Mathematics

I. A. Bakhtin

On the Problem of Extending Linear Positive Functionals

(Presented by Academician L. V. Kantorovich on 27 V 1967)

In the present paper a number of new theorems on the extension of linear positive functionals are given. The question of extending linear positive functionals in a Banach space with a solid cone was investigated by M. G. Krein (¹).

1. Consider, in a real Banach space \(E\) with cone \(K\) (¹), a linear functional \(f\) defined on some subspace \(E_f \subset E\). Denote by \(\mathcal L_f\) \((\mathcal L_f \subset E_f)\) the null subspace of the functional \(f\), by \(K_f\) the cone \(K \cap E_f\), and by \(K \oplus E_f\) and \(K \oplus \mathcal L_f\), respectively, the algebraic sums of the sets \(K\) and \(E_f\), and \(K\) and \(\mathcal L_f\). Suppose that on \(E_f\) the functional \(f\) is positive (¹): \(f(x) \geqslant 0\), if \(x \in K_f\). We ask under what conditions \(f\) can be extended from \(E_f\) to the whole space \(E\) with preservation of linearity and positivity. The answers to this question constitute the content of the paper.

Theorem 1. In order that a linear positive functional \(f(K_f \not\subset \mathcal L_f)\) can be extended from \(E_f\) to the whole space \(E\) with preservation of linearity and positivity, it is necessary and sufficient that

\[ \overline{K \oplus E_f} \ne \overline{K \oplus \mathcal L_f}. \]

Definition. An element \(u_0 \in K\) will be called an almost interior element of the cone \(K\) if, for every linear positive functional \(f \in E^*\) \((f \ne 0)\), the value \(f(u_0) > 0\).

Theorem 2. Suppose that the subspace \(E_f \subset E\) contains almost interior elements of the cone \(K\). Then, in order that a linear positive functional \(f \ne 0\) can be extended from \(E_f\) to the whole space \(E\) with preservation of linearity and positivity, it is necessary and sufficient that

\[ \overline{K \oplus \mathcal L_f} \ne E. \]

1. Below, \(E/\mathcal L_f\) denotes the quotient space, and \(K/\mathcal L_f\) the corresponding quotient set.

Theorem 3. Suppose that the subspace \(E_f\) contains almost interior elements of the cone \(K\). Then, in order that a linear positive functional \(f \ne 0\) can be extended, with preservation of linearity and positivity, from \(E_f\) to \(E\), it is necessary and sufficient that

\[ E/\mathcal L_f \ne \overline{K/\mathcal L_f}. \]

1. Theorem 4. In order that a linear positive functional \(f(K_f \not\subset \mathcal L_f)\) can be extended from \(E_f\) to the whole space \(E\), it is necessary and sufficient that there exist an element \(x_0 \in K_f\) and a number \(a_0 > 0\) such that, for all \(x \in K\),

\[ \rho(x+x_0,\mathcal L_f) \geqslant a_0. \]

1. Denote by \(E_f^{-}\) the set of all \(x \in E_f\) such that \(f(x) \leqslant 0\).

Theorem 5. In order that a linear positive functional \(f\) \((K_f \not\subset \mathcal L_f)\) can be extended from \(E_f\) to the whole space \(E\) with preservation of linearity and positivity, it is necessary and sufficient that there exist a number \(\beta_0>0\) such that for all \(x \in E_f^{-}\)

\[ \rho(x,K)\geqslant \beta_0\rho(x,\mathcal L_f). \]

1. In this section we shall assume that there exists a fixed number \(\delta_0>0\) such that for any pair of disjoint elements \(x,y\in E\) \((^2): |x|\wedge |y|=0\) the inequality

\[ \|x+y\|\geqslant \delta_0(\|x\|+\|y\|) \tag{1} \]

is satisfied.

Theorem 6. Suppose the following conditions are fulfilled:

a) the reproducing cone \(K\) is regular and minihedral \((^3)\);

b) inequality (1) is satisfied.

Then, in order that a linear positive functional \(f\) \((K_f\not\subset \mathcal L_f)\) can be extended from \(E_f\) to the whole space \(E\) with preservation of linearity and positivity, it is necessary and sufficient that there exist fixed elements \(x_0\in K_f\) and a number \(\gamma_0>0\) such that for every \(z\in \mathcal L_f\) \((z=z_+-z_-)\) the inequality

\[ \|z_-+(x_0-x_0\wedge z_+)\|\geqslant \gamma_0. \]

1. We now give a number of sufficient conditions for extendability of linear positive functionals.

Theorem 7. In order that a linear positive functional \(f\) \((K_f\ne \theta)\) can be extended from \(E_f\) to \(E\) with preservation of linearity and positivity, it is sufficient that there exist a number \(\beta_0>0\) such that for every \(x\in K\)

\[ \rho(x,\mathcal L_f)\geqslant \beta_0\|x\|. \]

Definition. The distance \(\rho(x,\mathcal L_f)\) will be called semimonotone if there exists a fixed number \(m_0>0\) such that for every pair \(x,y\in K\) and \(x\leqslant y\) \((y-x\in K)\) it follows that

\[ \rho(x,\mathcal L_f)\leqslant m_0\rho(y,\mathcal L_f). \]

Theorem 8. In order that a linear positive functional \(f\) \((K_f\not\subset \mathcal L_f)\) can be extended from \(E_f\) to \(E\) with preservation of linearity and positivity, it is sufficient that the distance \(\rho(x,\mathcal L_f)\) be semimonotone.

1. Theorem 9. Suppose the following conditions are fulfilled:

a) the reproducing cone \(K\) is normal and minihedral;

b) the functional \(f\ne 0\) in \(E_f\) is linear and positive;

c) in \(E_f\) the cone \(K\) is spatial \((^4)\);

d) for every \(y\in K_f\), every \(x\in K\) satisfying the condition \(x\leqslant y\) belongs to \(K_f\).

Then the functional \(f\) admits a linear positive extension from \(E_f\) to the whole space \(E\).

Theorem 10. Suppose the following conditions are fulfilled:

a) the reproducing cone \(K\) is regular and minihedral;

b) inequality (1) is satisfied;

c) in the subspace \(E_f\) there exist an almost interior element \(x_0\) of the cone \(K\) and a number \(a_0>0\) such that for every \(z\in \mathcal L_f\) \((z=z_+-z_-)\)

\[ \|x_0-x_0\wedge z_+\|\geqslant a_0. \]

Then the linear positive functional \(f\) \((K_f\not\subset \mathcal L_f)\) can be extended from \(E_f\) to the whole space \(E\) with preservation of linearity and positivity.

Theorem 11. Suppose that conditions a) and b) of Theorem 10 are satisfied and that there exist fixed numbers \(a, b > 0\) such that for every \(z \in \mathscr{L}_f\) \((z = z_+ - z_-)\) the inequalities

\[ a\|z\| \leq \|z_+\| \leq b\|z\|, \]

\[ a\|z\| \leq \|z_-\| \leq b\|z\| \]

hold.

Then the linear positive functional \(f\) \((K_f \not\subset \mathscr{L}_f)\) can be extended from \(E_f\) to the whole space with preservation of linearity and positivity.

1. Let us now dwell on the question of the existence of linear positive functionals that vanish on a fixed subspace.

Denote by \(H\) the closure of the linear hull \(\mathscr{L}(K)\) of the cone \(K\), by \(E_0\) a subspace in the space \(E\), and, finally, by \(\mathscr{L}_F\) the null subspace of the functional \(F\).

Theorem 12. Let \(E_0 \subset H\). Then, in order that there exist a linear positive functional \(F \ne 0\) such that \(E_0 \subset \mathscr{L}_F\), \(K \not\subset \mathscr{L}_F\), it is necessary and sufficient that

\[ \overline{K \oplus E_0} \ne H. \]

Theorem 13. Let the cone \(K\) contain almost interior elements. Then, in order that there exist a linear positive functional \(F \ne 0\) such that \(E_0 \subset \mathscr{L}_F\), it is necessary and sufficient that for every almost interior element \(x_0 \in K\) the inequality

\[ \inf_{x \in K} \rho(x + x_0, E_0) > 0 \]

hold.

Theorem 14. Suppose there exist a fixed number \(a_0 > 0\) and an almost interior element \(x_0 \in K\) such that for every \(y \in E\) one can indicate a functional \(f_y \in K^*\) \((\|f_y\| = 1)\) such that

\[ f_y(y) = 0, \qquad f_y(x_0) \geq a_0. \]

Then there exists a linear positive functional \(F \ne 0\) such that \(\mathscr{L}_F \supset E_0\).

Received
13 V 1967

References

\({}^{1}\) M. G. Krein, M. A. Rutman, UMN, 3, no. 1, 3 (1948).
\({}^{2}\) V. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, Moscow, 1961.
\({}^{3}\) M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Moscow, 1962.
\({}^{4}\) I. A. Bakhtin, Siberian Mathematical Journal, 6, no. 2, 262 (1965).