By Karl Prachar

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It is known that L2 (H, E) is a separable Hilbert space with the norm  1/2 1/2 B = i |Bhi | 2 = i,j (ej , Bhi )2  , and B does not depend on the choice of bases in H and E. Given a symmetric nonnegative nuclear operator Q in L(H, H), we denote by LQ (H, E) the set of all linear (bounded or unbounded) operators B defined on Q1/2 H, taking Q1/2 H into E and having the property BQ1/2 ∈ L2 (H, E). For B ∈ LQ (H, E) we define |B|Q = BQ1/2 . It is known that if B ∈ L2 (H, E), then |B| ≤ B , B ∈ LQ (H, E), and |B|Q ≤ |B|(trQ)1/2 .

5) is called Itˆ o’s formula for the square of the norm. For this purpose we place all processes v(t), h(t), v ∗ (t) in a single space. In those cases where the same vector belongs to various spaces we equip its norm with the symbol of the space in which it is considered. Suppose that the space V is a (possibly, non-closed) subspace of H, is dense in H in the norm of H, and |ϕ|H ≤ N |ϕ|V for all ϕ ∈ V , where N does not depend on v. Suppose that H is, in turn, a subspace of some Banach space V and that H is dense in V .

Then, for every g ∈ V , t ≥ 0, we have t t gv (s)ds ≤ |g|V 0 f (s)ds. 0 Therefore, from the properties of f (t), it follows that there exists a set Ω ⊂ Ω such t that P (Ω ) = 1 and for ω ∈ Ω , t ≥ 0, the mapping g → gv (s)ds is a bounded 0 linear functional on V . Under our assumptions, this functional can be written in the form g → gψ(t), where ψ(t) ∈ V , and, for ω ∈ Ω we have t v (s)ds. 16 then takes the following form, where, for simplicity of the formulation, we take τ = ∞; generalization to the case of arbitrary τ is obvious.

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Primzahlverteilung (Grundlehren der Mathematischen Wissenschaften 91) by Karl Prachar

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