UDC 513.88+517.948

MATHEMATICS

E. R. TSEKANOVSKII

ON THE RESOLVENT OF GENERALIZED SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS

(Presented by Academician L. S. Pontryagin on 3 V 1967)

In the present note a number of theorems on the resolvent of generalized self-adjoint extensions are considered.

I. Consider a triple of spaces \(G_{+}\subseteq G_{0}\subseteq G_{-}\), and let \(B\) be an arbitrary generalized self-adjoint operator acting from \(G_{+}\) into \(G_{-}\). We shall regard it as an operator in the space \(G_{-}\) with dense domain \(G_{+}\). Note that in the space \(G_{-}\) the operator \(B\) is, generally speaking, nonsymmetric. It is easy to see that all eigenvalues of the operator \(B\) \((Bf=\lambda f,\ f\in G_{+})\) are real. Denote \(\Delta_B(\lambda)=(B-\lambda I)G_{+}\).

Theorem 1. The number \(\lambda\) is an eigenvalue of the generalized self-adjoint operator \(B(G_{+}\to G_{-})\) if and only if
\[ \overline{\Delta_B(\overline{\lambda})}\ne G_{-}. \]

Proof. Let \(\lambda\) be an eigenvalue of \(B\), so that \(Bf=\lambda f\) \((f\ne0)\). In that case, for any \(g\in G_{+}\),
\[ (J^{-1}f,(B-\lambda I)g)_{-}=(f,J(B-\lambda I)g)_{+}= \]
\[ =(f,(B-\lambda I)g)_0=(Bf-\overline{\lambda}f,g)_0=0, \]
and, consequently, the vector \(J^{-1}f\ne0\) is orthogonal to \(\Delta_B(\lambda)\), which is possible only when
\[ \overline{\Delta_B(\lambda)}\ne G_{-}. \]

Now suppose that
\[ \overline{\Delta_B(\overline{\lambda})}\ne G_{-}. \]
In that case there exists a vector \(\alpha\in G_{-}\) orthogonal to the manifold \(\Delta_B(\lambda)\). Therefore, for any \(g\in G_{+}\),
\[ (\alpha,(B-\lambda I)g)_{-}=(J\alpha,J(B-\lambda I)g)_{+}= \]
\[ =(J\alpha,(B-\lambda I)g)_0=((B-\overline{\lambda}I)J\alpha,g)_0=0. \]
It follows from this that the vector \(f=J\alpha\ne0\) \((f\in G_{+})\) is an eigenvector of the operator \(B\), corresponding to the eigenvalue \(\lambda\) \((\lambda=\overline{\lambda})\). The theorem is proved.

Corollary. If \(\lambda\) is a nonreal number, then
\[ \overline{\Delta_B(\overline{\lambda})}=G_{-}. \]

II. Let \(A\) be a symmetric operator with defect index \((r,r)\) \((r<\infty)\), acting in \(G_0\). Consider the Hilbert space \(G_{+}=D_{A^*}\) with scalar product
\[ (f,g)_{+}=(A^*f,A^*g)_0+(f,g)_0\qquad (f,g\in D_{A^*}) \]
and construct, as was done in \((^{1,3})\), a triple of spaces \(G_{+}\subseteq G_{0}\subseteq G_{-}\). In \((^5)\) it was shown that \(A\) can be extended to \(G_{+}=D_{A^*}\) in such a way that the resulting extension \(A_{G_{+}}\ (G_{+}\to G_{-})\) is a generalized self-adjoint operator.

Theorem 2. If \(\lambda\) is a nonreal number and \(A_{G_{+}}\) is an arbitrary generalized self-adjoint extension of a symmetric operator with defect index \((r,r)\) \((r<\infty)\), then \(R_\lambda=(A_{G_{+}}-\lambda I)^{-1}\) continuously maps

maps the Hilbert space \(G_-\) onto the Hilbert space \(G_0\), and, moreover,

\[ \|R_\lambda\|\leqslant \frac{\sqrt{2}}{\sin\varphi_\lambda}(1+|\lambda|) \left(1+\frac{1}{|\operatorname{Im}\lambda|}\right) \quad (\varphi_\lambda\ne 0), \tag{1} \]

where \(\varphi_\lambda\) is the minimal angle between certain subspaces.

We outline the proof of this theorem. Denote
\[ \Delta_{A_{G_+}}(\lambda)=(A_{G_+}-\lambda I)G_+. \]
Then, if \(a=(A_{G_+}-\lambda I)f\) \((f\in G_+,\, a\in G_-)\), then

\[ \|a\|_-\geqslant |\operatorname{Im}\lambda|\,\|f\|_0^2/\|f\|_+. \tag{2} \]

Further,

\[ \|a\|_-=\sup_{\varphi\in G_+}\frac{|(\varphi,a)_0|}{\|\varphi\|_+} =\sup_{\varphi\in G_+}\frac{|(\varphi,(A_{G_+}-\lambda I)f)_0|}{\|\varphi\|_+} \geqslant \sup_{\psi\in D_A} \frac{|((A-\bar{\lambda}I)\psi,f)_0|} {\|A\psi\|_0+\|\psi\|_0}. \tag{3} \]

It follows that

\[ \|a\|_-\geqslant \sup_{\psi\in D_A} \frac{|((A-\bar{\lambda}I)\psi,f)_0|} {(1+|\lambda|)\left(1+\dfrac{1}{|\operatorname{Im}\lambda|}\right)\|(A-\bar{\lambda}I)\psi\|_0}. \]

Denote
\[ \Delta_A(\bar{\lambda})=(A-\bar{\lambda}I)D_A. \]
Obviously, \(\Delta_A(\bar{\lambda})\) is a subspace in \(G_0\). Then

\[ G_0=\Delta_A(\bar{\lambda})+\Delta_A(\lambda)^\perp. \tag{4} \]

As is known, \(\Delta_A(\bar{\lambda})=\mathfrak N_\lambda\), where \(\mathfrak N_\lambda\) is the eigenspace of the operator \(A^*\) corresponding to the eigenvalue \(\lambda\). From (2) and (3) it follows that

\[ \|a\|_-\geqslant \frac{1}{(1+|\lambda|)(1+1/|\operatorname{Im}\lambda|)}\,\|Pf\|_0, \tag{5} \]

where \(P\) is the projection operator onto \(\Delta_A(\bar{\lambda})\). From relations (2), (3), (4), and (5) it follows that

\[ \|(A_{G_+}-\lambda I)Pf\|_-\geqslant \frac{1}{(1+|\lambda|)(1+1/|\operatorname{Im}\lambda|)}\,\|Pf\|_0; \tag{6} \]

\[ \|(A_{G_+}-\lambda I)Qf\|_-\geqslant \frac{|\operatorname{Im}\lambda|}{\sqrt{|\lambda|^2+1}}\,\|Qf\|_0. \tag{7} \]

Here \(Q\) is the projection operator onto \(\mathfrak N_\lambda\). Denote

\[ G_-^{(1)}=(A_{G_+}-\lambda I)Pf,\qquad G_-^{(2)}=(A_{G_+}-\lambda I)Qf \quad (f\in G_+). \]

Then from (4)

\[ \Delta_{A_{G_+}}(\lambda)=G_-^{(1)}+G_-^{(2)}. \]

Denote by \(\cos\varphi_\lambda\) the cosine of the minimal angle \((^2)\) between the closures in \(G_-\) of the linear manifolds \(G_-^{(1)}\) and \(G_-^{(2)}\); taking into account relations (6) and (7), we obtain

\[ \|(A_{G_+}-\lambda I)f\|_-^2 \geqslant \frac{1-\cos^2\varphi_\lambda}{2}\, \frac{1} {(1+|\lambda|)^2\left(1+\dfrac{1}{|\operatorname{Im}\lambda|}\right)^2} \|f\|_0^2. \]

Since, by virtue of the corollary to Theorem 1, \(\overline{\Delta}_{A_{G_+}}(\lambda)=G_-\), relation (1) follows from the last relation.

Theorem 3. Every generalized self-adjoint extension \(A_G\) of a symmetric operator \(A\) with defect index \((r,r)\) \((r<\infty)\) does not admit a closure as an operator acting from \(G_0\) into \(G_-\).*

* Consequently, a fortiori, \(A_{G_+}\) does not admit a closure as an operator acting in \(G_-\) and having dense domain of definition \(G_+\).

Proof. We shall show that there exists a sequence \(f_n\) \((f_n \in G_+,\, n=1,2,\ldots)\), converging to zero in the metric \(G_0\), and such that \(A_{G_+}f_n\) converges in the metric \(G_-\), with
\[ \lim_{n\to\infty} A_{G_+}f_n \ne 0. \]

In (5) it was shown that every generalized self-adjoint extension \(A_{G_+}\) of the operator \(A\) has the form
\[ A_{G_+} f = A^* f + \sum_{k,j=1}^{r} \bigl[ a_{jk}(f,\hat e_j)_0 + b_{jk}(f,\hat q_j)_0 \bigr]\hat g_k + \sum_{k,j=1}^{r} \bigl[ c_{jk}(f,\hat e_j)_0 + d_{jk}(f\hat g_j)_0 \bigr]\hat e_k, \]
where the coefficient matrices satisfy the relations
\[ D=A^*,\qquad c_{kj}=\overline{c}_{jk},\qquad b_{kj}=\overline{b}_{jk}\quad (k\ne j), \]
\[ \operatorname{Im} c_{jj}=-\frac12,\qquad \operatorname{Im} b_{jj}=\frac12. \]

Let
\[ f_n=f_A^{(n)}+e_j, \]
where, in the metric \(G_0\),
\[ f_A^{(n)}\to(-e_j),\qquad f_A^{(n)}\in D_A \]
and
\[ e_j\in\mathfrak N_i,\qquad e_j=J\hat e_j \]
(see (5)). Thus \(f_n\to0\) in the metric \(G_0\). It can be shown that
\[ \alpha_n=A_{G_+}f_n \]
will converge in the metric \(G_-\).

Further,
\[ (\alpha_n,g)_0 = (f_A^{(n)},A^*g)_0 = i(e_j,g)_0 + \sum_{k=1}^{r} a_{jk}(\hat g_k,g)_0 + \sum_{k=1}^{r} c_{jk}(\hat e_k,g)_0. \]

Let
\[ \alpha=\lim_{n\to\infty}\alpha_n. \]
Then
\[ (\alpha,g)_0 = i(e_j,g)_0 - (e_j,A_g^*g)_0 + \sum_{k=1}^{r} a_{jk}(\hat g_k,g)_0 + \sum_{k=1}^{r} c_{jk}(\hat e_k,g)_0. \]

Analyzing the last relation, one can establish,* that
\[ \|\alpha\|_-=\sup_{\|g\|_+\le 1}|(\alpha,g)_0|\ne0. \]

Thus, we have indicated a sequence \(f_n\in G_+\) which tends to zero in the metric \(G_0\), while \(\alpha_n=A_{G_+}f_n\) does not tend to zero in the metric \(G_-\).

Theorem 4*. Let \(\widetilde A\) be a self-adjoint extension of a symmetric operator \(A\), and let \(A_{G_+}\) be a generalized self-adjoint extension of the operator \(A\) which is also an extension of \(\widetilde A\). Then, if \(\lambda\) is a regular point for \(\widetilde A\), then
\[ R_\lambda=(A_{G_+}-\lambda I)^{-1} \]
maps \(G_-\) continuously into \(G_0\).

We outline the proof of this theorem. Denote
\[ \|\alpha\| = \sup \frac{|(\varphi,(A_{G_+}-\lambda I)f)_0|}{\|\varphi\|_+} \ge \sup_{\psi\in D_{\widetilde A}} \frac{|((\widetilde A-\overline\lambda I)\psi,f)_0|} {\|\widetilde A\psi\|_0+\|\psi\|_0} \ge \]
\[ \ge \sup_{\psi\in D_{\sim}} \frac{|((\widetilde A-\overline\lambda I)\psi,f)_0|} {(1+|\lambda|)\bigl(1+\|(\widetilde A-\overline\lambda I)^{-1}\|_0\bigr)\|(\widetilde A-\overline\lambda I)\psi\|_0} = \]
\[ = \frac{1} {(1+|\lambda|)\bigl(1+\|(\widetilde A-\overline\lambda I)^{-1}\|_0\bigr)} \,\|f\|_0. \]

\[ \text{* In this theorem it is not required that the operator } A \text{ have finite defect numbers.} \]

It can be shown that if \(f\) is an arbitrary vector from \(G_0\), then in the range of values of \((A_{G_+}-\lambda I)\) there exists a sequence \(\alpha_n\), convergent in \(G_-\), for which \((A_{G_+}-\lambda I)^{-1}\alpha_n \to f\) in the metric of \(G_0\). Hence, and from the indicated inequalities, the assertion of the theorem follows.

Let us note, in conclusion, that theorems analogous to Theorems 2 and 3 can also be obtained for operators with unequal deficiency indices.

Donetsk State
University

Received
24 IV 1967

REFERENCES

\(^{1}\) Yu. M. Berezanskii, UMN, 18, 1 (1963).
\(^{2}\) I. Ts. Gokhberg, M. G. Krein, Introduction to the Theory of Non-Self-Adjoint Operators, “Nauka,” 1966.
\(^{3}\) E. R. Tsekanovskii, DAN, 165, No. 1 (1965).
\(^{4}\) E. R. Tsekanovskii, Mat. sbornik, 68 (110), 4 (1965).
\(^{5}\) E. R. Tsekanovskii, DAN, 178, No. 6 (1968).