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UDC 513.88 + 517.948
MATHEMATICS
E. R. TSEKANOVSKII
GENERALIZED EXTENSIONS OF UNBOUNDED OPERATORS
(Presented by Academician A. N. Kolmogorov on 27 III 1965)
I. Let \(H_0\) be a complete Hilbert space with scalar product \((f,g)_0\) and norm \(\|f\|_0\). Suppose that in \(H_0\) there is an everywhere dense linear set \(H_+\), which is a complete Hilbert space with respect to another scalar product \((f,g)_+\) and norm \(\|f\|_+\). We shall assume that
\[ \|f\|_0 \leq \|f\|_+ \qquad (f \in H_+). \]
We shall call the space \(H_+\) a space with positive* norm, and also a space of basic elements. We shall say that every antilinear functional \(a(f)\) on \(H_+\) is generated by a generalized element \(a\), and we shall write the value of the functional on the element \(f \in H_+\) as \(a(f) = (a,f)_0\). In what follows we use the notation \(\overline{(a,f)_0} = (f,a)_0\). Obviously, the totality of all generalized elements is a linear set. Since the space \(H_+\) is Hilbert, every antilinear functional on \(H_+\) can be represented in the form of a scalar product defined in \(H_+\), i.e.,
\[ (a,f)_0 = (a^*,f)_+ = (y_a,f)_+ \qquad (a^* = y_a,\ a^* \in H_+,\ f \in H_+). \tag{1} \]
Equality (1) generates a linear operator \(J\), mapping the set of generalized elements into the space \(H_+\). The range of the operator is, obviously, all of \(H_+\).
We introduce in the set of generalized elements \(H_-\) a scalar product, putting
\[ (a,\beta)_- = (y_a,y_\beta)_+ \qquad (a,\beta \in H_-). \tag{2} \]
The set \(H_-\) with the scalar product (2) is a Hilbert space.
It follows from (2) that the operator \(J\) is isometric, mapping \(H_-\) into \(H_+\). Since
\[ \|a\|_- = \sup_{f\in H_+} \frac{|(a,f)_0|}{\|f\|_+}, \]
the inequality holds
\[ \|f\|_- \leq \|f\|_0 \leq \|f\|_+. \]
Thus,
\[ H_+ \subseteq H_0 \subseteq H_-. \]
We shall call the space \(H_-\) a space with negative norm, and also the space of generalized elements of the Hilbert space \(H_0\). It is not difficult to prove that \(H_+\) is dense in \(H_-\).
II. Let \(\Omega(f,g)\) be a bilinear functional defined in \(H_+\). Then it is easy to show that
\[ \Omega(f,g) = (Bf,g)_0 \qquad (f,g \in H_+), \tag{3} \]
* In the exposition of Sec. I we mainly follow the paper of Yu. M. Berezanskii (²).
where \(B\) is a bounded linear operator acting from \(H_+\) into \(H_-\), which is uniquely determined by the bilinear functional \(\Omega\).
Let \(B\) be an arbitrary bounded linear operator acting from \(H_+\) into \(H_-\). The expression \((f, Bg)_0\) \((f, g \in H_+)\) is, obviously, a bilinear functional in \(H_+\). Then, according to (3), there is a uniquely determined bounded linear operator \(B^\times\), mapping \(H_+\) into \(H_-\), for which
\[ (f, Bg)_0 = (B^\times f, g)_0 \qquad (f, g \in H_+). \]
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.
III. Let \(T\) be a closed operator acting in the Hilbert space \(H_0\), for which
\[ T_0 \subset T, \qquad T^* \subset T_0^*, \]
where \(T_0\) is a symmetric operator with dense domain of definition.* Consider the Hilbert space \(H_+ = D_{T_0^*}\) with scalar product
\[ (f, g)_+ = (T_0^* f, T_0^* g)_0 + (f, g)_0 \qquad (f, g \in H_+). \]
Construct the triple of spaces \(H_+ \subseteq H_0 \subseteq H_-\). We now pose the following problem: can the operators \(T\) and \(T^*\) be extended linearly to all of \(H_+ = D_{T_0^*}\) so that the obtained extensions \(T_{H_+}\) and \(T^\times_{H_+}\) are adjoint to one another in the generalized sense? It turns out that this can always be done; moreover, the obtained generalized extensions \(T_{H_+}\) and \(T^\times_{H_+}\) will be linear bounded operators acting from \(H_+\) into \(H_-\), and, in contrast to \(T\) and \(T^*\), will have the same domain of definition \(H_+\). In the paper a number of properties of generalized extensions are established, their resolvent is investigated, and an estimate for its norm in the space \(H_-\) is found.
IV. Theorem 1. If the operator \(T\) belongs to the class \(\Omega\), then \(T\) and \(T^*\) can always be extended to all of \(H_+ = D_{T_0^*}\) so that the obtained extensions \(T_{H_+}(H_+ \to H_-)\) and \(T^\times_{H_+}(H_+ \to H_-)\) are adjoint to one another in the generalized sense.
Definition. A linear extension \(T_{H_+}\) of the operator \(T\) to \(H_+ = D_{T_0^*}\) will be called generalized if \(T^\times_{H_+}\) is an extension of \(T^*\) to \(H_+\). It follows from Theorem 1 that if the operator \(T\) belongs to the class \(\Omega\), then it has a nontrivial generalized extension \(T_{H_+}\), which, as examples show, is not determined uniquely by the operator. However, the following holds.
Theorem 2. Let \(T_{H_+}\) and \(T'_{H_+}\) be generalized extensions of the operator \(T\) to the space \(H_+ = D_{T_0^*}\), and let
\[ T_{H_+} = A + iB; \qquad T'_{H_+} = A' + iB', \]
where \(A, B, A', B'\) are linear generalized self-adjoint operators acting from \(H_+\) into \(H_-\). Then, if \(B = B'\), then \(A = A'\).
We shall regard the generalized extension of the operator \(T\) as an operator acting in the Hilbert space \(H_-\). This operator has in \(H_-\) the dense domain of definition \(H_+\).
Theorem 3. If \(\lambda\) is a regular point of an operator \(T\) belonging to the class \(\Omega\), then it is a point of regular type for any generalized extension \(T_{H_+}(H_+ \to H_-)\) of the operator \(T\).
* An operator \(T\) possessing the indicated properties will henceforth be assigned to the class \(\Omega\).
** In the theorem the existence is proved of a nontrivial generalized extension, different from that which is obtained if \(T\) and \(T^*\) are extended by zero.
Corollary 1. If \(\widetilde T_{H_+}\) is an arbitrary generalized extension of the operator \(T\), then at every point \(\lambda\) regular for \(T\) there exists the resolvent \((T_{H_+}-\lambda I)^{-1}\), whose closure is a bounded operator defined everywhere in \(H\), and for which
\[ \left\|\overline{(T_{H_+}-\lambda I)^{-1}}\right\| \le (1+|\lambda|)\left(1+\left\|(T-\lambda I)^{-1}\right\|_0\right). \tag{4} \]
Corollary 2. If the number \(\lambda\) is not a point of regular type for some generalized extension \(T_{H_+}\), then it belongs to the spectrum of the operator \(T\).
Corollary 3. If the operator \(T\) has no spectrum in the finite part of the plane, then every number \(\lambda\) is a point of regular type for every generalized extension \(T_{H_+}\), and, consequently, in this case there exists the resolvent \((T_{H_+}-\lambda I)^{-1}\), whose closure is a bounded operator in \(H\), and for which (4) holds.
I express my gratitude to Prof. M. S. Livshits and Prof. Yu. M. Berezanskii for valuable critical comments.
Kharkov Institute of Mining Machine Building,
Automation, and Computer Engineering
Received
7 IX 1964
CITED LITERATURE
- M. S. Livshits, Matem. sbornik, 19 (61), 239 (1946).
- Yu. M. Berezanskii, Uspekhi Mat. Nauk, 18, no. 1 (109) (1963).
- M. G. Krein, Collected Works of the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, no. 9 (1947).
- G. I. Kats, Ukrainian Mathematical Journal, 12, no. 1 (1960).
- E. R. Tsekanovskii, Doklady Akademii Nauk, 139, no. 1 (1961).