The Geometry of Infinite-Dimensional Spaces With a Bilinear Metric

Linear operators in spaces with indefinite metric and their applications

“…One can find criticism of similar incidents in [30,89,49], etc. We note that the corresponding linear transformations of finite-dimensional spaces were studied in sufficient detail in the second half of the previous century (Frobenius).…”

“…is called the canonical decomposition of the space ~ [49]. Analyzing the definition given above of a Krein space, we note that it presupposes the positive definiteness of the form [x, x] on ~+ and the form -[x, x] on 9-, and also the completeness of these spaces with respect to the norms I[ x [[ = ([x, x]) I/2 and ~ x [[ = (-[x, x]) I/2, respectively.…”

“…The operator f is simultaneously unitary and self-adjoint in the Hilbert space @ with metric (1.2): f*=~___p '-,. In most papers on the theory of operators in Krein spaces, the authors, in introducing these spaces, follow the opposite sequence of ideas to that of the account above: at the base lies a fixed Hilbert space @ with scalar product (x, y) and its canonical decomposition (I.i) and two subspaces @=~ orthogonal in the usual sense. This gives the basis for calling the metric [x, y] the J-metric, and the Hilbert space 4, equipped additionally with this metric, is called a p~-spaee [49]. This gives the basis for calling the metric [x, y] the J-metric, and the Hilbert space 4, equipped additionally with this metric, is called a p~-spaee [49].…”

“…Moreover, from Zorn's lemma follows the general maximality principle, stating that any lineal of each of the enumerated classes is contained in some (generally not unique) maximal lineal of the same class [368,49]. To them, in particular, relate neutral (~0), positive (~++ U {0}), and negative (s {0}) lineals 2 .…”

“…The simplest examples of maximal positive (negative) lineals are the subspaces @+(@-) in any canonical decomposition (I.I) of the space @ [49].…”