## 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 $$e_i$$).

### 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

Zbl 0607.47031
Full Text:

### References:

  G. Birkhoff,Lattice Theory, American Mathematical Society, Providence (1967).  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).  M. S. Matveichuk, ”Indefinite measures inJ-spaces,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 1, 24–26 (1989).  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  S. V. Docofeev and A. N. Sherstnev, ”Functions of frame type and their applications,”Izv. Vuzov. Matern., No. 4, 23–29 (1990).  K. R. Partasarati, ”Probability theory on closed subspaces of Hilbert space,”Matematika, No. 14:5, 102–122 (1970).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.