UDC 513.88+517.948

MATHEMATICS

E. R. Tsekanovskii

GENERALIZED SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS

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

In the present paper a new kind of extension of symmetric operators is introduced and studied. It turns out that any symmetric operator \(A\) can always be extended to the domain of definition of \(A^*\) in such a way that the extension obtained is symmetric with respect to a certain generalized “scalar product.” A number of properties of such extensions are established, and their classification and description are given. This makes it possible to characterize (for operators with finite and equal deficiency indices) all ordinary symmetric and self-adjoint extensions in terms of abstract boundary conditions, where the boundary conditions are a certain combination of generalized elements (functionals) “orthogonal” to the domain of definition of the symmetric operator*.

Let \(G_0\) be a certain complete Hilbert space with scalar product \((f,g)_0\). Suppose that in \(G_0\) there is an everywhere dense linear set \(G_+\), which is a complete Hilbert space with respect to another scalar product \((f,g)_+\). We shall assume that \(\|f\|_0 \leq \|f\|_+\) \((f \in G_+)\). The space \(G_+\), as is known \((^1)\), is called a space with positive norm, and also a space of basic elements. We shall say that every antilinear functional \(a(f)\) on \(G_+\) is generated by a generalized element \(a\), and we shall write \(a(f) \equiv (a,f)_0\) \((f \in G_+)\). In what follows we shall use the notation \(\overline{(a,f)}_0 = (f,a)_0\). Obviously, the totality \(G_-\) of all generalized elements is a linear set.

It is easy to see that

\[ (a,f)_0 = (Ja,f)_0 \qquad (f \in G_+, \ a \in G_-). \tag{1} \]

Equality (1) generates a linear operator \(J\) mapping the set \(G_-\) onto the space \(G_+\). Introduce in \(G_-\) a scalar product by setting

\[ (\alpha,\beta)_- = (J\alpha,J\beta)_+ \qquad (\alpha,\beta \in G_-). \]

The operator \(J\), obviously, is an isometric operator mapping \(G_-\) onto \(G_+\). Thus, \(G_+ \subseteq G_0 \subseteq G_-\). The space \(G_-\) will be called a space with negative norm, and also the space of generalized elements of the Hilbert space \(G_0\). It is not difficult to show \((^1)\) that \(G_+\) is dense in \(G_-\).

Let \(\Omega(f,g)\) be a bilinear functional in \(G_+\). Then, as is known \((^{2,3})\),

\[ \Omega(f,g) = (Bf,g)_0 \qquad (f,g \in G_+), \tag{2} \]

where \(B\) is a bounded linear operator acting from \(G_+\) into \(G_-\), which is uniquely determined by the functional \(\Omega\).

Now let \(B\) be an arbitrary bounded linear operator acting from \(G_+\) into \(G_-\). The expression \((f,Bg)_0\) \((f,g \in G_+)\) is, obviously, a bilinear functional in \(G_+\). Then, according to (2), uniquely

* In a somewhat different form, abstract boundary conditions occur in \((^4)\).

there exists a bounded linear operator \(B^{\times}\), also mapping \(G_+\) into \(G_-\), for which

\[ (f,Bg)_0=(B^{\times}f,g)_0\qquad (f,g\in G_+). \]

We shall call the operator \(B^{\times}\) the generalized adjoint operator with respect to \(B\). If \(B=B^{\times}\), then such an operator will be called a generalized self-adjoint operator.

Let \(A\) be a symmetric operator with dense domain, acting in the Hilbert space \(G_0\). In the linear set \(G_+=D_{A^*}\) introduce the scalar product

\[ (f,g)_+=(A^*f,A^*g)_0+(f,g)_0\qquad (f,g\in G_+). \tag{3} \]

Since \(A^*\) is closed, \(G_+\) is a complete Hilbert space. Construct the triple of spaces \(G_+\subseteq G_0\subseteq G_-\). According to von Neumann’s formulas,

\[ D_{A^*}=D_A+\mathfrak N_\lambda+\mathfrak N_{\bar\lambda}. \]

Theorem 1. In order that the subspaces \(D_A,\mathfrak N_\lambda,\mathfrak N_{\bar\lambda}\) be pairwise orthogonal in \(G_+\), it is sufficient that \(\lambda=\pm i\).

In what follows we shall use the representation of \(G_+\) in the form

\[ G_+=D_A\oplus \mathfrak N_i\oplus \mathfrak N_{-i}. \tag{4} \]

Denote \(\mathfrak M=\mathfrak N_i\oplus\mathfrak N_{-i}\). Every vector \(f\in D_{A^*}\), by virtue of (4), is uniquely represented in the form

\[ f=f_A+f_{\mathfrak N}\qquad (f_A\in D_A,\quad f_{\mathfrak N}\in \mathfrak M). \]

Theorem 2. In order that the symmetric operator \(A\) can be extended to the space \(G_+=D_{A^*}\) so that the resulting extension \(A_{G_+}\) \((G_+\to G_-)\) is a generalized self-adjoint operator, it is necessary and sufficient that there exist a linear operator \(P(\mathfrak M\to G_-)\) for which

\[ (Pf_{\mathfrak N},g_A)_0=(f_{\mathfrak N},Ag_A)_0,\qquad (Pf_{\mathfrak N},g_{\mathfrak N})_0=(f_{\mathfrak N},Pg_{\mathfrak N})_0 \]

\[ (f_{\mathfrak N},g_{\mathfrak N}\in\mathfrak M,\quad g_A\in D_A). \]

In this case

\[ A_{G_+}f=Af_A+Pf_{\mathfrak N}\qquad (f=f_A+f_{\mathfrak N},\ f_A\in D_A,\ f_{\mathfrak N}\in\mathfrak M). \]

Relying on Theorem 2, Theorem 3 is proved.

Theorem 3. Every closed symmetric operator \(A\) with dense domain can be extended to the whole space \(G_+=D_{A^*}\) so that the resulting extension \(A_{G_+}\) is a generalized self-adjoint operator acting from \(G_+\) into \(G_-\).

Let now the operator \(A\) have defect index \((r,r)\) \((r<\infty)\). Then, according to (4),

\[ f=f_A+\sum_{j=1}^{r}\xi_j e_j+\sum_{j=1}^{r}\eta_j g_j, \]

where \(\{e_j\}_1^r,\{g_j\}_1^r\) are orthonormal bases in the subspaces \(\mathfrak N_i,\mathfrak N_{-i}\). Introduce the notation

\[ \hat e_j=J^{-1}e_j,\qquad \hat g_j=J^{-1}g_j\quad (j=1,2,\ldots,r). \]

It is not difficult to verify that \(\hat e_j,\hat g_j\) \((j=1,2,\ldots,r)\) are generalized elements of the Hilbert space \(G_0\) and are “orthogonal” to \(D_A\), i.e.

\[ (\hat e_j,f_A)_0=(\hat g_j,f_A)_0=0\qquad (f_A\in D_A). \]

Theorem 4. Let \(A\) be a symmetric operator with defect index \((r,r)\) \((r<\infty)\), acting in the Hilbert space \(G_0\). In order

in order that the expression

\[ A_{G_+} f=A^* f+\sum_{j,k=1}^{r}\bigl[a_{jk}(f,\hat e_j)_0+b_{jk}(f,\hat g_j)_0\bigr]\hat g_k+ \tag{5} \]

\[ +\sum_{j,k=1}^{r}\bigl[c_{jk}(f,\hat e_j)_0+d_{jk}(f,\hat g_j)_0\bigr]\hat e_k \]

be a generalized self-adjoint extension of the operator \(A\) on \(G_+=D_{A^*}\), it is necessary and sufficient that the coefficient matrices satisfy the relations

\[ D=A^*,\qquad c_{nm}=\overline{c_{mn}},\qquad b_{nm}=\overline{b_{mn}}\quad (m\ne n), \tag{6} \]

\[ \operatorname{Im} c_{nn}=-\frac{1}{2},\qquad \operatorname{Im} b_{nn}=\frac{1}{2}. \]

On the basis of Theorem 2, one can give a complete description of all generalized self-adjoint extensions of a symmetric operator with equal finite defect numbers.

Theorem 5. Every generalized self-adjoint extension \(A_{G_+}\) of a symmetric operator \(A\) on \(G_+=D_{A^*}\) has the form (5), and the coefficient matrices satisfy the relations (6).

Definition. A generalized self-adjoint extension \(A_{G_+}\) of the operator \(A\) on \(G_+=D_{A^*}\) will be called weak if \(A_{G_+}\) is not an extension of any symmetric extension \(\tilde A\) of the operator \(A\).

Theorem 6. In order that the expression (5) with relations (6) be a weak extension \(A_{G_+}\) of the operator \(A\) on \(G_+=D_{A^*}\), it is necessary and sufficient that the matrix

\[ T=\begin{pmatrix} A & B\\ C & D \end{pmatrix} \]

be nonsingular.

It follows from Theorems 5 and 6 that every symmetric operator with defect index \((r,r)\) \((r<\infty)\) always has a weak generalized self-adjoint extension. The totality of all such extensions is described by expression (5) with conditions (6), where the matrix \(T\) is nonsingular.

Theorems analogous to Theorems 4, 5, and 6 can also be obtained for operators with finite but unequal defect numbers.

Theorem 7. In order that the expression (5), when the conditions (6) are fulfilled, be also an extension of some symmetric extension \(\tilde A\) \((\tilde A^*\ne \tilde A)\) of the operator \(A\), it is necessary and sufficient that

\[ \operatorname{rang}\begin{pmatrix} A & B\\ C & D \end{pmatrix}<r. \tag{7} \]

It follows from this theorem that the domain of definition of an ordinary symmetric extension \(\tilde A\) of the operator \(A\) consists of vectors \(f\in G_+=D_{A^*}\) for which

\[ \sum_{j=1}^{r}\bigl[a_{jk}(f,\hat e_j)_0+b_{jk}(f,\hat g_j)_0\bigr]=0, \]

\[ (k=1,2,\ldots,r) \]

\[ \sum_{j=1}^{r}\bigl[c_{jk}(f,\hat e_j)_0+d_{jk}(f,\hat g_j)_0\bigr]=0 \]

for some coefficient matrices satisfying the conditions (6) and (7).

Theorem 8. In order that the expression (5), when the conditions (6) are fulfilled, be an extension of some self-adjoint extension \(\tilde A\)

of the operator \(A\), it is necessary and sufficient that

\[ \operatorname{rang}\begin{pmatrix} A & B\\ C & D \end{pmatrix}=r. \tag{8} \]

Theorem 8 can be formulated somewhat differently, namely:

Theorem 9. In order that expression (5), under the conditions (6), be an extension of some self-adjoint extension \(\widetilde A\) of the operator \(A\), it is necessary and sufficient that there exist a unitary matrix \(U\) such that

\[ A+UB=0,\qquad C+UD=0. \tag{9} \]

The extensions considered in Theorems 7 and 8 will be called, respectively, medium and strong. From Theorems 8 and 9 there follows a curious result for matrices.

In order that the matrix (8) with conditions (6) have rank \(r\), it is necessary and sufficient that the conditions (9) be satisfied.

I express my gratitude to V. I. Matsayev for a number of valuable suggestions and for discussion of the present work.

Donetsk State
University

Received
26 IV 1967

REFERENCES

\(^{1}\) Yu. M. Berezanskii, UMN, 18, 1 (1963).
\(^{2}\) E. R. Tsekanovskii, DAN, 165, No. 1 (1965).
\(^{3}\) E. R. Tsekanovskii, Matem. sborn., 68 (110), 4 (1965).
\(^{4}\) N. Dunford, J. Schwartz, Linear Operators, Part II.