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On decompositions of Banach spaces into a sum of operator ranges

V. FonfV. Shevchik — 1999

Studia Mathematica

It is proved that a separable Banach space X admits a representation X = X 1 + X 2 as a sum (not necessarily direct) of two infinite-codimensional closed subspaces X 1 and X 2 if and only if it admits a representation X = A 1 ( Y 1 ) + A 2 ( Y 2 ) as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation X = T 1 ( Z 1 ) + T 2 ( Z 2 ) such that neither of the operator ranges T 1 ( Z 1 ) , T 2 ( Z 2 ) contains an infinite-dimensional closed subspace if and only...

On a functional-analysis approach to orthogonal sequences problems.

Sea T un operador lineal acotado e inyectivo de un espacio de Banach X en un espacio de Hilbert H con rango denso y sea {x} ⊂ X una sucesión tal que {Tx} es ortogonal. Se estudian propiedades de {Tx} dependientes de propiedades de {x}. También se estudia la ""situación opuesta"", es decir, la acción de un operador T : H → X sobre sucesiones ortogonales.

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