Operator ideals on non-commutative function spaces
Given two Banach spaces \(X\) and \(Y\), \(B(X,Y)\) (respectively \(\mathcal{K}(X,Y)\)) denotes the set of bounded (respectively compact) linear operators from \(X\) to \(Y\). We say that \(T\in B(X,Y)\) fixes a (complemented) copy of a subspace \(Z\subset X\) if \(Z\) is isomorphic to \(T(Z)\) (with \(Z\) and \(T(Z)\) complemented in \(X\) and in \(Y\), respectively). \(T\) is called strictly singular (respectively finitely strictly singular) if it does not fix any infinite-dimensional subspace (respectively if given \(\varepsilon>0\) there is a natural number \(N\) such that each subspace of dimension at least \(N\) contains an element \(x\) with \(\|Tx\|<\varepsilon\|x\|\). The strictly singular operators form an ideal \(\mathcal{SS}(X,Y)\subset B(X,Y)\) (in the ``usual'' sense: \(\mathcal{SS}(X,Y)\) contains the finite rank operators and if \(U\in B(X_0,X)\), \(V\in B(Y,Y_0)\), then \(VTU\in\mathcal{SS}(X_0,Y_0)\)), likewise for the finitely strictly singular operators \(\mathcal{FSS}(X,Y)\subset B(X,Y)\). The following inclusions hold and are, in general, strict: \[ \begin{cases}\mathcal{K}(X,Y)\subset\mathcal{FSS}(X,Y)\subset\mathcal{SS}(X,Y)\subset\mathcal{IN}(X,Y),\\ \mathcal{K}(X,Y)\subset\mathcal{SCS}(X,Y)\subset\mathcal{IN}(X,Y), \end{cases}\tag{1} \] where \(\mathcal{IN}\) denotes the set of inessential or Fredholm perturbation operators \(T\) defined by the fact that \(I+WT\) is Fredholm for every \(W\in B(Y,X)\) and where \(\mathcal{SCS}\) denotes the strictly cosingular operators \(T\) defined by the fact that \(Q_ZT\) is not surjective, with \(Q_Z\) the quotient map \(Y\to Y/Z\). In special cases, for example, when \(X\) and \(Y\) are Hilbert spaces or \(X=Y=l_p\) (\(1\leq p<\infty\)), the inclusions above are known to coincide. The authors study the ideals above when \(X\) and \(Y\) are certain non-commutative \(L_p\)-spaces; to be more precise, they mostly consider \(L_p\)-spaces associated to a hyperfinite von Neumann algebra \(A\) with a normal faithful finite trace or associated to \(B(H)\) (\(H\) an infinite-dimensional separable Hilbert space); in the latter case, the Schatten classes \(\mathcal{C}_p=\mathcal{C}_p(H)\) are obtained. Also, generalized Schatten classes \(\mathcal{C_E}\) associated to Lorentz sequence spaces \(\mathcal{E}=\mathcal{\ell}(w,p)\) are considered (where \(\mathcal{C_E}=\mathcal{C}_p(H)\) if \(\mathcal{E}=l_p\)). As usual, the lack of lattice structure does not necessarily exclude generalizations from the commutative setting to the non-commutative one, but makes proofs more cumbersome. To cope with this, the authors show and use the fact that if \(A\) satisfies the restrictions above (i.e., hyperfiniteness of \(A\) and finite trace), then \(L_p=L_p(A)\) (\(1<p<\infty\)) has the property that each weakly null seminormalized sequence contains an unconditional subsequence. Among other tools, they use the study of equiintegrable sets and the Kadec-Pełczyński dichotomy as done by \textit{Y. Raynaud} and \textit{Q.-H. Xu} [J. Funct. Anal. 203, No. 1, 149--196 (2003; Zbl 1056.46056)]. In the following, we describe some of the main results. To begin with, we cite three main results concerning \(L_p(A)\)-spaces with \(A\) as above. First, for \(1<p<\infty\), an operator \(T\) on \(L_p=L_p(A)\) (\(A\) as above) is not strictly singular (if and) only if it fixes a complemented copy of \(\ell_2\) or of \(\ell_p\). (For \(p>2\), this holds without the above restrictions on \(A\), by the Kadec-Pełczyński dichotomy.) Second, if \(T\) maps from \(L_{p_1}(A_1)\) to \(L_{p_2}(A_2)\) (\(A_1, A_2\) as \(A\) above, i.e., hyperfinite with finite traces) and if \(1<p_1,p_2<\infty\), \(p_1\neq p_2\), and either \(p_1\geq2\) or \(p_2>p_1\), then \(T\) is not strictly singular (if and) only if \(T\) fixes a complemented copy of \(\ell_2\). Thirdly, concerning the order structure, and still with \(A_1, A_2\) as above, the authors show that if \(1<p_2\leq p_1<\infty\), then \(\mathcal{SS}(L_{p_1},L_{p_2})_+=\mathcal{K}(L_{p_1},L_{p_2})_+\). The first of the just mentioned three results remains valid if \(L_p(A)\) is replaced by Schatten-Lorentz spaces \(\mathcal{C}_{\mathcal{\ell}(w,p)}\) and then the additional equivalence \(\mathcal{SS}(\mathcal{C}_{\mathcal{\ell}(w,p)})=\mathcal{IN}(\mathcal{C}_{\mathcal{\ell}(w,p)})\) holds. The authors give conditions for the inclusions in (1) to be strict when taking Schatten-Lorentz spaces for \(X\) and \(Y\). In the special case of Schatten spaces \(\mathcal{C}_p, \mathcal{C}_q\), it is shown that one has equality in the two conclusions \(\mathcal{K}(\mathcal{C}_p, \mathcal{C}_q)\subset\mathcal{FSS}(\mathcal{C}_p, \mathcal{C}_q)\subset\mathcal{SS}(\mathcal{C}_p, \mathcal{C}_q)\) if and only if \(1\leq q\leq2\leq p\leq\infty\). Moreover, concerning positive operators,{\parindent = 0mm \begin{itemize}\item{} for \(1\leq q<p<\infty\), one has \(\mathcal{K}(\mathcal{C}_p, \mathcal{C}_q)_+=B(\mathcal{C}_p, \mathcal{C}_q)_+\),\item{} for \(1\leq p<\infty\), one has \(\mathcal{K}(\mathcal{C}_p)_+=\mathcal{SS}(\mathcal{C}_p)_+\) and a positive \(T\) is in \(\mathcal{SS}(\mathcal{C}_p)_+\) iff it does not fix a complemented \(\ell_p\). \end{itemize}} Recall that \(T\in B(X,Y)\) is called Dunford-Pettis if it takes weakly null sequences to norm null ones. An operator \(T\in B(\mathcal{C}_1, Z)\) (\(Z\) a Banach space) is Dunford-Pettis iff each restriction of \(T\) to an \(\ell_2\)-copy of \(\mathcal{C}_1\) is compact. Finally, the authors give an instance where the product of two strictly singular operators is compact: if \(X\) is either an \(L_p=L_p(A)\) (with \(A\) as above), \(1<p<\infty\) or a \(\mathcal{C}_p(H)\), \(1\leq p\leq\infty\), then \(TS\) is compact if \(T,S\in\mathcal{SS}(X)\). The authors succeed in generalizing many facts from commutative \(L_p\)'s to non-commutative ones; for some results, the commutative analogues seem to have been unknown. Remarks show that many results are sharp.
scientific article; zbMATH DE number 6358019
scientific article