‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

We present supersymmetric positive definite scalar products together with natural Krein structures of supersymmetries.

It was shown that quantum metric fluctuations smear out the singularities of Green’s functions on the light cone [1], but it does not remove other ultraviolet divergences of quantum field theory. We have proved that the quantum field theory in Krein space, i.e. indefinite metric quantization, removes all divergences of quantum field theory with exception of the light cone singularity [2, 3]. In...

Let à be a self-adjoint extension in K̃ of a fixed symmetric operator A in K ⊆ K̃. An analytic characterization of the eigenvalues of à is given in terms of the Q-function and the parameter function in the Krein–Naimark formula. Here K and K̃ are Krein spaces and it is assumed that à locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are ...

In this paper, we study the finite horizon H∞ fixed-lag smoothing problem for linear descriptor systems. The key approach applied for deriving the H∞ fixed-lag smoother is the re-organization innovation analysis in Krein space. Under the Krein space, the H∞ fixed-lag smoothing is converted into an H2 estimation problem for the system with current and delayed measurements. By using innovation re...

Let 5 and 8 be Krein spaces. By S(3 ,8 ) we denote the (generalized Schur) class of all functions O, defined and holomorphic at z = 0 and with values in L(5,8), the space of bounded Linear operators from 5 to 8 (we write L ( 8 ) for L(3,S)). It is known, see Section 3 below, that any OeS(3,8) is in some neighborhood of z = 0 the characteristic function On of a unitary colligation A = (J?,S,@;U)...

This paper addresses the H , fixed-lag smoothing and prediction problems for linear continuous-time systems. We first present a solution to the optimal HZ estimation problem for linear continuous-time systems with instantaneous and delayed measurements. It is then shown that the H , fixedlag smoothing and prediction problems can be converted to the latter problem in Krein space. Therefore, the ...

Spectral points of positive and negative type, and type π+ and type π− for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. The main objective of the paper is to study these sign type properties in the non-selfadjoint case under various kinds of perturbations, e.g. compact perturbations and perturbations small in the gap metric....