Maximal nonnegative and $\theta$-accretive extensions of a positive definite linear relation

Abstract

Let $L_{0}$ be a closed linear positive definite relation ("multivalued operator") in a complex Hilbert space. Using the methods of the extension theory of linear transformations in a Hilbert space, in the terms of so called boundary value spaces (boundary triplets), i.e. in the form that in the case of differential operators leads immediately to boundary conditions, the general forms of a maximal nonnegative, and of a proper maximal $\theta$-accretive extension of the initial relation $L_{0}$ are established.

Maximally Аccretive and Nonnegative Extensions of a Nonnegative Linear Relation

“…the validity of this theorem follows from the results presented in [14] and Corollary 1, in particular, with (7), (8).…”

“…Therefore, by applying either the results obtained in [14] or Theorem 2 with B = M (0) = 0 , we conclude that L 1 + ε is maximally accretive (maximally nonnegative) if and only if…”

“…The present paper is a direct continuation of author's works [14,15]. Our aim is to establish the conditions of maximal nonnegativity and maximal accretivity of the proper extension of a closed linear nonnegative relation in a Hilbert space in the terms of "abstract boundary conditions.…”

“…It is known (see either [14] or Theorem 2 with B = M (0) = 0 ) that, for a certain contraction K ∈B(G),…”

Maximal accretive and nonnegative extensions of nonnegative linear relation

“…Ця стаття є безпосереднім продовженням праць автора [14,15]. Метою статті є встановлення умов максимальної невід'ємності та максимальної акретивності власного розширення замкненого лінійного невід'ємного відношення у гільбертовому просторі у термінах «абстрактних крайових умов».…”