UDC 517.944

MATHEMATICS

A. A. DEZIN

ON THE THEORY OF OPERATORS OF THE FORM $\dfrac{d}{dt}-A$

(Presented by Academician I. M. Vinogradov on 3 III 1965)

We shall consider the equation

\[ Lu \equiv \frac{du}{dt}-Au=f \tag{L} \]

with boundary condition

\[ \mu_1 u\big|_{t=0}-\mu_2 u\big|_{t=a}=0, \tag{$\Gamma$} \]

where, for $t\in[0,a]$, the functions $u(t), f(t)$ take values in a complex Hilbert space $\mathfrak H$; $\mu_1,\mu_2:\mathfrak H\to\mathfrak H$ are bounded linear operators, and $A:\mathfrak H\to\mathfrak H$ is an unbounded normal operator commuting with $d/dt$. Equation (L) is a very popular object of investigation, but in this case it is usually assumed either that $\mu_2=0$ and $A$ is the infinitesimal generator of a semigroup, or else one of the indicated assumptions is removed (for example, in ($^1,^2$)). In ($^3$) it was shown that the use of conditions of the form ($\Gamma$) makes it possible to find “good” problems also for “pathological” operators $L$. In the present article the same question is considered from a more general point of view. We shall be interested in two types of conditions ($\Gamma$). In the first (problems L[$\Gamma$-I]) $\mu_2=1$ and $\mu_1=\mu$ is a complex number. Conditions of this type make it possible to describe resolvable extensions ($^3$) of the operator $L$ in the case when the spectrum of the operator $A$ does not fill the whole complex plane $C$, and include the “classical” problems. In conditions of the second type (problems L[$\Gamma$-II]) $\mu_1=\mu^{-}(A)$, $\mu_2=\mu^{+}(A)$ are special projection operators constructed from the operator $A$. These conditions make it possible to describe a resolvable extension of the operator $L$ also for an operator $A$ with spectrum filling the whole plane $C$. It should be noted that the results described pertain to a rather special class of normal operators $A$: to differential operators with constant coefficients on the $n$-dimensional torus.

Let us pass to the consideration of problems of the first type. If $A$ is a numerical parameter, it is not difficult to write down an explicit formula giving the solution of problem L[$\Gamma$-I]. Introduce the notation:

\[ S_t\equiv S(A,t)=e^{tA}; \qquad \int_{\alpha}^{\beta} e^{(t-\tau)A}f(\tau)\,d\tau=I_{\alpha}^{\beta}(t,A)f. \]

Then the desired formula (under the assumption that $\mu-S_a\ne0$) is written in the form:

\[ u(t)=L^{-1}f(t)=\bigl(\mu I_0^t(t,A)+I_t^a(t,A)S_a\bigr)(\mu-S_a)^{-1}f. \tag{1} \]

If now $A$ is a normal operator (i.e. $A=R+iQ$, where $R,Q$ are commuting self-adjoint operators), formula (1) still retains its meaning under the assumption that $\mu$ is not a point of the discrete spectrum of the operator $S_a$.

Let $T$ be an $n$-dimensional torus obtained by identifying opposite faces of a cube with edges of length $2\pi$, referred to coordinates $(x_1,\ldots,x_n)$. On $T$ are defined a linear manifold $P$ of sufficiently smooth functions and the complex Hilbert space $H_x$ of functions with summable-

square with the usual scalar product and norm. If \(A(s)\) is a polynomial with constant complex coefficients

\[ A(s)=\sum_{|\alpha|\le m} A_\alpha s^\alpha,\qquad s^\alpha=s_1^{\alpha_1}\ldots s_n^{\alpha_n},\qquad |\alpha|=\alpha_1+\ldots+\alpha_n, \]

then \(\dot A\equiv \dot A(-iD)\) is the corresponding differential operator, defined on \(P\), and
\(\dot A(-iD)e^{is\cdot x}=A(s)e^{is\cdot x}\). By the operator \(A:H_x\to H_x\) we mean the closure of \(\dot A\) in \(H_x\).

The subsequent exposition contains a number of more or less obvious propositions, the proofs of which are omitted.

Proposition 1. The operator \(A\) is normal, i.e. \(AA^*=A^*A\).

Denote by \(\mathfrak S\) the set of integer vectors \((s_1,\ldots,s_n)\), where each \(s_i\) independently ranges over the set of values \(0,\pm1,\pm2,\ldots\), and by \(A(\mathfrak S)\) the set of values of the polynomial \(A(s)\) for \(s\in\mathfrak S\).

Proposition 2. The spectrum \(\sigma A\) of the operator \(A\) consists of the closure in \(C\) of the set \(A(\mathfrak S)\).

In considerations on \(T\), the role of the operational calculus that gives meaning to formula (1) is played by the use of the classical Fourier series.

Let now \(H=H_x\otimes H_t\), where \(H_t\) is the Hilbert space \(\mathscr L_2\) of functions defined on the interval \([0,a]\). By the operator \(L:H\to H\) we shall mean the closure in \(H\) of the operator of the left-hand side of equation (L), defined on smooth functions satisfying the conditions \((\Gamma)\). Or: a function \(u\in H\) belongs to \(\mathfrak D_L\) if there exist sequences \(\{u_i\}\) of smooth functions satisfying the conditions \((\Gamma)\), and a function \(f\in H\), such that \(|u-u_i,H|\to0\), \(|Lu_i-f,H|\to0\) as \(i\to\infty\). The corresponding solution of problem L-\(\Gamma\) will be called a strong solution, or simply a solution. The operator \(L\) is regular if \(L^{-1}\) exists, is bounded, and is defined on all of \(H\). In what follows, unless otherwise stated, the notation \(\mu=\infty\) is used. It is assumed that in this case the problem in question is that for which \(\mu_1=1,\ \mu_2=0\).

The set of values \(\mu\) for which the operator \(L\) is not regular will be called the \(\mu\)-spectrum of the problem L\(\Gamma\) and denoted by \(\Sigma(A,a)\). By \(\sigma A,\ \sigma S(A,a),\ \sigma L(A,a,\mu)\) we denote the spectra of the operators \(A,\ S_a,\ L\). The investigation of problem L\(\Gamma\) is the study of various relations and dependencies among the spectra listed.

Theorem 1. The equality
\[ \Sigma(A,a)=\sigma S(A,a) \]
holds.

The proof is quite elementary, but somewhat long. We give the main steps.

Proposition 3. The spectrum \(\sigma S(A,a)\) is the closure in \(C\) of the set
\[ \exp[aA(\mathfrak S)], \]
which forms the discrete spectrum \(\sigma_d S(A,a)\) of the operator \(S_a\).

Denote by \(\Sigma_d(A,a)\) the set of points \(\mu\) of the plane \(C\) such that for \(\mu\in\Sigma_d(A,a)\) the operator \(L^{-1}\) does not exist.

Proposition 4. The equality
\[ \Sigma_d(A,a)=\sigma_d S(A,a) \]
is valid.

Lemma 1. If \(\mu\notin\sigma S(A,a)\), then the operator \(L\) is regular.

The proof is based on the validity, in this case, of the energy inequality

\[ |u,H|\le c\,|Lu,H|, \tag{\(\Phi\)} \]

established by the methods used in \((^3)\).

Lemma 2. If
\[ \mu\in\sigma_c S(A,a)\equiv \sigma S(A,a)\setminus \sigma_d S(A,a), \]
then the operator \(L^{-1}\) exists, is defined on a set dense in \(H\), and is unbounded.

The existence of the operator \(L^{-1}\) is a trivial consequence of the uniqueness of the Fourier coefficients; unboundedness is established by considering a sequence \(\{f^k\}_1^\infty\) of right-hand sides
\[ f^k=\exp(is^k\cdot x), \]
where the sequence \(\{s^k\}_1^\infty\) of vectors from \(\mathfrak S\) is such that
\[ |\mu-\exp[aA(s^k)]|\to0 \]
as \(k\to\infty\). From Propositions 3 and 4 and Lemmas 1, 2, Theorem 1 follows.

In considering \(\sigma L(A,a,\mu)\) it is convenient to introduce the notation \(A_\lambda=\lambda E+A\).

Proposition 5. For given \(A,a,\mu\), a point \(\lambda \in \sigma_d L(A,a,\mu)\) (or \(\lambda \in \sigma_c L(A,a,\mu)\)) if and only if \(\mu \in \sigma_d S(A_\lambda,a)\) (or \(\mu \in \sigma_c S(A_\lambda,a)\)).

The following propositions clarify the special position of the Cauchy problem among the LG-I problems.

Proposition 6. If the point \(0\) (the point \(\infty\)) belongs to \(\sigma S(A,a)\), then \(\sigma L(A,a,0)\) (respectively \(\sigma L(A,a,\infty)\)) fills the whole plane \(C\). If \(\mu \ne 0,\infty\), then \(\sigma L(A,a,\mu)\) fills the whole plane if and only if the spectrum \(\sigma S(A,a)\) fills the whole plane.

Proposition 7. The points \(0,\infty\) cannot belong to \(\Sigma_d(A,a)\).

Thus, in the compact case under consideration (the manifold \(T\times[0,a]\) is compact), the uniqueness theory for the Cauchy problem is trivial. It follows from Proposition 6 that, if \(\sigma S(A,a)\) does not fill \(C\), one can always choose \(\mu\) so that the LG-I problem will determine a solvable extension of the operator \(L\).

Lemma 3. For \(n<3\) the spectrum \(\sigma A\) cannot fill the whole plane \(C\); for \(n\geqslant 3\) there exist operators \(A\) whose spectrum fills the whole plane.

The verification of the first part of the assertion is elementary. The second part is established by considering the operator for which
\[ A(s)=s_1+\alpha s_2+i(s_3+\beta s_2^2), \]
where \(\alpha,\beta\) are irrational. Indeed, the fractional parts of the pairs
\[ \{\alpha y,\ \beta y^2\},\qquad y=0,\pm1,\pm2,\ldots \]
are uniformly distributed in the unit square \((^4)\).

Thus, for \(n\geqslant 3\) the LG-I problem does not allow one to describe a solvable extension of the operator \(L\) for arbitrary \(A\).

Let us pass to the LG-II problem. Let the operator \(A\) be fixed and
\[ A(s)=r(s)+iq(s), \]
where \(r,q\) are real polynomials. We divide the set \(\mathfrak S\) into two subsets \(\mathfrak S^-\), \(\mathfrak S^+\), putting \(s\in\mathfrak S^-\) if \(r(s)\leqslant 0\) and \(s\in\mathfrak S^+\) if \(r(s)>0\). The partition of \(\mathfrak S\) induces a decomposition of \(H_x\) into the sum of orthogonal subspaces
\[ H_x=H_x^-\oplus H_x^+, \]
where \(H_x^-\) (\(H_x^+\)) is the closed linear span of the set of vectors of the form \(\exp(is\cdot x)\), \(s\in\mathfrak S^-\) (\(s\in\mathfrak S^+\)). Let \(\mu^-,\mu^+\) be the projection operators in \(H_x^-, H_x^+\), respectively.

Theorem 2. For any operator \(A\) with constant coefficients on \(T\), the LG-II problem in which \(\mu_1=\mu^-\), \(\mu_2=\mu^+\), determines a solvable extension of the operator \(L\).

For the proof it is enough to note that the coefficients of the expansion
\[ u(x,t)=\sum_{\mathfrak S} u_s(t)e^{is\cdot x} \]
of an arbitrary smooth solution of the described problem will be determined from the equations
\[ u_s(t)=\int_0^t e^{(t-\tau)A(s)} f_s(\tau)\,d\tau,\qquad s\in\mathfrak S^-, \]
\[ u_s(t)=-\int_t^a e^{(t-\tau)A(s)} f_s(\tau)\,d\tau,\qquad s\in\mathfrak S^+, \]
where \(f_s(\tau)\) are the coefficients of the corresponding expansion of the right-hand side. An estimate of the form
\[ |u_s(t)|^2 \leqslant |f_s(t)H_t|^2 \int_0^t e^{2(t-\tau)r(s)}\,d\tau \]
in the case \(s\in\mathfrak S^-\), and an analogous estimate for the case \(s\in\mathfrak S^+\), make it possible (by virtue of the corresponding sign of \(r(s)\)) to establish the energy inequality-

property \((\Phi)\), whence the existence and uniqueness theorem for the solution for every \(f \in H\) follows from (3) in the usual way.

The choice of the operators \(\mu_1, \mu_2\) in problem LG-II, when describing the solvable extension, can of course be varied within fairly broad limits, but the use in the considerations of subspaces of the type \(\mathfrak S^{-}, \mathfrak S^{+}\) (for \(\sigma A=C\)) seems to be dictated by the essence of the matter.

We note in conclusion that the approach described to the study of equation (L) induces a certain special classification of differential operators \(A\) with constant coefficients and of boundary-value problems LG-I according to the spectral properties of the operator \(S_a\). The methods used evidently admit generalizations also to equations more general than (L).

REFERENCES

1. S. Agmon, L. Nirenberg, Comm. Pure and Appl. Math., 16, No. 2, 121 (1963).
2. G. I. Laptev, Author’s abstract of candidate dissertation, Voronezh, 1964.
3. A. A. Dezin, DAN, 148, No. 5 (1963).
4. D. V. S. Kassele, Introduction to the Theory of Diophantine Approximations, IL, 1961.