MATHEMATICS

A. V. KUZHEL

ON THE REDUCTION OF UNBOUNDED NON-SELF-ADJOINT OPERATORS TO TRIANGULAR FORM

(Presented by Academician A. N. Kolmogorov on 4 XII 1957)

1. The question of reducing a broad class of bounded non-self-adjoint operators to triangular form was solved by M. S. Livshits in papers \((^{1,2})\). An analogous problem is considered here for unbounded operators.

Denote by \(G_A\) the set of vectors \(f\) belonging to the domain of definition \(D_A\) of the operator \(A\) and such that \((Af,g)=(f,Ag)\) for every \(g\in D_A\). Let \(A_0\) be the operator coinciding with \(A\) on \(G_A\) and defined only on \(G_A\).

Definition. A closed operator \(A\) with everywhere dense domain of definition \(D_A\) will be called a quasi-Hermitian operator of rank \(r\) (in what follows, a \(K^r\)-operator) if \(A_0\) is a Hermitian operator with defect index \((r,r)\) \((r>0)\) and \(\dim D_A=r\pmod {G_A}\). We shall consider only those \(K^r\)-operators for which there exists at least one point \(\lambda\) regular together with its conjugate \((\operatorname{Im}\lambda\ne0)\). Without loss of generality one may assume that \(\lambda=i\). The class of \(K^r\)-operators contains all bounded non-self-adjoint operators, as well as many integral, differential, and other operators.

A \(K^r\)-operator \(A\) will be called a \(K^r_{\mathrm I}\)-operator if \(D_A=D_{A^*}\); a \(K^r_{\mathrm{II}}\)-operator if \(\overline{G}_A=H\), and a \(K^r_{\mathrm{III}}\)-operator if \(D_A\ne D_{A^*}\), \(\overline{G}_A\ne H\). We note that the class of \(K^r_{\mathrm{II}}\)-operators coincides with the class of quasi-self-adjoint extensions \((^3)\) of symmetric operators with equal defect numbers.

The largest invariant subspace \((^4)\) of the operator \(A\), in which \(A=A^*\), will be denoted by \(H_A\) and called the supplementary subspace of the operator \(A\). The operator \(A_p\), coinciding with \(A\) on the manifold \((H\ominus H_A)\cap D_A\), will be called the simple part of the operator \(A\). The operator \(A\) is called simple if \(A=A_p\).

2. Let \(r=1\). Consider a certain \(K^1\)-operator \(A\) and put
\[ B=iR_{-i}-iR_{-i}^*-2R_{-i}^*R_{-i},\qquad R_{-i}=(A+iI)^{-1}. \]
The operator \(B\) may be represented in the form
\[ Bf=J(f,g)g\quad (f\in H), \]
where \(J=\pm1\), \(g\in\mathfrak N_{-i}=H\ominus\Delta_{A_0}(-i)\), \(\Delta_{A_0}(-i)=(A_0+iI)G_A\).

The function
\[ \omega_A(\lambda)=1-(1-\lambda i)\bigl((A^*-iI)(A^*-\lambda I)^{-1}g,g\bigr)J \]
will be called characteristic for the operator \(A\).

Theorem 1. Simple \(K^1\)-operators \(A_1\) and \(A_2\) are unitarily equivalent if and only if their characteristic functions coincide.

In the upper half-plane the function \(\omega_A(\lambda)\) satisfies the inequality
\[ \omega_A(\lambda)J\overline{\omega_A(\lambda)}<J. \]
Using this property and the known representation of a bounded function regular in the unit disk, one can show,

that \(\omega_A(\lambda)\) can be represented in the form

\[ \omega_A(\lambda)=\prod_{k=1}^{N}\frac{L(k)-\lambda}{\overline{L(k)}-\lambda}\,\gamma(k)\, \exp\left[i\int_{0}^{\nu}\frac{1+\lambda\alpha(t)}{\alpha(t)-\lambda}\,dt\right]\exp[i\lambda]\mu, \]

where \(N\leq\infty\); \(\nu<\infty\); \(\mu\geq0\);

\[ L(k)=\alpha_k+\frac{i}{2}\beta_k^2J;\qquad \gamma(k)=\frac{\overline{L(k)}-i}{L(k)-i}\,|L(k)-i|. \]

\(\{\alpha_k\}\), \(\{\beta_k\}\) are certain sequences of real numbers; \(\alpha(t)\) is a nondecreasing numerical function.

Let \(H_1=l_2,\ H_2=L_2(0,\nu),\ H_3=L_2(0,\mu)\).

Define in \(H_k\) \((k=1,2,3)\) the operators \(A_k\) as follows:

\[ A_1f_1=\left\{f_1(k)L(k)+i\sum_{j=k+1}^{N} f_1(j)\beta(j)J\beta(k)\right\}\qquad (f\in H_1), \]

\[ A_2f_2=\alpha(x)f_2(x)+i\int_x^\nu f_2(t)\gamma(t)J\overline{\gamma(x)}\,dt\qquad (f_2\in H_2), \]

where \(\gamma(x)=\alpha(x)+i\); moreover, we shall assume that \(f_2\in D_{A_2}\) if \(A_2f_2\in H_2\), and

\[ \int_0^\nu f_2(t)\gamma(t)\,dt<\infty. \]

Further,

\[ A_3f_3=\frac{1}{i}\frac{df_3(x)}{dx}, \]

where \(f_3\in D_{A_3}\) if:

1) \(f_3(x)\) is absolutely continuous on \([0,\mu]\); 2) \(f_3(x), f_3'(x)\in H_3\); 3) \(f_3(\mu)=0\).

Consider now in \(H=H_1\oplus H_2\oplus H_3\) the manifold \(D_A\), consisting of vectors \(f\) of the form

\[ P_1f=f_1+\left[\frac{\sqrt2}{2}i\int_0^\nu f_2(t)\gamma(t)\,dt-\frac12 f_3(0)e^{J\nu}\right]g, \tag{1} \]

\[ P_2f=f_2(x)-\frac12 f_3(0,\,-\nu+x)\frac{\alpha(x)-i}{\alpha(x)+i},\qquad P_3f=f_3(x), \]

where \(f_k\in D_{A_k}\ (k=1,2,3)\);

\[ g=\left\{\frac{\beta_k}{L(k)+i}\prod_{j=k+1}^{N}\frac{\overline{L(j)}+i}{L(j)+i}\,\gamma(j)\right\}; \]

\(P_k\) is the projection operator onto \(H_k\) \((k=1,2,3)\).

On the manifold \(D_A\) define the operator \(A\) as follows: if the vector \(f\) is determined by the equalities (1), then

\[ P_1Af=A_1f_1-i\left[\frac{\sqrt2}{2}i\int_0^\nu f_2(t)\gamma(t)\,dt-\frac12 f_3(0)e^{J\nu}\right]g, \]

\[ P_2Af=A_2f_2+\frac{i}{2}Jf_3(0)e^{J(\nu+x)}\frac{\alpha(x)-i}{\alpha(x)+i},\qquad P_3Af=A_3f_3. \]

We shall call the operator \(A\) a triangular model of the \(K^1\)-operator \(A\).

The following assertions hold:

1) The operator \(A\) is a \(K^1\)-operator.

2) The characteristic function of the operator \(A\) coincides with the characteristic function of the operator \(A\).

3) The nonreal spectrum of the operator \(A\) consists of the set of numbers \(L(k)\), which are eigenvalues of the operator \(A\); the real spectrum coincides with the set of values assumed by the function \(\alpha(x)\).

From Theorem 1 and assertion 2) it follows:

Theorem 2. For every \(K^1\)-operator \(A\) there exists an isometric operator \(V\), mapping \(H \ominus H_A\) onto \(\mathbf H \ominus \mathbf H_A\) one-to-one and such that the simple part \(A_p\) of the operator \(A\) is carried under this mapping into the simple part \(\mathbf A_p\) of its triangular model.

We note that the operators \(A_1, A_2, A_3\), by means of which the operator \(\mathbf A\) is defined, are \(K^1\)-operators. Here the operators \(A_1\) and \(A_2\) outwardly resemble the discrete and continuous parts of the triangular model of bounded non-self-adjoint operators (1). If in equality (1) \(\mu=0\) and

\[ \sum_{k=1}^{N}\beta_k+\int_0^\nu |\alpha(x)|^2\,dx<\infty, \]

then the operator \(A\) is a \(K^1_1\)-operator and can be transformed to the same form as that of the triangular model for bounded operators. Conversely, if \(A\) is a \(K^1_1\)-operator, then in equality (1) \(\mu=0\) and

\[ \sum_{k=1}^{N}\beta_k^2+\int_0^\nu |\alpha(x)|^2\,dx<\infty. \]

Consequently, the triangular model for \(K^1_1\)-operators coincides (in outward form) with the model for bounded operators.

An operator \(A\), defined in \(H=H_1\oplus H_2\), will be called a coupling of the operators \(A_1\) and \(A_2\), acting in \(H_1\) and \(H_2\), respectively, if \(A|_{D_A\cap H_1}=A_1\), \([A^*|_{D_{A^*}\cap H_2}]^*=A_2\). The operator considered earlier is a coupling of the operators \(A_1, A_2, A_3\). In the general case, the triangular model of a \(K^r\)-operator \(A\) \((r<\infty)\) is a coupling of the operators \(A_1,A_2,A_3,A_4\), which are defined in certain (specified) spaces of vector-functions as follows:

\[ A_1 f_1=\left\{ f_1(k)L(k)+i\sum_{j=k+1}^{N} f_1(j)\sigma(j)J\sigma^*(k)\right\},\qquad A_2 f_2=B_2^{-1}Q_2JB_2^{-1}f_2, \]

where \(Jf_2=f_2(x)J\),

\[ Q_2 f_2=\frac{1}{i}\frac{df_2(x)}{dx}\quad (0\leq x\leq \mu), \]

and \(f_2\in D_{Q_2}\) if:

1) \(f_2(x)\) is absolutely continuous on \([0,\mu]\);
2) \(f_2(x), f_2'(x)\in H_2\);
3) \(f_2(\mu)=0\).

The operator \(B_2\) is defined by the equality \(B_2 f_2=f_2(x)P(x)\); moreover, the operator \(A_2\) is considered in the subspace \(\widetilde H_2=H_2\ominus H_0\), where \(H_0\) is the largest subspace annihilating the operator \(B_2\).

The operator \(A_3\) is defined by the equality

\[ A_3 f_3=\alpha(x)f_3(x)+i\int_x^\nu f_3(t)\gamma(t)\omega(t)\omega^{-1}(x)J\gamma^*(x)\,dt. \]

The operator \(A_4\) is defined in the same way as the operator \(A_2\).

The elements by means of which the operators \(A_k\) \((k=1,2,3,4)\) are defined are taken from the multiplicative representation of the characteristic matrix-function of the \(K^r\)-operator \(A\) for which the model is constructed.

For lack of space we are unable to give the definition of the characteristic matrix-function and to give a complete description of the triangular model for an arbitrary \(K^r\)-operator \((r<\infty)\). We note, however, that in the general case the definition of the characteristic matrix-function differs little from the case \(r=1\). The multiplicative representation mentioned may be obtained easily by using the main theorem of V. P. Potapov (5). For every \(r<\infty\) the same assertions hold as in the case \(r=1\).

It is interesting to note that in the triangular model for \(K^1\)-operators, in contrast to the triangular model for bounded operators, there appears—

a substantially new element—the differentiation operator. In the general case, two essentially new operators appear in the triangular model \((A_2\) and \(A_4)\), reminiscent of the differentiation operator. The presence of two, and not one, such operators for \(r > 2\) is connected with the nonpermutability of the matrices entering into the multiplicative representation of the characteristic matrix-function for \(K^r\)-operators.

In conclusion I take the opportunity to express my deep gratitude to my supervisor, Prof. M. S. Livshits.

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
19 VIII 1957

References

1. M. S. Livshits, DAN, 84, No. 5 (1952).
2. M. S. Livshits, Matem. sborn., 34 (76), 1 (1954).
3. N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators, 1950, p. 394.
4. M. A. Naimark, Izv. AN SSSR, ser. matem., 4, No. 1, 56 (1940).
5. V. P. Potapov, Tr. Moskovsk. matem. obshch., 4 (1955).