In tutta questa parte supporremo sempre K = ℂ. Nella Sezione 2.3 abbiamo costruito F(f): V → V per una trasformazione lineare f: V → V che sia normale rispetto ad un fissato prodotto hermitiano su V, con F: Spec(f) → ℂ una assegnata funzione.

L’oggetto di questo capitolo è lo studio delle soluzioni di un sistema differenziale del primo ordine del tipo (5.1) $$ \frac{{dx}} {{dt}} = :\dot x = f(t,x), $$ dove supporremo sempre che f sia una mappa (5.2) $$ f:I \times \Omega \ni (t,x) \mapsto f(t,x) = \left[ {\begin{array}{*{20}c} {f_1 (t,x)} \\ \vdots \\ {f_n (t,x)} \\ \end{array} } \right] \in \mathbb{R}^n , $$ con I intervallo aperto di ℝ e Ω aperto di ℝ n , soddisfacente d’ora innanzi almeno le seguenti condizioni di regolarità:

Definizione 2.1.1. Dati uno spazio vettoriale V su K (K = ℝ oppure ℂ) ed una ap- plicazione lineare f: V → V, si dice che λ ∈ K è un autovalore dif se esiste v ∈ V, v ≡ 0, tale che f(v) = l</font >v. f(v) = \lambda v. ((2.1))

Il presente volume fornisce strumenti dell'algebra lineare con la prospettiva infinito-dimensionale e tratta equazioni/sistemi differenziali ordinari. Il libro di testo è di livello avanzato, rivolto a studenti della laurea magistrale o del dottorato di ricerca, suddiviso in due parti.

We study the problem of perturbations of C ∞ C^\infty -hypoelliptic operators by lower order terms. After giving several examples which show many different possibilities, we then prove a stability result which shows that a hypoelliptic linear partial differential operator P P which loses finitely many derivatives and whose formal adjoint P ∗ P^* is still hypoelliptic (but with no assumption on the loss of derivatives) remains hypoelliptic with the same loss of derivatives after perturbation by a lower order linear partial differential operator (whose order depends on the loss of derivatives).

We study the local and semi-global solvability of a class of operators with multiple characteristics. The main point is a result on propagation of singularities. It is shown that the solvability occurs with a loss of derivatives related to the order of characteristics.

We show that for a class of hyperbolic second order operators with double characteristics the Ivrii–Petkov condition is also sufficient for the well-posedness of the Cauchy problem in C ∞.

We study the validity of Hörmander's inequality for some classes of pseudo-differential operators for which the transversal ellipticity condition does not hold.

We give here a family of second order examples tailored to those by Kohn and by Christ, which are (C ) hypoelliptic and lose an arbitrarily large (fixed) number of derivatives.

A sufficient condition is given in order to ensure the Hormander inequality for a (formally) self-adjoint transversally elliptic pseudodifferential operator P in the case of the symplectic form changing rank on the characteristic manifold.

We study a possible extension to the infinite-dimensional case of the classicalLyapunov lemma for matrices. More precisely, for a fixed elliptic system A ofdifferential operators of order m, we consider the operator equationTA + A*T = Q, where Q is any given classical pseudodifferential system oforder m, and T is sought as a classical pseudodifferential system of order0.

We give necessary and sufficient conditions for the lower bound {fx55-01} to hold for any compact setK ⊂X, an open set ofR n , andP =P* ∃ ψ phg 4 (X) with p(x, ξ) ~ q 2 2 + p3 + p2 + ..., q2 beingtransversally elliptic with respect to the characteristic manifold Σ =q 2 -1 (0).

On etudie les developpements asymptotiques des solutions distribution de certaines equations hyperboliques fuchsiennes

(1983). Pseudodifferential operators of mellin type. Communications in Partial Differential Equations: Vol. 8, No. 5, pp. 477-544.