MATHEMATICS

M. G. KREIN

ON THE THEORY OF LINEAR NON-SELF-ADJOINT OPERATORS

(Presented by Academician S. L. Sobolev on 16 IX 1959)

In what follows, $\mathfrak H$ denotes a separable Hilbert space; $\mathfrak R$ is the linear ring of all linear bounded operators acting in $\mathfrak H$; $\mathfrak J$ is the two-sided ideal in $\mathfrak R$ of all completely continuous operators; $\mathfrak S$ is the two-sided ideal in $\mathfrak R$ of all operators $A \in \mathfrak J$ such that $\operatorname{Sp}(A^*A)^{1/2} < \infty$ (i.e., having trace $\operatorname{Sp} A$ absolutely convergent).

If $A \in \mathfrak S$, then the quantity $\det(I + A)$ is meaningful, defined as the limit, as $n \to \infty$, of the determinants $\left|\delta_{jk} + (A\varphi_j,\varphi_k)\right|_1^n$, where $\{\varphi_j\}_1^\infty$ is any orthonormal basis in $\mathfrak H$; this limit always exists and does not depend on the choice of basis. It is easily shown that if $A, B \in \mathfrak J$, $AB \in \mathfrak S$ and $BA \in \mathfrak S$, then $\det(I + AB) = \det(I + BA)$.

If $A, B \in \mathfrak R$, $B - A \in \mathfrak S$, then the determinant

$$ D_{B/A}(\lambda) = \det[(I-\lambda B)(I-\lambda A)^{-1}] = \det[I + \lambda(B-A)(I-\lambda A)^{-1}] $$

is meaningful for all complex $\lambda$ for which $(I-\lambda A)^{-1}\in \mathfrak R$ exists. In (1) it was already noted that if $A \in \mathfrak R$, $B - A \in \mathfrak S$, $C - B \in \mathfrak S$, then $D_{C/A}(\lambda) = D_{C/B}(\lambda)D_{B/A}(\lambda)$.

An operator $A \in \mathfrak R$ is called dissipative if, in its decomposition into Hermitian components $A = A_R + iA_J$, the imaginary component $A_J = (A - A^*)/2i$ is a nonnegative operator: $(A_J f, f) \geq 0$ $(f \in \mathfrak H)$.

The following generalization of a theorem of M. S. Livshits (²) and M. S. Brodskii (³) on the characteristic operator of a dissipative operator holds.

Theorem 1. Let $A = G + iH$ $(H = A_J)$ be a dissipative operator, $B = G + iF$, where $-H \leq F \leq H$.

Then, for $\operatorname{Im}\lambda < 0$, the operator

$$ W_\lambda = I + i(H-F)^{1/2}(A-\lambda I)^{-1}(H-F)^{1/2} $$

is nonexpanding, i.e. $\|W_\lambda f\| \leq \|f\|$ $(f \in \mathfrak H)$.

The theorem follows from the easily verified identity

$$ I - W_\lambda^*W_\lambda = (H-F)^{1/2}[R_\lambda^*(H+F)R_\lambda - 2\operatorname{Im}\lambda R_\lambda^*R_\lambda](H-F)^{1/2}, $$

where $R_\lambda = (A-\lambda I)^{-1}$. If, in addition, the condition $H \in \mathfrak S$ is fulfilled (i.e. $\operatorname{Sp} H < \infty$), then, putting $T = A - B = H - F$, we shall have

$$ D_{B/A}(\lambda) = \det[I - i\lambda T(I-\lambda A)^{-1}] = \det[I - i\lambda T^{1/2}(I-\lambda A)^{-1}T^{1/2}] = \det W_{1/\lambda}. $$

Theorem 2. If the operators $A$ and $B$ satisfy the conditions of the preceding theorem and $\operatorname{Sp} H < \infty$, then $|D_{B/A}(\lambda)| \leq 1$ for $\operatorname{Im}\lambda > 0$.

If $A = G + iH \in \mathfrak R$, $H \in \mathfrak S$, then, putting $A_1 = G + iH_1$, $H_1 = H_+ + H_-$, where $H_+$ and $H_-$ are the orthogonal nonnegative operators from the decomposition $H = H_+ - H_-$, we shall have

$$ D_{G/A}(\lambda) = D_{G/A_1}(\lambda)D_{A_1/A}(\lambda) = D_{G/A_1}(\lambda)/D_{A/A_1}(\lambda), $$

where, by Theorem 2, the functions \(D_{G/A_1}(\lambda)\) and \(D_{A/A_1}(\lambda)\) will have modulus \(\leqslant 1\) for \(\operatorname{Im}\lambda>0\). Hence

Theorem 3. If \(A=G+iH\in\mathfrak R,\ H\in\mathfrak S\), then inside the upper (lower) half-plane \(\operatorname{Im}\lambda>0\) \((\operatorname{Im}\lambda<0)\) the function \(D_{G/A}(\lambda)\) can be represented as a quotient of two holomorphic bounded functions.

2. By the multiplicity of an \(x\)-number (characteristic number) of an operator \(A\in\mathfrak S\) is meant the dimension of the corresponding root subspace of the operator \(A\). By \(n(r;A)\), \(n_{\pm}(r;A)\) are denoted, respectively, the exact number (i.e., counting multiplicity) of the \(x\)-numbers of the operator \(A\) in the circle \(|\lambda|\leqslant r\), in the interval \((0,r]\) or \([-r,0]\).

An operator \(A\in\widetilde{\mathfrak S}\) is called Volterra if it has no \(x\)-numbers.

A number of assertions of Theorem 4 below are easily derived from Theorem 3 and from a theorem of the author \((^4)\), according to which an entire function \(f(\lambda)\), representable inside each of the two half-planes \(\operatorname{Im}\lambda>0\) and \(\operatorname{Im}\lambda<0\) as a quotient of two bounded holomorphic functions, always has the properties:

\[ 1)\quad \ln |f(\lambda)|=O(|\lambda|)\quad \text{as } \lambda\to\infty; \qquad 2)\quad \int_{-\infty}^{\infty}\frac{|\ln |f(\lambda)||}{1+\lambda^2}\,d\lambda<\infty . \tag{1} \]

Theorem 4. If the operator \(A=G+iH\) is Volterra and \(H\in\mathfrak S\), then the entire function \(f(\lambda)=D_{G/A}(\lambda)\exp(-i\lambda\,\operatorname{Sp}H)\) has the properties (1) and is representable in the form

\[ f(\lambda)=\prod_j (1-\lambda/a_j)e^{\lambda/a_j}, \]

where \(\{a_j\}\) is the complete sequence of \(x\)-numbers of the operator \(A_*\). For this sequence there exists the common limit

\[ \frac{h}{\pi} =\lim_{r\to\infty}\frac{n_+(r;G)}{r} =\lim_{r\to\infty}\frac{n_-(r;G)}{r}, \]

where \(|\operatorname{Sp}H|\leqslant h\leqslant \operatorname{Sp}|H|\;(=\operatorname{Sp}H_+ + \operatorname{Sp}H_-)\). If, in particular, the operator \(A\) is dissipative, then \(h=\operatorname{Sp}H\).

A weaker assertion was formulated in \((^5)\). The general method set forth in \((^6)\) (see also \((^1)\)) makes it possible to draw the following conclusion from Theorem 4.

Theorem 5. If the operator \(A=G+iH\in\mathfrak S\) is dissipative, \(\operatorname{Sp}H<\infty\), and at least one of the two conditions is satisfied:

\[ 1)\quad \lim_{r\to\infty}\frac{n_+(r;G)}{r}=0; \qquad 2)\quad \lim_{r\to\infty}\frac{n_-(r;G)}{r}=0, \]

then the system of root vectors of the operator \(A\) is complete in \(\mathfrak H\).

This theorem, being a strengthening of Theorem 1 from \((^1)\), in essence already follows from the considerations given in the present article.

3. If \(A=G+iH\in\mathfrak S\) and \(H\in\mathfrak S\), then

\[ D_{A^*/A}(\lambda)=D_{A^*/G}(\lambda)D_{G/A}(\lambda) =D_{G/A}(\lambda)/\overline{D}_{G/A}(\lambda). \tag{2} \]

Let us explain that we write \(g(\lambda)=\overline{f}(\lambda)\) if \(g(\bar\lambda)=\overline{f(\lambda)}\). From (2) it follows that \(|D_{A^*/A}(\lambda)|=1\) for \(\operatorname{Im}\lambda=0\). Hence, from Theorem 3, one obtains without difficulty:

Theorem 6. If the operator \(A=G+iH\in\mathfrak S\) and \(H\in\mathfrak S\), then

\[ D_{A^*/A}(\lambda)=e^{2ia\lambda}\prod_j \frac{1-\lambda/\bar\lambda_j}{1-\lambda/\lambda_j}, \]

where \(\{\lambda_j\}\) is the complete sequence of characteristic values of the operator \(A\),

\[ a=\operatorname{Sp} H-\sum_j \operatorname{Im}\left(\frac{1}{\lambda_j}\right). \tag{3} \]

1. As is known, the logarithmic length \(L_g\) of a measurable set \(\Delta\subset(1,\infty)\) is the integral over \(\Delta\) of \(dr/r\). In the paper \((^7)\) W. K. Hayman established an important theorem which, in particular, contains the following proposition.

Let \(u(\lambda)\) be a nonnegative superharmonic function in the open upper half-plane, and let
\[ h=\inf (u(\lambda)/\operatorname{Im}\lambda), \]
where \(\lambda\) ranges over this half-plane. Then there exists a set \(\Delta\subset(1,\infty)\) with \(L_g(\Delta)<\infty\) such that, as \(\rho\to\infty\), outside \(\Delta\), uniformly in \(\theta\) \((0<\theta<\pi)\), the limiting relation
\[ \lim (u(\rho e^{i\theta})/\rho)=h\sin\theta \]
will hold.

This proposition, in combination with Theorems 3 and 6, and also with Jensen’s formula:

\[ N(\rho;G)-N(\rho;A)=\int_0^\rho \frac{n(r;G)}{r}\,dr-\int_0^\rho \frac{n(r;A)}{r}\,dr =\frac{1}{2\pi}\int_0^{2\pi}\ln|D_{G/A}(\rho e^{i\theta})|\,d\theta \]

leads to the following conclusion:

Theorem 7. If \(A=G+iH\in\mathfrak{S}\), \(H\in\mathfrak{S}\), then, as \(\rho\to\infty\), outside a suitable set \(\Delta\) with \(L_g(\Delta)<\infty\), the asymptotic relation

\[ N(\rho;G)-N(\rho;A)=c\rho+o(\rho), \tag{4} \]

holds, where the constant \(c\ge 2|a|/\pi\) (the quantity \(a\) is defined in (3)). If the operator \(A\) is dissipative, then \(c=2a/\pi\).

Hence, and from a theorem of M. S. Livshits \((^2)\) (see also \((^8)\), it follows that

Theorem 8. In order that the system of root vectors of a completely continuous dissipative operator \(A=G+iH\) with \(H\in\mathfrak{S}\) be complete, it is necessary and sufficient that
\[ N(\rho;G)-N(\rho;A)=o(\rho), \]
as \(\rho\to\infty\), outside a suitable set \(\Delta\) with \(L_g(\Delta)<\infty\).

It should be pointed out that in his recent paper \((^9)\) B. Ya. Levin developed methods that allowed him to show (without using infinite determinants) that, in the case of a dissipative operator \(A=G+iH\in\mathfrak{S}\) with \(\operatorname{Sp}H<\infty\), the asymptotic inequality
\[ N(\rho;G)-N(\rho;A)\le \rho\,\operatorname{Sp}H+o(\rho) \]
holds as \(\rho\to\infty\), outside some set \(\Delta\) with \(L_g(\Delta)<\infty\). A joint discussion of the considerations of paper \((^9)\) showed that a little must be added to them in order, for the indicated case, to obtain relation (4) with some \(c\le 2a/\pi\) (for this case Theorem 7 asserts that \(c=2a/\pi\)). After this the author reexamined the methods of his paper \((^1)\) and found that, after supplementing them somewhat, one can arrive at the results of the present paper.

Odessa Civil Engineering Institute

Received
16 IX 1959

CITED LITERATURE

\(^1\) M. G. Krein, Uspekhi Mat. Nauk, 14, no. 3 (87), (1959).
\(^2\) M. S. Livshits, Mat. Sb., 34 (76), 1 (1954).
\(^3\) M. S. Brodskii, Mat. Sb., 39 (81), 2 (1956).
\(^4\) M. G. Krein, Izv. AN SSSR, Ser. Mat., 11, 309 (1947).
\(^5\) I. Ts. Gohberg, M. G. Krein, DAN, 128, no. 2 (1959).
\(^6\) V. B. Lidskii, DAN, 119, no. 6 (1958); Tr. Moskov. Mat. Obshch., 8, 83 (1959).
\(^7\) W. K. Hayman, J. Math. Pures et Appl., 35, 115 (1956).
\(^8\) B. R. Mukminov, DAN, 99, no. 4 (1954).
\(^9\) B. Ya. Levin, Sb. Tr. Kharkov. Inst. Inzh. Zh.-D. Transporta, 1959.