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Embeddings of Müntz spaces in \(L^{\infty}(\mu)\). (English) Zbl 1420.46020

The paper has two sections with results (Sections 2 and 3).
In Section 2, the embedding \(i_\mu^\Lambda\) of Müntz spaces \(M_\infty^\Lambda\) in \(L^\infty ([0,1],dx)\) into \(L^\infty ([0,1],\mu)\) is studied. Here \(\Lambda\) is the sequence on which the Müntz space is formed, \(dx\) is the Lebesgue measure and \(\mu\) is some non-zero positive Borel measure. This embedding always exists, can well be the null-operator (point measures), has norm at most 1 and “typically” has norm 1 (Proposition 2.1). The embedding is compact exactly when \(\mu(1-\varepsilon,1)=0\) for some \(\varepsilon>0\) (Proposition 2.3). Moreover, its essential norm (= distance to the compact operators) is 1 unless it is compact (Theorem 2.5). In fact, \(i_\mu^\Lambda\) is compact as soon as it is weakly compact or Dunford-Pettis or integral (Corollary 2.7).
In Section 3, the general embedding \(i_{\mu_1,\mu_2}\) of \(L_\infty([0,1],\mu_1)\) into \(L^\infty ([0,1],\mu_2)\) is studied in the case that \(\mu_2\) is absolutely continuous with respect to \(\mu_1\) (so that equivalence classes are the same in the two \(L^\infty\)-spaces). It is readily seen that \(\|i_{\mu_1,\mu_2}\|=1\) (Lemma 3.1). The paper culminates in proving that \(i_{\mu_1,\mu_2}\) is compact exactly when it is weakly compact exactly when \(\mu_2\) is a finite linear combination of point measures and that it has either essential norm 1 or 0 (Theorems 3.3 and 3.4).

MSC:

46B25 Classical Banach spaces in the general theory
47B07 Linear operators defined by compactness properties
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