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2 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2 Michalak A.

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Notes on binary trees of elements in C(K) spaces with an application to a proof of a theorem of H. P. Rosenthal

Michalak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2006

Vol. 46, [Z] 2

233-244

A Banach space X contains an isomorphic copy of C([0, 1]), if it contains a binary tree (en) with the following properties (1) e_n = e_2n + e_2n+1 and (2) [formuła]. We present a proof of the following generalization of a Rosenthal result: if E is a closed subspace of a separable C(K) space with separable annihilator and S : E - X is a continuous linear operator such that S* has nonseparable range, then there exists a subspace Y of E isomorphic to C([0, 1]) such that S|Y is an isomorphism, based on the fact.

On continuous linear operators on D[0, 1) with nonseparable ranges

Michalak A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2003

[Z] 43(2)

221-248

D[0,1) is the Banach space (under the sup norm) of all scalar functions defined on the interval [0,1) that are right-continuous at each point of [0,1), and have a left-hand limit at each point of (0,1]. The main result of the paper is that a continuous linear operator S : D[0,1) -> D[0,1) has a nonseparable range if and only if S fixes an isomorphic copy of D[0,1).