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Opérateurs diagonaux dans les espaces à bases. (Diagonal operators in spaces with a basis). (French) Zbl 0631.46013

For a Banach space \(X\) with a Schauder basis \((e_ n)_{n\geq 1}\), we consider the space \(L_ d(X)\) of diagonal operators (whose matrix representation is diagonal) and the space \(K_ d(X)\) of compact and diagonal operators. If \(\bar e_ n=e^*_ n\otimes e_ n\) is the projection operator on the nth coordinate, then \((\bar e_ n)_{n\geq 1}\) is a Schauder basis for \(K_ d(X)\), and permits a description of \(L_ d(X)\). We prove that if \((e_ n)_{n\geq 1}\) is shrinking or boundedly complete, then \((\bar e_ n)\) is shrinking; a “reverse” implication is true. We also prove that if \(K_ d(X)\) is a subspace of \(c_ 0\) then the basis \((e_ n)_{n\geq 1}\) has an unconditional subsequence.

MSC:

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47A66 Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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