MATHEMATICS

V. A. SHCHEL’NOV

ON MULTIVALUED LINEAR OPERATORS IN A LOCALLY CONVEX SPACE

(Presented by Academician V. I. Smirnov, 28 XII 1959)

In the present note we consider some properties of general, generally speaking multivalued, linear operators in locally convex separated spaces.

Let (X, Y) be two locally convex spaces. Any subset of the product ((X,Y)) of the spaces (X) and (Y) may be regarded as the graph (\Gamma_A) of some operator (A) with domain (D_A \subset X) and range (R_A \subset Y). We shall call an operator (A) linear if its graph (\Gamma_A) is a linear set. We shall call a linear operator (A) open if it maps every neighborhood of zero in (D_A) into a neighborhood of zero in (R_A). An operator inverse to an open one will be called continuous. We define the closure (\overline{A}) of an operator (A) as the operator whose graph is the closure of the graph of the given operator.

If (M) is a vector subspace in (X), and (X^) is the space conjugate to (X), then by (M^0) we shall denote the vector subspace in (X^) orthogonal to (M). By the operator adjoint to the given linear operator (A) we shall mean the operator (A^) with domain (D_{A^} \subset Y^*) and graph

[
\Gamma_{A^}=\Gamma_{-A}^{0}\subset (X^,Y^*).
]

It is easy to verify that

[
\Gamma_{A^}={(x^,y^)\in (X^,Y^): \langle Ax,y^\rangle=\langle x,x^*\rangle,\ x\in D_A}.
]

Theorem 1. (1^\circ). In order that the operator (A) be open, it is necessary, and if (A^{-1}(0)=\overline{A}^{-1}(0)), also sufficient, that the operator (\overline{A}) be open.

(2^\circ). If the set (R_A) is closed, then (R_A=R_{\overline{A}}).

(3^\circ). If the operator (A) is open and the set (R_A) is closed, then
[
A^{-1}(0)=\overline{A}^{-1}(0).
]

(4^\circ). Let the operator (A) be open and the set (R_A) closed. Then, in order that the operator (A) be closed, it is necessary and sufficient that the set (A^{-1}(0)) be closed.

(5^\circ). In order that the operator (A) be closed, it is necessary and sufficient that
[
R_A=R_{\overline{A}}
\quad\text{and}\quad
A^{-1}(0)=\overline{A}^{-1}(0).
]

Theorem 2.

[
A^(0)=D_A^0,\qquad A^{-1}(0)=R_A^0,
]

[
\overline{A}(0)=D_{A^}^0,\qquad \overline{A}^{-1}(0)=R_{A^}^0.
]

We shall prove the first relation. Obviously,

[
(D_A,Y)=(0,Y)+\Gamma_A.
]

Hence

[
(D_A^o,0)=(X^,0)\cap \Gamma_A^o=(A^(0),0),
]

and the required equality is proved.

Corollaries. 1°. In order that the operator (A) admit a single-valued closure, it is necessary and sufficient that (D_{A^}) be weakly dense in (Y^).

2°. In order that the operator (A^*) be single-valued, it is necessary and sufficient that (D_A) be dense in (X).

Theorem 3. Let (A) be a linear closed operator and (\Gamma_A\subset (X,Y)).

I. In order that the operator (A) be weakly open, it is necessary and sufficient that (R_{A^*}) be weakly closed.

II. Suppose that (X,Y) are Fréchet spaces. Then the following five properties are equivalent:

1) (A) is strongly open;
2) (A) is weakly open;
3) (R_A) is closed;
4) (A^) is weakly open;
5) (R_{A^}) is weakly closed.

III. Let (X) and (Y) be Banach spaces. Then each of the five properties in item II is equivalent to either of the following two:

6) (A^) is strongly open;
7) (R_{A^}) is strongly closed.

The proofs of Theorems 1 and 3 systematically use the results of the monograph ((^1)).

Theorem 3 is proved by reduction to the case of a linear single-valued continuous operator defined on the whole space. For this purpose one considers the linear single-valued continuous operator (V), defined on (X) as follows:

[
V(x)=\varphi(x,0),
]

where (\varphi) is the canonical mapping of the space ((X,Y)) onto the quotient space ((X,Y)/\Gamma_A). Next, the connections between the properties of the operators (A,A^) and (V,V^) are investigated. The theorem follows from the corresponding assertions for the operator (V).

Remark 1. Applying Theorems 1 and 3 to the operator (A^{-1}) is equivalent to replacing in these theorems the sets (A^{-1}(0),\overline{A}^{-1}(0),R_A,R_{\overline A},R_{A^}), respectively, by the sets (A(0),\overline A(0),D_A,D_{\overline A},D_{A^}), and the word “open” by the word “continuous.”

Remark 2. Taking into account Theorems 1 and 2, one can obtain some assertions of Theorem 3 for an arbitrary (not necessarily closed) linear operator.

Theorem 4. Let (X) be a Fréchet space; (M) and (N) vector subspaces in (X); (M_1) and (N_1) vector subspaces in (X^*).

1°. If the sum (M+N) is closed, then the sum (M^o+N^o) is weakly closed.
2°. If the sum (M_1+N_1) is weakly closed, then the sum (M_1^o+N_1^o) is closed.
3°. If (X) is a Banach space, then, in order that the sum (M^o+N^o) be weakly closed, it is necessary and sufficient that it be strongly closed.

For the proof of the theorem it is enough to apply Theorem 3 to the operator (A) having graph

[
\Gamma_A=\Gamma_I\cap ((M,M)+(0,N)),
]

where (\Gamma_I) is the graph of the identity operator in (X).

Theorem 4 is equivalent to the properties of Theorem 3 that pertain to the sets (R_A) and (R_{A^*}).

If (A) and (B) are linear operators, then, under appropriate conditions, the operators are defined in a natural way: the sum (A+B) and the product (BA). The following theorem is proved on the basis of Theorem 3.

Theorem 5. Let (A) and (B) be closed linear operators acting in a Fréchet space. Then:

I.
1°. If (D_A + D_B) is closed, then ((A+B)^ = A^ + B^).
2°. If (D_{A^} + D_{B^}) is weakly closed, then (\Gamma_{A+B} = \Gamma^\circ_{-(A^+B^)}).*

II.
1°. If (R_A + D_B) is closed, then ((BA)^ = A^B^).
2°. If (D_{A^} + R_{B^}) is weakly closed, then (\Gamma_{BA} = \Gamma^\circ_{-A^B^}).*

If the operators (A) and (B) act in Banach spaces, then the sets (D_{A^}+D_{B^}) and (D_{A^}+R_{B^}) are weakly closed if and only if they are strongly closed.

REFERENCES

1. N. Bourbaki, Topological Vector Spaces, IL, 1959.