Some Properties Of Pg -Frames In Banach Spaces

For a Banach space [Formula: see text] and [Formula: see text], we introduce the notion of [Formula: see text]-frames for the family [Formula: see text]. For [Formula: see text] and a strict contraction [Formula: see text], we reveal that the family [Formula: see text] is a [Formula: see text]-Bessel sequence. For an invertible operator [Formula: see text], we characterize the [Formula: see text]-frame for the family [Formula: see text]. For a [Formula: see text]-frame [Formula: see text], we show that spectral radius [Formula: see text]. We portray a family [Formula: see text], which is never a [Formula: see text]-frame. We reveal the [Formula: see text]-frame relationship between two similar operators. We demonstrate the stability of the [Formula: see text]-frame [Formula: see text] under sufficiently small perturbations. In addition, we introduce the concept of the tensor product of the [Formula: see text]-frame for the family [Formula: see text] equipped with projective norm. We demonstrate that the tensor product of two [Formula: see text]-frames [Formula: see text] and [Formula: see text] is a [Formula: see text]-frame for [Formula: see text] and vice-versa. We relate the tensor product [Formula: see text] with the spectral radius [Formula: see text]. For [Formula: see text] and [Formula: see text], we provide a necessary and sufficient condition for the family [Formula: see text] to become a [Formula: see text]-frame for [Formula: see text].