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On \(M\)-ideals of compact operators in Lorentz sequence spaces. (English) Zbl 0997.46016

Let \(K(X,Y)\) be the subspace of compact operators between Banach spaces \(X\) and \(Y\) in \(L(X,Y)\), the Banach space of bounded linear operators. \(K(X,Y)\) is an \(M\)-ideal in \(L(X,Y)\) if there exists a linear projection \(P\) on \(L(X,Y)^*\) with \(\ker P=K(X,Y)^\bot=\{f\in L(X,Y)^*: f(S)=0\) \(\forall S\in K(X,Y)\}\) satisfying\(\|f\|=\|Pf\|+\|f-Pf\|\) for every \(f\in L(X,Y)^*\). It is well known that \(K(\ell_p,\ell_q)\) is an \(M\)-ideal for all \(p,q\in (1,\infty)\), but \(K(\ell_1,\ell_q)\) is never an \(M\)-ideal. In particular, if \(p>q\) then \(K(\ell_p,\ell_q)=L(\ell_p,\ell_q)\); i.e., \(K(\ell_p,\ell_q)\) is a trivial \(M\)-ideal [see J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces, I. Sequence spaces”, Berlin-Heidelberg-New York (1977; Zbl 0362.46013)]. Combining existing results and known methods in the paper the following result is shown (Theorem 2.1):
Let \(1\leq p,q<\infty, X=d(v,p), Y=d(w,q)\), where \(d(v,p)\) and \(d(w,q)\) are Lorentz sequence spaces. Then
(a) \(K(X,Y)\) is a trivial \(M\)-ideal (i.e., \(K(X,Y)=L(X,Y)\)) whenever \(p>q\) and \(w\notin \ell_{p/(p-q)}\),
(b) \(K(X,Y)\) is not an \(M\)-ideal whenever \(p>q\) and \(w\in \ell_{p/(p-q)}\),
(c) \(K(X,Y)\) is not an \(M\)-ideal whenever \(1=p\leq q\).
The main results of the paper are the following theorems.
{Theorem 3.4.} If \(1<p\leq q<\infty\) and \[ \sup_n {W_n\over{V_n^{q/p}}}<\sup_{m,n}{W_n\over{V_{m+n}^{q/p}-V_m^{q/p}}}, \] then \(K(d(v,p),d(w,q))\) is not \(M\)-ideal, where \(V_n=\sum_{k=1}^n v_k, W_n=\sum_{k=1}^n w_k, v=(v_k), w=(w_k)\).
Theorem 4.2. If \(s>1, 1<p\leq q/s\), and \(v=(v_k)\) satisfies \(V_m^s+V_n^s\leq V_{m+n}^s, m,n=1,2,\ldots\), then \(K(d(v,p),d(w,q))\) is an \(M\)-ideal for any \(w=(w_k)\). Moreover, \(K(d(v,p),d(w,q))\) is not complemented in \(L(d(v,p),d(w,q))\).

MSC:

46B45 Banach sequence spaces
46B20 Geometry and structure of normed linear spaces
47B07 Linear operators defined by compactness properties

Citations:

Zbl 0362.46013
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References:

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