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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
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References:
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