MATHEMATICS

G. LANGER

ON \(J\)-HERMITIAN OPERATORS

(Presented by Academician L. S. Pontryagin on 23 IV 1960)

1. Let \(H\) be a separable Hilbert space with scalar product \((x,y)\) \((x,y\in H)\); let \(P\) and \(Q\) be orthogonal projections in \(H\), with \(P+Q=I\) (\(I\) is the identity mapping), and let \(J=P-Q\). We define in \(H\) the indefinite scalar product \([x,y]\) by the formula (cf. \(\left(^{1-4}\right)\))

\[ [x,y]=(Jx,y)\quad (x,y\in H). \tag{1} \]

By an operator in \(H\) we mean a linear bounded mapping of \(H\) into \(H\). An operator \(A\) in \(H\) will be called \(J\)-Hermitian \(\left(^{4}\right)\) if \([Ax,y]=[x,Ay]\) for all \(x,y\in H\). An operator \(A\) is \(J\)-Hermitian if and only if \(A=JA^{*}J\), where \(A^{*}\) is the operator adjoint to \(A\). Hence it follows that, for a \(J\)-Hermitian operator \(A\),

\[ \operatorname{Im} A=\frac{1}{2i}(A-A^{*})=\frac{1}{i}(PAQ+QAP), \tag{2} \]

and, conversely, every operator satisfying condition (2) is \(J\)-Hermitian. An operator \(U\) in \(H\) is called \(J\)-unitary \(\left(^{3,4}\right)\) if there exists an everywhere-defined inverse operator in \(H\) and \([Ux,Uy]=[x,y]\) for all \(x,y\in H\).

The spectrum \(\sigma(A)\) of an operator \(A\) and its parts \(\sigma_p(A)\), \(\sigma_c(A)\), \(\sigma_r(A)\), as well as the resolvent set \(\rho(A)\), are defined as in \(\left(^{5}\right)\). Then it is not difficult to prove the following proposition:

Theorem 1. If the operator \(A\) in \(H\) is \(J\)-Hermitian, then:

1) the set \(\sigma(A)\) is symmetric with respect to the real axis (cf. \(\left(^{2}\right)\)).

2) From \(\lambda\in\sigma_p(A)\) it follows that \(\overline{\lambda}\in\sigma_p(A)\cup\sigma_r(A)\).

3) From \(\lambda\in\sigma_r(A)\) it follows that \(\overline{\lambda}\in\sigma_p(A)\).

Corollary. For a \(J\)-Hermitian operator \(A\), the set \(\sigma_p(A)\cup\sigma_r(A)\) is symmetric with respect to the real axis, and \(\sigma_r(A)\) contains no real points.

1. L. S. Pontryagin proved (see also \(\left(^{2,6,7}\right)\)) that for a \(J\)-Hermitian operator \(A\), under the condition that \(\dim PH=\varkappa\) \((0<\varkappa<\infty)\), there always exists a nonnegative invariant \(\varkappa\)-dimensional subspace in which the spectrum of \(A\) has a nonnegative imaginary part. We generalize this theorem to \(J\)-Hermitian operators with a completely continuous imaginary part (for \(\varkappa=\infty\)).

A. We denote

\[ T^{+}=\{x:x\in H,\ [x,x]>0\}\cup\{0\}; \]

\[ T^{-}=\{x:x\in H,\ [x,x]<0\}\cup\{0\};\qquad T^{0}=\{x:x\in H,\ [x,x]=0\}. \]

If \(T\) is a subset of \(H\), then a subspace contained in \(T\) will be called a maximal subspace of \(T\) if every other subspace containing the given one is not wholly contained in \(T\).

We shall consider operators \(E\) in \(H\) with the properties:

\[ EP=E,\qquad PE=P. \tag{3} \]

Obviously, \(E^{2}=E\). If the range of \(E\) is contained in \(T^{+}\cup T^{0}\) and conditions (3) are satisfied, then \(EH\) is a maximal subspace of \(T^{+}\cup T^{0}\). A simple metric characteristic of such operators \(E\) is given by:

Lemma 1. If for an operator \(E\) in \(H\) conditions (3) are satisfied, then \(EH\subset T^{+}\cup T^{0}\) if and only if \(\|E\|\leq \sqrt{2}\).

B. Here and everywhere below in Section 2 we assume \(\dim PH=\dim QH=\infty\). In the range \(PH\) of the orthogonal projector \(P\), introduce an orthonormal basis \(x_{1},x_{2},\ldots\). By \(P_{n}\) denote the orthogonal projector from \(H\) onto the linear span of the vectors \(x_{1},x_{2},\ldots,x_{n}\) \((n=1,2,\ldots)\). Let \(I_{n}=P_{n}+Q,\ J_{n}=P_{n}-Q,\ H_{n}=I_{n}H\). The definite (respectively, indefinite) scalar product in \(H\) induces a definite (respectively, indefinite) scalar product in \(H_{n}\), which we shall again denote by \((x,y)\) (respectively, \([x,y]\)) \((x,y\in H_{n})\). The relation \([x,y]=(J_{n}x,y)\) holds. An operator \(U_{n}\) (respectively, \(A_{n}\)) in \(H_{n}\) is again called \(J_{n}\)-unitary (respectively, \(J_{n}\)-Hermitian) if \(U_{n}^{-1}\) exists as an operator in \(H_{n}\), and \([U_{n}x,U_{n}y]=[x,y]\) (respectively, \([A_{n}x,y]=[x,A_{n}y]\)) for all \(x,y\in H_{n}\).

Lemma 2. If \(U_{n}\) is a \(J_{n}\)-unitary operator in \(H_{n}\), and for an operator \(E_{n}\) in \(H_{n}\) the conditions \(E_{n}P_{n}=E_{n},\ P_{n}E_{n}=P_{n}\) and \(\|P_{n}E_{n}x\|-\|QE_{n}x\|\geq 0\) for \(x\in H_{n}\) are satisfied, then the operator \(P_{n}U_{n}^{-1}E_{n}P_{n}\) maps the subspace \(P_{n}H_{n}\) one-to-one onto itself.

B. Denote by \(\mathfrak T\) the totality of all operators \(E_{n}\) in \(H_{n}\) possessing the properties

\[ \|P_{n}E_{n}x\|-\|QE_{n}x\|\geq 0\qquad (x\in H_{n}); \tag{4} \]

\[ P_{n}E_{n}=P_{n},\qquad E_{n}P_{n}=E_{n}. \tag{5} \]

The set \(\mathfrak T\) is nonempty, since \(P_{n}\in\mathfrak T\). It is easily proved that \(\mathfrak T\) is convex and closed in the weak topology. Moreover, from Lemma 1, for \(E_{n}\in\mathfrak T\) it follows that \(\|E_{n}\|\leq \sqrt{2}\). But the unit sphere in the ring of operators in \(H^{n}\) is weakly compact \((^{5})\). Therefore \(\mathfrak T\), as a weakly closed part of a weakly compact set, is also compact in the weak operator topology.

G. If \(U_{n}\) is a \(J_{n}\)-unitary operator in \(H_{n}\), then, by Lemma 2, there exists a bounded one-to-one mapping \((P_{n}U_{n}^{-1}E_{n}P_{n})^{-1}\) of the subspace \(P_{n}H_{n}\) onto \(P_{n}H_{n}\).

Consider the mapping \(D\) defined on \(\mathfrak T\):

\[ D(E_{n})=U_{n}^{-1}E_{n}P_{n}(P_{n}U_{n}^{-1}E_{n}P_{n})^{-1}P_{n},\qquad E_{n}\in\mathfrak T. \]

Then \(P_{n}D(E_{n})=P_{n},\ D(E_{n})P_{n}=D(E_{n}),\ \|P_{n}D(E_{n})x\|-\|QD(E_{n})x\|\geq 0\) for \(x\in H_{n}\), i.e. \(D(E_{n})\in\mathfrak T\). Moreover, the mapping \(D\) is continuous in the weak operator topology, since \(P_{n}H_{n}\) is finite-dimensional. By the Schauder–Tikhonov theorem \((^{5})\), the mapping \(D\) has at least one fixed point. Thus, there exists an operator \(E_{n}^{(0)}\in\mathfrak T\) such that \(D(E_{n}^{(0)})=E_{n}^{(0)}\), and, consequently,

\[ U_{n}E_{n}^{(0)}=E_{n}^{(0)}U_{n}E_{n}^{(0)}. \tag{6} \]

Thus, \(E_{n}^{(0)}H_{n}\) is an \(n\)-dimensional nonnegative invariant subspace for \(U_{n}\).

Using the Cayley transform \(({}^{2,8})\), it follows from this that for every \(J_n\)-Hermitian operator \(A_n\) in \(H_n\) there exists an operator \(E_n^{(0)}\) with the properties:

\[ \|P_n E_n^{(0)}x\|-\|QE_n^{(0)}x\|\geqslant 0 \quad (x\in H_n); \tag{4′} \]

\[ E_n^{(0)}P_n=E_n^{(0)},\qquad P_nE_n^{(0)}=P_n; \tag{5′} \]

\[ E_n^{(0)}A_nE_n^{(0)}=A_nE_n^{(0)}. \tag{6′} \]

Inequality \((4′)\), by virtue of \((5′)\) and Lemma 1, is equivalent to \(\|E_n^{(0)}\|\leqslant \sqrt{2}\).

D. Let \(A\) be a \(J\)-Hermitian operator in \(H\) with a completely continuous imaginary part, and let \(I_n, J_n, H_n\) be defined as in B. Further, let \(E_n=I_nAI_n\). To the operator \(A_n\) there corresponds in \(H_n\) an operator \(E_n^{(0)}\) with properties \((4′)\), \((5′)\), \((6′)\). Define an operator \(E_n'\) in \(H\) by the equality \(E_n'=E_n^{(0)}I_n\). Then for \(E_n'\) relations analogous to \((4′)\) and \((5′)\) are valid, and

\[ E_n'AE_n'x=I_nAE_n'x\quad (x\in H). \tag{7} \]

From \(\|E_n'\|\leqslant\sqrt{2}\) there follows the existence of a weakly convergent subsequence \((E_{n_\nu}')\), i.e. there exists an operator \(E_0\) in \(H\) such that \(E_{n_\nu}'\to E_0\) (as \(\nu\to\infty\)) in the weak operator topology. Again \(\|E_0\|\leqslant\sqrt{2}\) and \(E_0P=E_0,\; PE_0=P\). From \((7)\), taking into account relation \((2)\), and also the fact that \(\operatorname{Im} A\) is completely continuous, passing to the limit as \(\nu\to\infty\), we obtain \(E_0AE_0=AE_0\). Hence, finally, we obtain:

Theorem 2. Let \(A\) be a \(J\)-Hermitian operator in \(H\) with a completely continuous imaginary part. Then there exists an (idempotent) operator \(E_0\) in \(H\) with the properties

\[ \|E_0\|\leqslant\sqrt{2},\qquad PE_0=P,\qquad E_0P=E_0,\qquad AE_0=E_0AE_0. \]

Consequently, there exists a maximal subspace \(E_0H\) of \(T^+\cup T^0\), invariant with respect to \(A\).

Corollary. If \(A\) satisfies the conditions of Theorem 2 and \(E_0^+=JE_0^*J\), then the subspace \((I-E_0^+)H\) is also invariant with respect to \(A\). \((I-E_0^+)H\) is a maximal subspace of \(T^-\cup T^0\).

Starting from the subspace \(E_0H\), one can construct an infinite-dimensional subspace \(M\), invariant with respect to \(A\), such that the spectrum of \(A\) in \(M\) has a nonnegative imaginary part.

1. In this section we consider the case when \(\varkappa=1\).

The operator \(A\) is called simple \(({}^{9})\) if there exists a vector \(x_1\) such that c.l.s. \([A^p x_1,\; p=0,1,\ldots]=H\), i.e. if \(A\) is a cyclic operator \(({}^{10})\) with generating element \(x_1\). Every \(J\)-Hermitian operator in \(H\) in the case \(\varkappa=1\) is representable as the orthogonal sum of a simple \(J\)-Hermitian operator with generating vector \(x_1\in PH\) and an (ordinary) Hermitian operator. Therefore we shall assume that \(A\) is a simple operator, and denote \(\lambda_1=(Ax_1,x_1)\).

Using the model constructed in \(({}^{9})\), pp. 24–27, we obtain:

Theorem 3. Let \(A\) be a simple \(J\)-Hermitian operator in the space \(H\) \((J=P-Q,\; \dim PH=1)\). Then \(A\) is unitarily equivalent to the operator

\[ \widetilde{A}\{\gamma,g(t)\} = \left\{ \lambda_1\gamma-\int_{-\infty}^{\infty} g(t)\,d\tau(t),\; tg(t)+\gamma \right\} \]

in the space \(\widetilde{H}=R_1\times L_\tau^2\) with scalar product
\[ (\{\varphi,f\},\{\gamma,g\}) = 2\left[\varphi\overline{\gamma}+\int f(t)\overline{g(t)}\,d\tau(t)\right]. \]
To the element \(x_1\) there corresponds the element
\[ \left\{\frac{1}{\sqrt{2}},\,0\right\}\in\widetilde{H}. \]

\(\tau(t)\) is defined as

\[ \tau(t)=(E_t QAx_1,\ QAx_1), \tag{8} \]

where \(E_t\) is the resolution of the identity for \(\dfrac{A+A^*}{2}\).

With the aid of the representation of a simple \(J\)-Hermitian operator \(A\) established in Theorem 3, for the case \(\dim PH=1\) one can prove certain assertions concerning the spectrum of \(A\). For the eigenvalues of the operator the following holds:

A number \(\lambda\) is an eigenvalue of \(A\) if and only if it satisfies the equation

\[ \int \frac{d\tau(t)}{t-\lambda}+\lambda_1-\lambda=0 \]

and, moreover,

\[ \int \frac{d\tau(t)}{|t-\lambda|^2}<\infty . \]

The limiting points of \(\sigma(A)\) can only be points of \(\sigma\!\left(\dfrac{A+A^*}{2}\right)\), since the imaginary part of \(A\) is finite-dimensional \((*)\). In view of (8), the relation

\[ \sigma\!\left(\frac{A+A^*}{2}\right)=S_\tau\cup\{\lambda_1\} \]

holds, where \(S_\tau\) is the set of growth points of the function \(\tau\). From (9) it follows that the limiting points \(S_\tau\) coincide with the limiting points of \(\sigma\!\left(\dfrac{A+A^*}{2}\right)\).

Theorem 4. Under the assumptions of Theorem 3, every limiting point \(S_\tau\) that is not an eigenvalue of \(A\) belongs to \(\sigma_c(A)\).

Examples show that isolated growth points \(\tau(t)\) which are isolated eigenvalues of \(\sigma\!\left(\dfrac{A+A^*}{2}\right)\) may belong to \(\rho(A)\).

I express my gratitude to Prof. M. G. Krein and I. S. Iokhvidov for the literature indicated and for their help in preparing this communication.

Institute of Pure Mathematics
Dresden Higher Technical School
Dresden
German Democratic Republic

Received
15 IV 1960

REFERENCES

1. L. S. Pontryagin, Izv. AN SSSR, ser. matem., 8, No. 6 (1944).
2. I. S. Iokhvidov, M. G. Krein, Tr. Moskovsk. matem. obshch., 5, 367 (1956).
3. Yu. P. Ginzburg, DAN, 117, No. 2 (1957).
4. Yu. P. Ginzburg, Nauchn. zap. fiz.-matem. fak. Odessk. ped. inst., 22, 1 (1953).
5. N. Dunford, J. T. Schwartz, Linear Operators, I, N. Y., 1958.
6. M. G. Krein, UMN, 5, 2 (36) (1950).
7. M. L. Brodskii, UMN, 14, No. 1 (1959).
8. I. S. Iokhvidov, Zapiski N.-i. inst. matem. i mekh., Kharkovsk. gos. univ. i Kharkovsk. matem. obshch., 21 (1949).
9. M. S. Brodskii, M. S. Livshits, UMN, 13, No. 1 (1958).
10. I. S. Iokhvidov, M. G. Krein, Tr. Moskovsk. matem. obshch., 8, 413 (1959).