Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1957. Volume 115, No. 2
MATHEMATICS
I. M. GLAZMAN
ON A CERTAIN ANALOGUE OF THE THEORY OF EXTENSIONS OF HERMITIAN OPERATORS AND A NONSYMMETRIC ONE-DIMENSIONAL BOUNDARY-VALUE PROBLEM ON THE HALF-AXIS
(Presented by Academician A. N. Kolmogorov on 11 II 1957)
- Let \(H\) be a Hilbert space; let \(J\) be some conjugation operator in \(H\) (i.e., an operator defined everywhere in \(H\), satisfying the conditions \((Jf,Jg)=(g,f)\) and \(J^2 f=f\) for any \(f\) and \(g\) in \(H\) \((^1)\).
Definition 1. A linear operator acting in \(H\) is called \(J\)-symmetric if
\[ (Af,Jg)=(f,JAg) \tag{1} \]
for all \(f\) and \(g\) in the domain of definition \(D_A\) of the operator \(A\).
An example of a \(J\)-symmetric operator in \(\mathscr L^2(0,\infty)\) is, in particular, the differential operator \(L'\) generated by the operation
\[ l[y]=\sum_{k=0}^{n}(-1)^k\frac{d^k}{dx^k}\left[p_{n-k}(x)\frac{d^k y}{dx^k}\right] \tag{2} \]
on finite functions \(\varphi(x)\), absolutely continuous together with their derivatives up to order \((2n-1)\), satisfying the condition \(\varphi(0)=\varphi'(0)=\ldots=\varphi^{(2n-1)}(0)=0\). Here it is assumed that the coefficients of the operation (2) are complex-valued functions satisfying the usual smoothness conditions; here and below, by the operator \(J\) in the space \(\mathscr L^2(0,\infty)\) is meant the operator of complex conjugation, defined by the equality \(J[f(x)]=\overline{f(x)}\) for any function \(f(x)\) from \(\mathscr L^2(0,\infty)\).
If the domain of definition \(D_A\) of a \(J\)-symmetric operator \(A\) is dense in \(H\), then the operator \(A^*\) exists, and from (1) there follows the relation
\[ JAJ\subset A^*. \tag{3} \]
Definition 2. A \(J\)-symmetric operator \(A\) with domain of definition dense in \(H\) is called \(J\)-self-adjoint if
\[ JAJ=A^*. \]
An example of a \(J\)-self-adjoint operator in \(L^2(0,\infty)\) is, in particular, the nonself-adjoint differential operator of second order investigated in the fundamental paper of M. A. Naimark \((^2)\).
It follows from (3) that every \(J\)-symmetric operator with domain of definition dense in \(H\) admits a closure. In particular, the operator \(L'\) admits a closure, which we shall denote by \(L\) and call the differential operator with minimal domain of definition generated by the operation (2).
Let us note that a symmetric operator may in special cases turn out to be \(J\)-symmetric, and a self-adjoint operator \(J\)-self-adjoint—
(for example, the operator \(L\) in the case of real coefficients of the differential operation (2) and its real\({}^{(4)}\) self-adjoint extensions).
Definition 3\({}^{(3)}\). A linear operator \(A\) acting in \(H\) is called dissipative if
\[ \operatorname{Im}(Af,f)\geqslant 0 \tag{4} \]
for every \(f\in D_A\).
The operator \(L\) constructed above will be dissipative under the condition
\[ \operatorname{Im} p_k(x)\geqslant 0\qquad (k=0,1,\ldots,n;\; 0\leqslant x<\infty). \tag{5} \]
If \(A\) is a closed dissipative operator and \(\operatorname{Im}\lambda<0\), then the resolvent \(R_\lambda=(A-\lambda I)^{-1}\) exists and \(\|R_\lambda\|\leqslant |\operatorname{Im}\lambda|^{-1}\); moreover, the linear manifold \(\Delta_\lambda=(A-\lambda I)D_A\) is closed, and the dimension \(m\) of its orthogonal complement does not depend on \(\lambda\). This dimension \(m\) will be called the defect number of the operator \(A\).
Theorem 1. In order that a dissipative \(J\)-symmetric operator with domain dense in \(H\) be \(J\)-self-adjoint, it is necessary and sufficient that its defect number be equal to zero.
With the aid of the pair of formulas
\[ Af+if=g,\qquad Af-if=Vg \]
we introduce the Cayley transform \(V\) of a \(J\)-symmetric operator \(A\) with domain dense in \(H\) and defect number \(m\geqslant 0\).
From (1) and (4) it follows that \(V\) is a \(J\)-symmetric operator in \(H\), defined on the subspace \(\Delta_{-i}\subset H\) \((\operatorname{def}\Delta_{-i}=m)\), and \(\|V\|\leqslant 1\).
If \(\widetilde V\) is a \(J\)-symmetric extension of the operator \(V\) and \(\|\widetilde V\|\leqslant 1\), then from the equality \(\widetilde Vh=h\) it follows that \(h=0\), and the operator \(\widetilde A\), defined by the pair of formulas
\[ f=\frac{1}{2i}(g-\widetilde Vg),\qquad \widetilde Af=\frac{1}{2}(g+\widetilde Vg), \]
is a \(J\)-symmetric dissipative extension of the operator \(A\).
Modifying the method of M. G. Krein\({}^{(5)}\) for extending bounded symmetric operators with non-dense domain in \(H\), we obtain a \(J\)-symmetric extension \(\widetilde V\) of the operator \(V\) to all of \(H\), satisfying the condition \(\|\widetilde V\|\leqslant 1\). Hence Theorem 2 follows.
Theorem 2. Every \(J\)-symmetric dissipative operator \(A\) with domain dense in \(H\) admits an extension to a \(J\)-self-adjoint dissipative operator \(\widetilde A\).
Let us note that from the \(J\)-self-adjointness of the operator \(A\) there follows the \(J\)-self-adjointness of its resolvent \(R_\lambda\) at regular points \(\lambda\), which is essential for the proof of Theorem 5 (see below).
2. From what has been set forth, with the aid of the methods applied in \({}^{(4)}\), we obtain a series of theorems generalizing the known propositions \({}^{(1,4)}\) on differential operators of the form (2) with real coefficients to the case of complex-valued coefficients with nonnegative imaginary part.
Theorem 3. If conditions (5) are fulfilled, then the number \(m\) of linearly independent solutions of the equation
\[ \sum_{k=0}^{n}(-1)^k\frac{d^k}{dx^k}\left[p_{n-k}(x)\frac{d^k y}{dx^k}\right]=\lambda y \qquad(\operatorname{Im}\lambda<0), \]
belonging to \(\mathscr L^2(0,\infty)\), does not depend on \(\lambda\) and satisfies the inequality \(m \geqslant n\).
Theorem 4. The operator \(L\) with minimal domain of definition, generated by operation (2) with coefficients satisfying conditions (5), can be extended to a \(J\)-self-adjoint dissipative differential operator.
Theorem 5. The resolvent \(R_\lambda\) of any \(J\)-self-adjoint dissipative extension of the operator \(L\) for \(\operatorname{Im}\lambda < 0\) is a bounded integral operator defined on all of \(\mathscr L^2(0,\infty)\),
\[ R_\lambda g=\int_0^\infty \Gamma(x,s;\lambda)\,g(s)\,ds \]
with kernel
\[ \Gamma(x,s;\lambda)=\Gamma(s,x;\lambda). \]
For \(m<2n\) the resolvent kernel satisfies the condition
\[ \int_0^\infty |\Gamma(x,s;\lambda)|^2\,ds<\infty. \]
For \(m=2n\),
\[ \int_0^\infty\int_0^\infty |\Gamma(x,s;\lambda)|^2\,dx\,ds<\infty; \]
so that in this case the resolvent \(R_\lambda\) is a completely continuous operator.
For \(n=1\), Theorems 3–5 give a generalization of the well-known results of H. Weyl \({}^{(5)}\) to a differential equation of the form
\[ -y''+q(x)y-\lambda y=g(x)\qquad (0\leqslant x<\infty), \]
where \(q(x)\) is a complex-valued function summable on every finite interval, with nonnegative imaginary part \(\operatorname{Im}q(x)\).
Kharkov Polytechnic Institute
named after V. I. Lenin
Received
24 IX 1956
CITED LITERATURE
\({}^{1}\) N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators, 1950.
\({}^{2}\) M. A. Naimark, Tr. Moscow Math. Soc., 3, 181 (1954).
\({}^{3}\) B. R. Mukminov, DAN, 99, No. 4 (1954).
\({}^{4}\) I. M. Glazman, Uspekhi Mat. Nauk, 5, issue 6 (40) (1950).
\({}^{5}\) M. G. Krein, Matem. sbornik, 20 (62): 3 (1947).
\({}^{6}\) H. Weyl, Math. Ann., 68, 220 (1910).