Injections of Banach spaces with closed image of the unit ball (Q1069100)

Let X, Y, Z be Banach spaces, U a closed 1-ball in Y, \((A_ n)\) a sequence in L(X,Z), \(Q=\{x\in X:\) \(\lim_{n}A_ nx\) exists\(\}\), \(\tilde Y\) the Banach space (Q,\(\||.\||)\), where \(\|| x\|| =\max \{\| x\|\), \(\sup_{n}\| A_ nx\| \}\) (x\(\in Q)\) and \(\tilde U\) the closed 1-ball in \(\tilde Y.\) If T is an injection of Y into X which is not an isomorphic imbedding and TU is closed, then Y contains an (infinite dimensional) subspace which is isometric to the dual of a Banach space with basis. If, in addition, there exists a subspace \(Y_ 1\subset Y\) such that \(Y^*_ 1\) is separable and \(T| Y_ 1\) is not an isomorphic imbedding, then there are subspaces \(Y_ 2\subset Y_ 1\) and \(Y_ 3\subset Y\) such that \(Y_ 2\subset Y_ 3\), \(Y_ 3\) is isometric to \(Y_ 2^{**}\) and the natural imbedding of \(Y_ 2\) into \(Y_ 3\) coincides with the natural isometric imbedding of \(Y_ 2\) into \(Y_ 2^{**}.\) If Q is not closed and TU is closed, then the limit operator \(A_ 0\) is closed (and necessarily unbounded) on Q. (Here \(Y=\tilde Y.)\) Other related results are also presented.