James Rovnyak

Part I of the paper considered infinite orthogonal sums of regular subspaces in a Kreĭn space (that is, of subspaces which are themselves Kreĭn spaces). How precisely these sums should be defined and conditions for when such sums are themselves regular were determined. These included, for example, a boundedness condition for the sum of the corresponding orthogonal projections. The same problem is addressed here for (quasi-)pseudo-regular subspaces. Such subspaces are orthogonal direct sums of a regular space and isotropic, or neutral, subspaces. Alternate characterizations of such subspaces are given, and infinite orthogonal sums are examined via unconditional, or Moore-Smith, sums of operator ranges.

The regular subspaces of a Kreĭn space have well-known properties analogous to the closed subspaces of a Hilbert space. A boundedness condition on sums of projection operators extends this analogy to a natural notion of infinite orthogonal direct sums. The boundedness condition is shown to have several equivalent forms and consequences for Cartesian products. Moore-Smith convergence and a theorem of T. H. Hildebrandt play key roles in the formulation of results and proofs.

A well-known first-order necessary condition for a point to be a local minimizer of a given function is the non-negativity of the dot product of the gradient and a vector in a feasible direction. This paper proposes a series of alternative first-order necessary conditions and corresponding first-order sufficient conditions that seem not to appear in standard texts. The conditions assume a nonzero gradient. The methods use extensions of the notions of gradient, differentiability, and twice differentiability. Examples, including one involving the Karush–Kuhn–Tucker (KKT) theorem, illustrate the scope of the conditions.

Infinite orthogonal families of regular subspaces of a Kreĭn space exhibit a wide range of behavior. An elementary method is used to show that conditions on sums of projections produce behavior similar to that of orthogonal closed subspaces of a Hilbert space.

This note identifies errors in the joint paper [Trans. Amer. Math. Soc. 355 (2003), 813–836] by the authors and T. Constantinescu. The Main Theorem requires a stronger hypothesis. The applications to interpolation problems are affected by this change; some survive, but others do not and are withdrawn.