On characterizations of a some classes of Schauder frames in Banach spaces

Abstract In this paper, we prove the following results. There exists a Banach space without basis which has a Schauder frame. There exists an universal Banach space B (resp. $\tilde{B}$) with a basis (resp. an unconditional basis) such that, a Banach X has a Schauder frame (resp. an unconditional Schauder frame ) if and only if X is isomorphic to a complemented subspace of B (resp. $\tilde{B}$). For a weakly sequentially complete Banach space, a Schauder frame is unconditional if and only if it is besselian. A separable Banach space X has a Schauder frame if and only if it has the bounded approximation property. Consequenty, The Banach space L(H, H) of all bounded linear operators on a Hilbert space H has no Schauder frame. Also, if X and Y are Banach spaces with Schauder frames then, the Banach space $ X\widehat{\otimes}_{\pi}Y$ (the projective tensor product of X and Y) has a Schauder frame. From the Faber-Schauder system we construct a Schauder frame for the Banach space C [0,1] (the Banach space of continuous functions on the closed interval [0,1]) which is not a Schauder basis of C [0,1]. Finally, we give a positive answer to some open problems related to the Schauder bases (In the Schauder frames setting). MSC2020 Classification: 46B04 , 46B10 , 46B15 , 46B25 , 46B45