further, \(\psi \) is said to be irregular, if \(\psi (B)=0\) for every \(B\in \mathcal L_{\infty}\cap \mathcal L_s\). The following result is proved in the paper: For every \(\psi \in \mathcal L_s^*\) there exists a uniquely determined regular \(\psi _1\in \mathcal L_s^*\) and an irregular \(\psi _2\in \mathcal L_s^*\) such that \(\psi =\psi _1+\psi _2\) (Theorem 1).

Identifying the Banach space \(\mathcal N^2_s\) of symmetric bounded bilinear functionals on \(\mathbb{H}\) with \(\mathcal L_s\), a second order differential operator associated to \(\psi \in (\mathcal N^2_s)^*\) is defined as follows: For \(x\in \mathbb{H}\), let \(\mathcal G_x\) stand for real functionals \(f\) on \(\mathbb{H}\) possessing the second order Fréchet derivative \(f''(x)\in \mathcal N_s^2\) at \(x\). For \(\psi \in (\mathcal N^2_s)^*\) and \(x\in \mathbb{H}\), put \(\mathbf D^{\psi}_x{:}f\to \psi (f''(x))\), \(f\in \mathcal G_x\). Then \(\mathbf D^{\psi}=\{\mathbf D^{\psi}_x;\;x\in \mathbb{H}\}\) is called a second order differential operator on \(\mathbb{H}\). Thus, by Theorem 1, every differential operator can be, in an obvious sense, uniquely split into a regular and an irregular part.

Given an orthonormal basis \(E=\{e_j;\;j\in \mathbb{N}\}\) and \(x\in \mathbb{H}\), define \[ L^Ef(x)=\lim _{n\to \infty}\frac 1n\sum _{j=1}^nf''(x)(e_j,e_j) \] for those \(f\in \mathcal G_x\), for which the limit exists. It is shown that the Lévy Laplacian \(L^E\) can be extended to an irregular differential operator on \(\mathbb{H}\).

In the paper, properties of irregular operators are studied. In particular, it turns out that irregular differential operators actually behave as the first order differential operators. Certain invariance properties are shown to characterize irregular operators (Theorem 4). Further, all differential operators enjoying the maximum principle are described (Theorem 5). The last part of the paper deals with the Dirichlet and the Poisson problem for irregular differential operators enjoying the maximum principle. For a class of domains and a class of boundary conditions (or right hand sides for the Poisson problem), uniqueness and existence are established (Theorem 7). Relations to domains and functionals investigated by G. E. Šilov are also studied. The paper under review generalizes and extends results recently obtained for the Lévy Laplacian by A. B. Mingarelli and S. Wang in [Differ. Eqn. Dyn. Syst. 1, 23-34 (1993; Zbl 0885.35142)].

Reviewer: Ivan Netuka (Praha)

##### MSC:

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |

46C99 | Inner product spaces and their generalizations, Hilbert spaces |

46B28 | Spaces of operators; tensor products; approximation properties |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

31C45 | Other generalizations (nonlinear potential theory, etc.) |