By Bacco M., Mocellin V.

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**Sample text**

But for p = 1, however, ∗ I is true since we have strong convergence Pm → I here. For 50% of Pm → p = ∞, only Pm → I is true, provided X , and hence E exists. The following two propositions show that this behaviour is a rule. 73. If p = 1, then Aτ → A implies the strong convergence Aτ → A. 7. Applicability vs. Stability 47 P Proof. 65 that the ﬁnal part of (Aτ ) is bounded and that (Aτ − A)Pm ⇒ 0 as τ → ∞ for every m ∈ N. Consequently, for every K ∈ K(E) and every m ∈ N, (Aτ − A)K ≤ (Aτ − A)Pm K + Aτ − A Qm K , where the ﬁrst term tends to zero as τ → ∞, and the second term can be made as small as desired if we choose m large enough since K is compact and Qm → 0 strongly as m → ∞.

3 P-convergence vs. 6 we see that the relation between K(E) and K(E, P) essentially determines the relation between P- and ∗-strong convergence. In analogy to Figure 1 on page 15, we will brieﬂy study this relation here, depending on the space E = p (Zn , X). e. e. if dim X < ∞. 16, Aτ → A always K implies Aτ → A, and both are equivalent if E is reﬂexive; that is if X is reﬂexive (recall that every ﬁnite-dimensional space X is reﬂexive) and 1 < p < ∞. Consequently, we get that P K ∗ A Aτ → A =⇒ Aτ → A ⇐⇒ Aτ → K P ∗ Aτ → A =⇒ Aτ → A =⇒ Aτ → A if 1 < p < ∞ and X reﬂexive, if dim X < ∞.

In particular, the latter holds with p = 1 and q = ∞ if and only if a ∈ W(T); that is, |ak | < ∞, which is a proper subclass of C(T). So this is certainly an example of the dependence of BDOp on p. 40. The impression that, for every A ∈ BDOp , it holds that Am ⇒ A where [Am ] is just the restriction of [A] to a ﬁnite number of diagonals, is false in general! As a counter-example, take p = 2, n = 1, X = C, and A = L(a), where a ∈ C(T) is such that the sequence of partial Fourier sums m t∈T → a k tk k=−m is not uniformly convergent to a as m → ∞.