In an infinite-dimensional \(J\)-space every measure is bounded.

*(English. Russian original)*Zbl 0880.46018
J. Math. Sci., New York 73, No. 5, 583-591 (1995); translation from Issled. Prikl. Mat. 18, 96-111 (1992).

According to T. Ya. Azizov and I. S. Iokhvidov [“Foundations of the theory of linear operators in spaces with an indefinite metric” (in Russian), Moscow (1986; Zbl 0607.47031)] a \(J\)-space first of all is a space \(\mathcal H\) with an indefinite metric \(\langle x,x\rangle\) and a \(\mathbb{Z}_2\)-grading, i.e. a decomposition \({\mathcal H}_+\oplus{\mathcal H}_-\) into subspaces with positive resp. negative metric. Spaces with this characteristics are often referred to as Krein spaces. One may choose an operator \(J\) such that \((x,y)=\langle Jx,y\rangle\) turns \(\mathcal H\) into a pre-Hilbert space. As for a \(J\)-space, one assumes that \(\mathcal H\) is in fact a Hilbert space. Let \(P\) be the set of \(J\)-selfadjoint bounded projections \(p\) on \(\mathcal H\). Then \(\mu:P\to\mathbb{R}\) is called a measure if \(\mu(p)=\sum \mu(p_i)\) for any (countable) partition \(p=\sum p_i\), where \(p_i\) is some orthogonal family. The measure is bounded if \(\mu(p)< c|p|\) for some constant \(c\). One result is the following: assuming that \(\dim{\mathcal H}_\pm=\infty\), then the measure \(\mu\) is bounded iff \(\mu(p)= \text{tr}(Ap)\) for some \(J\)-selfadjoint nuclear operator \(A\). But the main result is: In an infinite-dimensional \(J\)-space, every measure is bounded. The proof is rather lengthy and uses properties of frame functions (a frame function \(f:S\to\mathbb{R}\) of weight \(w\), where \(S\) is the unit sphere of some Hilbert space, satisfies \(\sum f(e_i)=W\) for any orthonormal basis \(e_i\)).

Reviewer: G.Roepstorff (Aachen)

##### MSC:

46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |

47B50 | Linear operators on spaces with an indefinite metric |

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

##### Keywords:

spaces with an indefinite metric; \(J\)-space; \(\mathbb{Z}_ 2\)-grading; Krein spaces; \(J\)-selfadjoint bounded projections; measure
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\textit{M. S. Matveichuk}, J. Math. Sci., New York 73, No. 5, 1 (1992; Zbl 0880.46018); translation from Issled. Prikl. Mat. 18, 96--111 (1992)

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##### References:

[1] | G. Birkhoff,Lattice Theory, American Mathematical Society, Providence (1967). |

[2] | T. Ya. Azizov and I. S. Iokhvidov,Foundations of the Theory of Linear Operators in Spaces with an Indefinite Metric [in Russian], Nauka, Moscow (1986). |

[3] | M. S. Matveichuk, ”Indefinite measures inJ-spaces,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 1, 24–26 (1989). |

[4] | A. N. Sherstnev, ”On the concept of a charge in a noncommutative treatment of measure theory,”Veroyain. Met. i Kibern., No. 10–11, 68–72 (1974). · Zbl 0303.28015 |

[5] | S. V. Docofeev and A. N. Sherstnev, ”Functions of frame type and their applications,”Izv. Vuzov. Matern., No. 4, 23–29 (1990). |

[6] | K. R. Partasarati, ”Probability theory on closed subspaces of Hilbert space,”Matematika, No. 14:5, 102–122 (1970). |

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